\section{Introduction}

		Image restoration covers some fundamental settings in image processing such as denoising, deblurring and super-resolution. Over the past few years, image restoration methods have demonstrated impressive improvements in both visual quality and distortion measures such as peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) \citep{wang2004image}. It was noticed, however, that improvement in accuracy, as measured by distortion, does not necessarily lead to improvement in visual
		quality, referred to as perceptual quality. Furthermore, the lower the distortion of an estimator, the more the distribution of its outputs generally deviates from the distribution of the signals it attempts to estimate.
		This phenomenon, known as the \emph{perception-distortion tradeoff} \citep{blau2018perception}, has captured significant attention, where it implies that faithfulness to ground truth images comes at the expense of perceptual quality, namely the deviation from statistics of natural images. Several works have extended the perception-distortion tradeoff to settings such as lossy compression \citep{blau2019rethinking} and classification \citep{liu2019classification}.

	Despite the increasing popularity of performing comparisons on the perception-distortion plane, the exact characterization of the minimal distortion that can be achieved under a given perception constraint remains an important open question.
	Although \citet{blau2018perception} investigated the basic properties of this \emph{distortion-perception function}, such as monotonicity and convexity, little is known about its precise nature. While a general answer to this question is unavailable, in this paper, we derive a closed form expression for the distortion-perception (DP) function for the mean squared-error (MSE) distortion and the Wasserstein-$2$ perception index.

	Our main contributions are:
		\emph{(i)} We prove that the DP function is \emph{always} quadratic in the perception constraint $P$,
		regardless of the underlying distribution (Theorem \ref{Thm:=00005Bthe-Distortion-Perception-funct}).
			\emph{(ii)} We show that it is possible to construct estimators on the DP curve from the estimators at the two extremes of the tradeoff (Theorem~\ref{Thm::extrapol}): The one that globally minimizes the MSE, and a minimizer of the MSE under a perfect perceptual quality constraint. The latter can be obtained as a stochastic transformation of the former.
			\emph{(iii)} In the Gaussian setting, we further provide a closed form expression for optimal estimators and for the corresponding DP curve (Theorems \ref{thm:Gaussian1} and \ref{thm:Gaussian_not_unique}).
			We show this Gaussian DP curve is a lower bound on the DP curve of any distribution having the same second order statistics. Finally, we illustrate our results, numerically and visually, in a super-resolution setting in Section \ref{sec::numerical}. The proofs of all the theorems in the main text are provided in Appendix \ref{APPsec::proofs}.

Our theoretical results shed light on several topics that are subject to much practical activity. Particularly, in the domain of image restoration, numerous works target perceptual quality rather than distortion (\emph{e.g.} \citep{wang2018esrgan,lim2017enhanced,ledig2017photo}). However, it has recently been recognized that generating a single reconstructed image often does not convey to the user the inherent ambiguity in the problem. Therefore, many recent works target \emph{diverse} perceptual image reconstruction, by employing randomization among possible restorations \citep{lugmayr2020srflow,bahat2020explorable,prakash2021removing,abid2021generative}. Commonly, such works perform sampling from the posterior distribution of natural images given the degraded input image. This is done \emph{e.g.}~using priors over image patches \citep{friedman2021posterior}, conditional generative models \citep{ohayon2021high,prakash2020divnoising}, or implicit priors induced by deep denoiser networks \citep{kawar2021stochastic}. Theoretically, posterior sampling leads to perfect perceptual quality (the restored outputs are distributed like the prior). However, a fundamental question is whether this is optimal in terms of distortion. As we show in Section~\ref{sec::The MSE--Wasserstein-2 tradeoff}, posterior sampling is often not an optimal strategy, in the sense that there often exist perfect perceptual quality estimators that achieve lower distortion.

Another topic of practical interest, is the ability to \emph{traverse the distortion-perception tradeoff} at test time, without having to train a different model for each working point. Recently, interpolation has been suggested for controlling several objectives at test-time. \citet{shoshan2019dynamic} propose using interpolation in some latent space in order to approximate intermediate objectives. \citet{wang2018esrgan} use per-pixel interpolation for balancing perceptual quality and fidelity. Studies of network parameter interpolation are presented by \citet{wang2018esrgan,wang2019deep}.  \citet{deng2018enhancing} produces a low distortion reconstruction and a high perceptual quality one, and then uses style transfer to combine them. An important question, therefore, is which strategy is optimal. In Section~\ref{sec::optimalestimators} we show that for the MSE--Wasserstein-2  tradeoff, linear interpolation leads to optimal estimators. We also discuss a geometric connection between interpolation and the fact that estimators on the DP curve form a geodesic in Wasserstein space.

	\section{Problem setting and preliminaries}
	\label{Sec::D-P::optimalTransport}

	\subsection{The distortion-perception tradeoff}
	Let $X,Y$ be random vectors taking values in $\R^{n_{x}}$ and $\R^{n_{y}}$, respectively. We consider the problem of constructing an estimator $\hat{X}$ of $X$ based on $Y$. Namely, we are interested in determining a conditional distribution $p_{\hat{X}|Y}$ such that $\hat{X}$ constitutes a good estimate of $X$.

	In many practical cases, the goodness of an estimator is associated with two factors: (i) the degree to which $\hat{X}$ is close to $X$ on average (low distortion), and (ii) the degree to which the distribution of $\hat{X}$ is close to that of $X$ (good perceptual quality). An important question, then, is \emph{what is the minimal distortion that can be achieved under a given level of perceptual quality?} and \emph{how can we construct estimators that achieve this lower bound?}
	In mathematical language, we are interested in analyzing the distortion-perception (DP) function (defined similarly to the perception-distortion function of \cite{blau2018perception})
	\begin{equation}
		D(P)=\min_{p_{\hat{X}|Y}}\left\{\EE[d(X,\hat{X})] \;:\; d_{p}(p_X,p_{\hat{X}})\leq P\right\}.\label{eq:D_P::General_definition}
	\end{equation}
	Here, $d:\R^{n_x}\times\R^{n_x}\rightarrow \R^{+}\cup\{0\}$ is some distortion criterion, $d_{p}(\cdot,\cdot)$ is some divergence between probability measures, and $p_{\hat{X}}$ is the probability measure on $\R^{n_x}$ induced by $p_{\hat{X}|Y}$ and $p_Y$. We assume that $\hat{X}$ is independent of $X$ given $Y$.

	As discussed in \citep{blau2018perception}, the function $D(P)$ is monotonically non-increasing and is convex whenever $d_{p}(\cdot,\cdot)$ is convex in its second argument (which is the case for most popular divergences). However, without further concrete assumptions on the distortion measure $d(\cdot,\cdot)$ and the perception index $d_p(\cdot,\cdot)$, little can be said about the precise nature of $D(P)$.

    Here, we focus our attention on the squared-error distortion $d(x,\hat{x})=\|x-\hat{x}\|^{2}$ and the Wasserstein-2 distance $d_p(p_X,p_{\hat{X}})=W_2(p_X,p_{\hat{X}})$, with which \eqref{eq:D_P::General_definition} reads
	\begin{equation}
		D(P)=\min_{p_{\hat{X}|Y}}\left\{\EE[\|X-\hat{X}\|^2] \;:\; W_2(p_X,p_{\hat{X}})\leq P\right\}.
		\label{eq:MSE::D_P::Definition}
	\end{equation}
    Throughout this paper we assume that all distributions have finite first and second moments. In addition, from Theorem \ref{Thm::extrapol} below it will follow that the minimum is indeed attained, so that \eqref{eq:MSE::D_P::Definition} is well defined. It is well known that the estimator minimizing the mean squared error (MSE) without any constraints, is given by $X^{*}=\EE[X|Y]$. This implies that $D(P)$ monotonically decreases until $P$ reaches
    $P^*\triangleq W_2(p_X,p_{X^*})$,
    beyond which point $D(P)$ takes the constant value $D^*\triangleq \EE[\|X-X^{*}\|^{2}]$.
     This is illustrated in Fig.~\ref{fig:DP_func}.
     It is also known that in this case $D(0)\leq 2D^*$ since the posterior sampling estimator $p_{\hat{X}|Y}=p_{X|Y}$ achieves $W_2(p_X,p_{\hat{X}})=0$ and $\EE[\|X-\hat{X}\|^2]=2D^*$ \citep{blau2018perception}. However, apart for these rather general properties, the precise shape of the DP curve has not been determined to date, and neither have the estimators that achieve the optimum in \eqref{eq:MSE::D_P::Definition}. This is our goal in this paper.

\begin{figure}
		\centering
		\includegraphics[width=0.44\linewidth]{plot/DP_func_star} \includegraphics[bb=39bp 62bp 420bp 300bp,clip,scale=0.75,width=0.55\linewidth]{plot/geodesic_noraster}

		\caption{\textbf{Left: The distortion-perception function.} When using the MSE distortion and the Wasserstein-2 perception index, the minimal possible distortion, $D^*$, is achieved by the estimator $X^{*}=E[X|Y]$. The perception index attained by this estimator is $P^*$. At the other extreme of the tradeoff, we know that the distortion at $P=0$ is bounded from above by $2D^*$.\label{fig:DP_func} \textbf{Right:} \label{fig:geodesic} The minimal distortion $D(P)$ for a given perception
	index $P<P^*$ can be achieved by an estimator with a distribution $\gamma_{P}$
	lying on a straight line (or geodesic) defined by the geometry of
	the probabilities space.
	Given $P$, $\gamma_{P}$ achieves $W_{2}(p_X,\gamma_{P})=P$ and $W_{2}(p_{X^*},\gamma_{P})=P^*-P$,
	hence $D(P)=D^*+W_2^2(p_{X^*},\gamma_{P})=D^*+(P^*-P)^{2}.$}
	\end{figure}

	\subsection{The Wasserstein and Gelbrich Distances}

	Before we present our main results, we briefly survey a few properties of the Wasserstein distance, mostly taken from \citep{panaretos2020invitation}. The Wasserstein-$p$ ($p\geq 1$) distance between measures $\mu$ and $\gamma$ on a separable Banach space $\mathcal{X}$ with norm $\| \cdot \|$ is defined by
	\begin{equation}\label{eq:Wp_def}
	W_{p}^{p}(\mu,\gamma)\triangleq\inf\left\{ \EE_{(U,V)\sim\nu}[\|U-V\|^{p}]\;:\;\nu\in\Pi(\mu,\gamma)\right\},
	\end{equation}
	where $\Pi(\mu,\gamma)$ is the set of all probabilities on $\mathcal{X}\times\mathcal{X}$ with marginals $\mu$ and $\gamma$. A joint probability $\nu$ achieving the optimum in \eqref{eq:Wp_def} is often referred to as \emph{optimal plan}. The Wasserstein space of probability measures is defined as
	\[
	\mathcal{W}_{p}(\mathcal{X})\triangleq\left\{\gamma:\int_{\mathcal{X}}\|x\|^{p}d\gamma<\infty\right\},
	\]
	and $W_p$ constitutes a metric on $\mathcal{W}_{p}(\mathcal{X})$.

    For any $(m_{1},\Sigma_{1}),(m_{2},\Sigma_{2})\in \R^{d}\times \SSS_{+}^{d}$
	(where $\SSS_{+}^{d}$ is the set of symmetric positive semidefinite matrices in $\R^{d\times d})$, the Gelbrich distance is defined as
	\begin{equation}
	G^{2}((m_{1},\Sigma_{1}),(m_{2},\Sigma_{2}))\triangleq\Vert m_{1}-m_{2}\Vert_{2}^{2}+\mathrm{Tr}\left\{\Sigma_{1}+\Sigma_{2}-2\left(\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}\right)^{\frac{1}{2}}\right\}.\label{eq:Gelbrich_dist}
	\end{equation}
	The root of a PSD matrix is always taken to be PSD. For any two probability measures $\mu_{1},\mu_{2}$ on $\R^{d}$ with means and covariances $(m_{1},\Sigma_{1}),(m_{2},\Sigma_{2})$, from \citep[Thm. 2.1]{gelbrich1990formula} we have that
	\begin{equation}\label{eq:W2_greater_G2}
	W_{2}^{2}(\mu_{1},\mu_{2})\geq G^{2}((m_{1},\Sigma_{1}),(m_{2},\Sigma_{2})).
	\end{equation}

	When $\mu_1=\mathcal{N}(m_{1},\Sigma_{1})$ and $\mu_2=\mathcal{N}(m_{2},\Sigma_{2})$ are Gaussian distributions on $\R^{d}$, we have that $W_{2}(\mu_{1},\mu_{2})= G((m_{1},\Sigma_{1}),(m_{2},\Sigma_{2}))$.
	This equality is obvious for non-singular measures but is true for any two Gaussian distributions \citep[p.~18]{panaretos2020invitation}.
	If $\Sigma_1$ and $\Sigma_2$ are non-singular, then the distribution attaining the optimum in \eqref{eq:Wp_def} corresponds to
	\begin{equation}\label{eq:optimal_dist_for_gauss}
	U\sim \mathcal{N}(m_1,\Sigma_1),\;\; V=m_{2}+T_{1\rightarrow 2}(U-m_{1}),
	\end{equation}
	where
	\begin{equation}\label{eq::T1->2:def}
	T_{1\rightarrow 2}=\Sigma_{1}^{-\frac{1}{2}}\left(\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}\right)^{\frac{1}{2}}\Sigma_{1}^{-\frac{1}{2}}
	\end{equation}
	is the optimal transformation pushing forward from $\mathcal{N}(0,\Sigma_{1})$ to $\mathcal{N}(0,\Sigma_{2})$ \citep{knott1984optimal}. This transformation satisfies $\Sigma_{2}=T_{1\rightarrow 2}\Sigma_{1}T_{1\rightarrow 2}.$ For a discussion on singular distributions, please see App.~\ref{Appsec::Supp}.

	\section{Main results}

	\subsection{The MSE--Wasserstein-2 tradeoff}
	\label{sec::The MSE--Wasserstein-2 tradeoff} The DP function \eqref{eq:MSE::D_P::Definition} depends, of course, on the underlying joint probability $p_{XY}$ of the signal $X$ and measurements $Y$. Our first key result is that this dependence can be expressed solely in terms of $D^*$ and $P^*$. In other words, knowing the distortion and perception index attained by the minimum MSE estimator $X^*$, suffices for determining $D(P)$ for any $P$.

	\begin{thm}[The DP function]
		\label{Thm:=00005Bthe-Distortion-Perception-funct} The DP function \eqref{eq:MSE::D_P::Definition} is given by
		\begin{equation}
		D(P)=
		D^*+\left[(P^*-P)_{+}\right]^{2},
		\label{eq:Thm:DP_function::Main_result}
		\end{equation}
		where $(x)_+=\max(0,x)$. Furthermore, an estimator achieving perception index $P$ and distortion $D(P)$ can always be constructed by applying a (possibly stochastic) transformation to $X^*$.
	\end{thm}

	Theorem~\ref{Thm:=00005Bthe-Distortion-Perception-funct} is of practical importance because in many cases constructing an estimator that achieves a low MSE (\emph{i.e.} an approximation of $X^*$) is a rather simple task. This is the case, for example, in image restoration with deep neural networks. There, it is common practice to train a network by minimizing its average squared error on a training set. Now, measuring the MSE of such a network on a large test set allows approximating $D^*$. We can also obtain an approximation of at least a lower bound on $P^*$ by estimating the second order statistics of $X$ and $X^*$. Specifically, recall that $P^*$ is lower bounded by the Gelbrich distance between $(m_X,\Sigma_X)$ and $(m_{X^*},\Sigma_{X^*})$, which is given by $(G^*)^2\triangleq\mathrm{Tr}\{\Sigma_{X}+\Sigma_{X^*}-2(\Sigma_{X}^{1/2}\Sigma_{X^*}\Sigma_{X}^{1/2})^{1/2}\}$ (see \eqref{eq:W2_greater_G2}). Given approximations for $D^*$ and $G^*$, we can approximate a lower bound on the DP function for any $P$,
	\begin{equation}\label{eq:Thm2::LwrBound_G_Sx_S*}
	D(P)\geq D^*+[(G^*-P)_+]^{2}.
	\end{equation}
	The bound is attained when $X$ and $Y$ are jointly Gaussian.

	\paragraph{Uniqueness} A remark is in place regarding the uniqueness of an estimator achieving \eqref{eq:Thm:DP_function::Main_result}. As we discuss below, what defines an optimal estimator $\hat{X}$ is its joint distribution with $X^*$. This joint distribution may not be unique, in which case the optimal estimator is not unique. Moreover, even if $p_{\hat{X} X^{*}}$ is unique, the uniqueness of the estimator is not guaranteed because there may be different conditional distributions $p_{\hat{X}|Y}$ that lead to the same optimal $p_{\hat{X} X^{*}}$. In other words, given the optimal $p_{\hat{X} X^{*}}$, one can choose any joint probability $p_{\hat{X} Y X^* }$ that has marginals $p_{\hat{X} X^{*}}$ and $p_{YX^* }$. One option is to take the estimator $\hat{X}$ to be a (possibly stochastic) transformation of $X^*$, namely $p_{\hat{X}|Y}=p_{\hat{X}|X^*}p_{X^*|Y}$. But there may be other options.
	In cases where either $Y$ or $\hat{X}$ are a deterministic transformation of $X^*$ (\emph{e.g.} when $X^*$ has a density, or is an invertible function of $Y$), there is a unique joint distribution $p_{\hat{X} Y X^{*}}$ with the given marginals \cite[Lemma 5.3.2]{ambrosio2008gradient}. In this case, if $p_{\hat{X}X^{*}}$ is unique then so is the estimator $p_{\hat{X}|Y}$.

	\paragraph{Randomness} 	Under the settings of image restoration, many methods encourage diversity in their output by adding randomness \citep{lugmayr2020srflow,bahat2020explorable,prakash2021removing}. In our setting, we may ask under what conditions there exists an optimal estimator $\hat X$ which is a deterministic function of $Y$. For example, when $p_Y=\delta_0$ but $X$ has some non-atomic distribution, it is clear that no deterministic function of $Y$ can attain perfect perceptual quality. It turns out that a sufficient condition for the optimal $\hat X$ to be a deterministic function of $Y$ is that $X^*$  have a density. We discuss this in App.~\ref{APPsec::proofs} and explicitly illustrate it in the Gaussian case (see Sec.~\ref{Sec::GaussianSetting}), where if $X^*$ has a non-singular covariance matrix then $\hat X$ is a deterministic function of~$Y$.

	\paragraph{When is posterior sampling optimal?} Many recent image restoration methods attempt to produce diverse high perceptual quality reconstructions by sampling from the posterior distribution $p_{X|Y}$ \citep{friedman2021posterior, ohayon2021high,kawar2021stochastic}. As discussed in \citep{blau2018perception}, the posterior sampling estimator attains a perception index of $0$ (namely $W_{2}(p_X,p_{\hat{X}})=0$) and distortion $2D^*$. But an interesting question is: when is this strategy optimal? In other words, in what cases do we have that the DP function at $P=0$ equals precisely $2D^*$ and is not strictly smaller? Note from the definition of the Wasserstein distance \eqref{eq:Wp_def}, that $(P^*)^2=W_{2}^{2}(p_X,p_{X^{*}})\leq \EE[\|X-X^*\|^2]=D^*$. Using this in \eqref{eq:Thm:DP_function::Main_result}
 shows that the DP function at $P=0$ is upper bounded by
	\begin{equation}\label{eq:Thm2::UprBound_2Dmin}
	D(0)=D^*+(P^*)^2\leq2D^*,
	\end{equation}
	and the upper bound is attained when $(P^*)^2=D^*$. To see when this happens, observe that
	\begin{equation}\label{eq:Sandwitch}
	    \mathrm{Tr}\left\{\Sigma_{X}+\Sigma_{X^*}-2(\Sigma_{X}^{\frac{1}{2}}\Sigma_{X^*}\Sigma_{X}^{\frac{1}{2}})^{\frac{1}{2}}\right\}=(G^*)^2\leq (P^*)^2\leq D^*=\mathrm{Tr}\{\Sigma_X-\Sigma_{X^*}\}.
	\end{equation}
	We can see that when $\mathrm{Tr}\{\Sigma_{X^*}\}=\mathrm{Tr}\{(\Sigma_{X}^{1/2}\Sigma_{X^*}\Sigma_{X}^{1/2})^{1/2}\}$, the leftmost and rightmost sides become equal, and thus $(P^*)^2=D^*$. To understand the meaning of this condition, let us focus on the case where $\Sigma_{X}$ and $\Sigma_{X^*}$ are jointly diagonalizable. This is a reasonable assumption for natural images, where shift-invariance induces diagonalization by the Fourier basis \citep{unser1984approximation}. In this case, the condition can be written in terms of the eigenvalues of the matrices, namely $\sum_{i}\lambda_i(\Sigma_{X^*})=\sum_{i} \sqrt{\lambda_i(\Sigma_{X^*})\lambda_i(\Sigma_{X})}$. This condition is satisfied when each  $\lambda_i(\Sigma_{X^*})$ equals either $\lambda_i(\Sigma_{X})$ or $0$. Namely, the $i$th eigenvalue of the error covariance of $X^*$, which is given by $\Sigma_X-\Sigma_{X^*}$, is either $\lambda_i(\Sigma_{X})$ or $0$. We conclude that posterior sampling is optimal when there exists a subspace $\mathcal{S}$ spanned by some of the eigenvectors of $\Sigma_X$, such that the projection of $X$ onto $\mathcal{S}$ can be recovered from $Y$ with zero error, but the projection of $X$ onto $\mathcal{S}^\perp$ cannot be recovered at all (the optimal estimator is trivial). This is likely not the case in most practical scenarios. Therefore, it seems that \emph{posterior sampling is often not optimal}. That is, posterior sampling can be improved upon in terms of MSE without any sacrifice in perceptual quality.

	\subsection{Optimal estimators}
	\label{sec::optimalestimators}
	While Theorem~\ref{Thm:=00005Bthe-Distortion-Perception-funct} reveals the shape of the DP function, it does not provide a recipe for constructing optimal estimators on the DP tradeoff. We now discuss the nature of such estimators.

	Our first observation is that since $\hat{X}$ is independent of $X$ given $Y$, its MSE can be decomposed as $\EE[\|X-\hat{X}\|^2]=\EE[\|X-X^*\|^2+\EE[\|X^*-\hat{X}\|^2]$ (see App.~\ref{APPsec::proofs}). Therefore, the DP function \eqref{eq:MSE::D_P::Definition} can be equivalently written as
	\begin{equation}\label{eq:MSE::D_P::Alternative}
    D(P)=D^*+\min_{p_{\hat{X}|Y}}\left\{\EE[\|\hat{X}-X^*\|^2] \;:\; W_2(p_X,p_{\hat{X}})\leq P\right\}.
	\end{equation}
	Note that the objective in \eqref{eq:MSE::D_P::Alternative} depends on the MSE between $\hat{X}$ and $X^*$, so that we can perform the minimization on $p_{\hat{X}|X^*}$ rather than on $p_{\hat{X}|Y}$ (once we determine the optimal $p_{\hat{X}|X^*}$ we can construct a consistent $p_{\hat{X}|Y}$ as discussed above).

	Now, let us start by examining the leftmost side of the curve $D(P)$, which corresponds to a perfect perceptual quality estimator (\emph{i.e.}~$P=0$). In this case, the constraint becomes $p_{\hat{X}}=p_X$. Therefore,
	\begin{equation}
    D(0)=D^*+\min_{p_{\hat{X}X^*}}\left\{\EE[\|\hat{X}-X^*\|^2] \;:\; p_{\hat{X}X^*}\in\Pi(p_X,p_{X^*})\right\},
	\end{equation}
	where $\Pi(p_X,p_{X^*})$ is the set of all probabilities on $\R^{n_x}\times\R^{n_x}$ with marginals $p_X,p_{X^*}$. One may readily recognize this as the optimization problem underlying the Wasserstein-2 distance between $p_X$ and $p_{X^*}$. This leads us to the following conclusion.
	\begin{thm}[Optimal estimator for $P=0$]
	\label{thm:perfect_perception}Let $\hat{X}_0$ be an estimator achieving perception index $0$ and MSE $D(0)$. Then its joint distribution with $X^*$ attains the optimum in the definition of $W_2(p_X,p_{X^*})$. Namely, $p_{\hat{X}_0 X^*}$ is an optimal plan between $p_X$ and $p_{X^*}$.
	\end{thm}

	Having understood the estimator $\hat{X}_0$ at the leftmost end of the tradeoff, we now turn to study optimal estimators for arbitrary $P$. Interestingly, we can show that Problem \eqref{eq:MSE::D_P::Alternative} is equivalent to (see App.~\ref{APPsec::proofs})
	\begin{equation}\label{eq:ObjectiveArbitraryP}
	D(P)=D^*+\min_{p_{\hat{X}}}\left\{ W^2_2(p_{\hat{X}},p_{X^{*}})\;:\; W_{2}(p_{\hat{X}},p_X)\leq P\right\}.
	\end{equation}
	Namely, an optimal $p_{\hat{X}}$ is closest to $p_{X^*}$ among all distributions within a ball of radius~$P$ around $p_X$, as illustrated in Fig.~\ref{fig:geodesic}.
	Moreover, $p_{\hat{X}X^*}$ is an optimal plan between $p_{\hat{X}}$ and $p_{X^*}$.
	As it turns out, this somewhat abstract viewpoint leads to a rather practical construction for $\hat{X}$ from the estimators $\hat{X}_0$ and $X^*$ at the two extremes of the tradeoff.  Specifically, we have the following result, proved in App.~\ref{APPsec::proofs}.
	\begin{thm}[Optimal estimators for arbitrary $P$]
		\label{Thm::extrapol}Let $\hat{X}_0$ be an estimator achieving perception index~$0$ and MSE $D(0)$. Then for any $P\in[0,P^*]$,
		the estimator
		\begin{equation}
		\hat{X}_P=\left(1-\frac{P}{P^*}\right)\hat{X}_{0}+\frac{P}{P^*}X^{*}\label{eq:hat_X::extrapolated}
		\end{equation}
		is optimal for perception index $P$. Namely, it achieves perception index $P$ and distortion $D(P)$.
	\end{thm}

	Theorem~\ref{Thm::extrapol} has important implications for perceptual signal restoration. For example, in the task of image super-resolution, there exist many deep network based methods that achieve a low MSE \citep{lim2017enhanced,ulyanov2018deep,shocher2018zero}. These provide an approximation for $X^*$. Moreover, there is an abundance of methods that achieve good perceptual quality at the price of a reasonable degradation in MSE (often by incorporating a GAN-based loss) \citep{ledig2017photo,wang2018esrgan,shaham2019singan}. These constitute approximations for $\hat{X}_0$. However, achieving results that strike other prescribed balances between MSE and perceptual quality commonly require training a different model for each setting. \citet{shoshan2019dynamic} and \citet{navarrete2018multi} tried to address this difficulty by introducing new training techniques that allow traversing the distortion-perception tradeoff at test time. But, interestingly, Theorem~\ref{Thm::extrapol} shows that in our setting such specialized training methods are not required. Having a model that leads to low MSE and one that leads to good perceptual quality, it is possible to construct any other estimator on the DP tradeoff, by simply averaging the outputs of these two models with appropriate weights. We illustrate this in Sec.~\ref{sec::numerical}.

	\subsection{The Gaussian setting}
	\label{Sec::GaussianSetting}
	When $X$ and $Y$ are jointly Gaussian, it is well known that the minimum MSE estimator $X^*$ is a linear function of the measurements $Y$. However, it is not \emph{a-priori} clear whether all estimators along the DP tradeoff are linear in this case, and what kind of randomness they possess. As we now show, equipped with Theorem~\ref{Thm::extrapol}, we can obtain closed form expressions for optimal estimators for any $P$. For simplicity, we assume here that $X$ and $Y$ have zero means and that $\Sigma_X,\Sigma_{Y} \succ0$.

	It is instructive to start by considering the simple case, where $\Sigma_{X^*}$ is non-singular (in Theorem~\ref{thm:Gaussian1} below we address the more general case of a possibly singular $\Sigma_{X^*}$). It is well known that
	\begin{equation}\label{eq:SigmaXstar}
	    X^*=\Sigma_{XY}\Sigma_Y^{-1}Y,\qquad \Sigma_{X^*}=\Sigma_{XY}\Sigma_{Y}^{-1}\Sigma_{YX}.
	\end{equation}

	Now, since we assumed that $\Sigma_X,\Sigma_{X^*}\succ0$, we have from Theorem~\ref{thm:perfect_perception} and \eqref{eq:optimal_dist_for_gauss},\eqref{eq::T1->2:def} that
	\begin{equation}
	    \hat{X}_0 = \Sigma_{X^*}^{-\frac{1}{2}}\left(\Sigma_{X^*}^{\frac{1}{2}}\Sigma_{X}\Sigma_{X^*}^{\frac{1}{2}}\right)^{\frac{1}{2}}\Sigma_{X^*}^{-\frac{1}{2}} X^*.
	\end{equation}
	Finally, we know that $P^*=G^*$, which is given by the left-hand side of \eqref{eq:Sandwitch}. Substituting these expressions into \eqref{eq:hat_X::extrapolated}, we obtain that an optimal estimator for perception $P\in[0,G^*]$ is given by
	\begin{equation}\label{eq:GaussXpInvertible}
	   \hat{X}_P=\left(\left(1-\frac{P}{G^*}\right)\Sigma_{X^*}^{-\frac{1}{2}}\left(\Sigma_{X^*}^{\frac{1}{2}}\Sigma_{X}\Sigma_{X^*}^{\frac{1}{2}}\right)^{\frac{1}{2}}\Sigma_{X^*}^{-\frac{1}{2}} +\frac{P}{G^*}I\right) \Sigma_{XY}\Sigma_Y^{-1}Y.
	\end{equation}
	As can be seen, this optimal estimator is a deterministic linear transformation of $Y$ for any $P$.

	The setting just described does not cover the case where $Y$ is of lower dimensionality than $X$ because in that case $\Sigma_{X^*}$ is necessarily singular (it is a $n_x\times n_x$ matrix of rank at most $n_y$; see \eqref{eq:SigmaXstar}). In this case, any deterministic linear function of $Y$ would result in an estimator $\hat{X}$ with a rank-$n_y$ covariance. Obviously, the distribution of such an estimator cannot be arbitrarily close to that of $X$, whose covariance has rank $n_x$. What is the optimal estimator in this more general setting, then?

	\begin{thm}[Optimal estimators in the Gaussian case]\label{thm:Gaussian1}
		Assume $X$ and $Y$ are zero-mean jointly Gaussian random vectors with $\Sigma_X,\Sigma_{Y} \succ0$. Denote $T^{*}\triangleq T_{p_X\rightarrow p_{X^*}}=\Sigma_{X}^{-1/2}(\Sigma_{X}^{1/2}\Sigma_{X^*}\Sigma_{X}^{1/2})^{1/2}\Sigma_{X}^{-1/2}$.
		Then for any $P\in[0,G^{*}]$, an estimator with perception index $P$ and MSE $D(P)$ can be constructed as
		\begin{equation} \label{eq::X_P_Gauss:interpolation}
		\hat{X}_P=\left(\left(1-\frac{P}{G^{*}}\right)\Sigma_{X}^{\frac{1}{2}}\left(\Sigma_{X}^{\frac{1}{2}}\Sigma_{X^*}\Sigma_{X}^{\frac{1}{2}}\right)^{\frac{1}{2}}\Sigma_{X}^{-\frac{1}{2}}\Sigma_{X^*}^{\dagger}+\frac{P}{G^{*}}I\right)\Sigma_{XY}\Sigma_{Y}^{-1}Y+\left(1-\frac{P}{G^{*}}\right)W,
		\end{equation}
		where $W$ is a zero-mean Gaussian noise with covariance $\Sigma_{W}=\Sigma_{X}^{1/2}(I-\Sigma_{X}^{1/2}T^{*}\Sigma_{X^*}^{\dagger}T^{*}\Sigma_{X}^{1/2})\Sigma_{X}^{1/2}$, which is independent of $Y,X$, and $\Sigma_{X^*}^{\dagger}$ is the pseudo-inverse of $\Sigma_{X^*}$.
	\end{thm}
	Note that in this case, we indeed have a random noise component that shapes the covariance of $\hat{X}_P$ to become closer to $\Sigma_X$ as $P$ gets closer to $0$. It can be shown (see App.~\ref{APPsec::proofs}) that when $\Sigma_{X^*}$ is invertible, $\Sigma_W=0$ and \eqref{eq::X_P_Gauss:interpolation} reduces to \eqref{eq:GaussXpInvertible}. Also note that, as in \eqref{eq:GaussXpInvertible}, the dependence of $\hat{X}_P$ on $Y$ in \eqref{eq::X_P_Gauss:interpolation} is only through $X^*=\Sigma_{XY}\Sigma_{Y}^{-1}Y$.

	As mentioned in Sec.~\ref{sec::The MSE--Wasserstein-2 tradeoff}, the optimal estimator is generally not unique. Interestingly, in the Gaussian setting we can explicitly characterize a \emph{set} of optimal estimators.
	\begin{thm}[A set of optimal estimators in the Gaussian case] \label{thm:Gaussian_not_unique}
		Consider the setting of Theorem~\ref{thm:Gaussian1}. Let $\Sigma_{\hat X_0 Y}\in \R^{n_{x}\times n_{y}}$ satisfy
		\begin{equation}
		\Sigma_{\hat X_0 Y}\Sigma_{Y}^{-1}\Sigma_{YX}=\Sigma_{X}^{\frac{1}{2}}(\Sigma_{X}^{\frac{1}{2}}\Sigma_{X^*}\Sigma_{X}^{\frac{1}{2}})^{\frac{1}{2}}\Sigma_{X}^{-\frac{1}{2}},\label{eq:Thm_Gaussian_General::M_cond1-2-1-1}
		\end{equation}
		and $W_{0}$ be a zero-mean Gaussian noise with covariance
		\begin{equation}
		\Sigma_{W_{0}}=\Sigma_{X}-\Sigma_{\hat X_0 Y}\Sigma_{Y}^{-1}\Sigma_{\hat X_0 Y}^{T}\succeq0
		\label{eq:Thm_Gaussian_General::M_cond2-2-1-1}
		\end{equation}
		that is independent of $X,Y$. Then, for any $P\in[0,G^*]$, an optimal estimator with perception index $P$ can be obtained by
		\begin{equation}
		\hat{X}_P=\left(\left(1-\frac{P}{G^{*}}\right)\Sigma_{\hat X_0 Y}+\frac{P}{G^{*}}\Sigma_{XY}\right)\Sigma_{Y}^{-1}Y+\left(1-\frac{P}{G^{*}}\right)W_{0}.
		\end{equation} The estimator given in \eqref{eq::X_P_Gauss:interpolation}
		is one solution to \eqref{eq:Thm_Gaussian_General::M_cond1-2-1-1}-\eqref{eq:Thm_Gaussian_General::M_cond2-2-1-1},
		but is generally not unique.
	\end{thm}

\section{A geometric perspective on the distortion-perception tradeoff}
\label{sec::geometric}
	In this section we provide a geometric point of view on our main results. Specifically, we show that the results of Theorems \ref{Thm:=00005Bthe-Distortion-Perception-funct} and \ref{Thm::extrapol} are a consequence
	of a more general geometric property of the space $\mathcal{W}_{2}(\R^{n_{x}})$.
	In the Gaussian case, this is simplified to a geometry of covariance
	matrices.

	Recall from \eqref{eq:ObjectiveArbitraryP} that the optimal $p_{\hat{X}}$ is the one closest to $p_{X^*}$ (in terms of Wasserstein distance) among all measures at a distance $P$ from $p_X$. This implies that to determine $p_{\hat{X}}$, we should traverse the geodesic between $p_{X^*}$ and $p_{X}$ until reaching a distance of $P$ from $p_X$.	Furthermore, $p_{\hat{X}X^*}$ should be the optimal plan between $p_{\hat{X}}$ and $p_{X^*}$. Interestingly, geodesics in Wasserstein spaces take a particularly simple form, and their explicit construction also turns out to satisfy the latter requirement. Specifically,
	let $\gamma,\mu$ be measures in $\mathcal{W}_{2}(\R^{d})$, let $\nu\in\Pi(\gamma,\mu)$	be an optimal plan attaining $W_2(\gamma,\mu)$, and let $\pi_{i}$ denote the projection $\pi_{i}:\R^{d}\times \R^{d}\rightarrow \R^{d}$
	such that $\pi_{i}((x_{1},x_{2}))=x_{i},\;i=1,2$. Then, the curve
	\begin{equation}
	\gamma_{t}\triangleq\left[(1-t)\pi_{1}+t\pi_{2}\right]\#\nu,\quad t\in[0,1]\label{eq:Geodesic:nu}
	\end{equation}
	is a constant-speed geodesic from $\gamma$ to $\mu$ in $\mathcal{W}_{2}(\R^{d})$
	\citep{ambrosio2008gradient}, where $\#$ is the push-forward operation\footnote{For measures $\gamma,\mu$ on $\mathcal{X},\mathcal{Y}$, we say that a measurable transform $T:\mathcal{X}\rightarrow\mathcal{Y}$ pushes $\gamma$ forward to $\mu$ (denoted  $T\#\gamma=\mu$) iff $\gamma(T^{-1}(B))=\mu(B)$ for any measurable $B \subseteq \mathcal{Y}.$}. Particularly,
	\begin{equation}
	W_{2}(\gamma_{t},\gamma_{s})=|t-s|W_{2}(\gamma,\mu),
	\end{equation}
	and it follows that $\ensuremath{W_{2}(\gamma_{t},\gamma)=tW_{2}(\gamma,\mu)}$
	and $\ensuremath{W_{2}(\gamma_{t},\mu)=(1-t)W_{2}(\gamma,\mu)}$. Furthermore, if $\gamma_t,t\in [0,1]$ is a constant-speed geodesic with $\gamma_0=\gamma,\gamma_1=\mu$, then the optimal plans between $\gamma,\gamma_t$ and between $\gamma_t,\mu$ are given by \begin{equation}
	    \left[\pi_1,(1-t)\pi_{1}+t\pi_{2}\right]\#\nu,\quad \left[(1-t)\pi_{1}+t\pi_{2},\pi_2\right]\#\nu,
	\end{equation}
	respectively, where $\nu\in\Pi(\gamma,\mu)$ is some optimal plan.
	Applying \eqref{eq:Geodesic:nu} to $(\hat X_0,X^*)\sim \nu$ with $t=P/P^*$, we obtain \eqref{eq:hat_X::extrapolated}, where we show that the obtained estimator achieves $\EE[\|\hat {X}_P-X^*\|^2]=(1-t)^2 W^2_2(p_X,p_{X^*})$. This explains the result of Theorem \ref{Thm::extrapol}.

	It is worth mentioning that this geometric interpretation is simplified under some common settings. For example, when $\gamma$ is absolutely continuous (w.r.t.~the Lebesgue measure),
	we have a measurable map $T_{\gamma\rightarrow\mu}$ which is
	the solution to the optimal transport problem with the quadratic cost \citep[Thm 1.6.2, p.16]{panaretos2020invitation}. The geodesic \eqref{eq:Geodesic:nu} then takes the form
	\begin{equation}
	\gamma_{t}=[Id+t(T_{\gamma\rightarrow\mu}-Id)]\#\gamma,\quad t\in[0,1].
	\end{equation}
	Therefore, in our setting, if $\gamma=p_{X^*}$ has a density, then we can obtain $\hat {X}_P$ by the deterministic transformation $[X^*+\left( 1-\frac{P}{P^*} \right) \left(T_{p_{X^*}\rightarrow p_{X}}(X^*)-X^*\right)]$ (see Remark about randomness in Sec.~\ref{sec::The MSE--Wasserstein-2 tradeoff}).

	Further simplification arises when $\gamma,\mu$ are centered non-singular Gaussian measures, in which case $T_{\gamma\rightarrow\mu}$
	is the linear and symmetric transformation \eqref{eq::T1->2:def}.
	Then, $\gamma_{t}$ is a Gaussian measure with covariance $\Sigma_{\gamma_{t}}=T_{t}\Sigma_{\gamma}T_{t}$,
	where $T_{t}\triangleq[I+t(T_{\gamma\rightarrow\mu}-I)].$
	Therefore, in the Gaussian case, the shortest path (\ref{eq:Geodesic:nu})
	between distributions is reduced to a trajectory in the geometry of covariance matrices induced by the Gelbrich distance \citep{takatsu2010wasserstein}. If additionally $\Sigma_{\gamma}$ and $\Sigma_{\mu}$ commute, then the Gelbrich distance is further reduced to the $\ell^2$-distance between matrices, as we discuss in App.~\ref{appsec::commute}.

\section{Numerical illustration} \label{sec::numerical}

In this Section we evaluate $12$ super resolution algorithms on the BSD100 dataset\footnote{All codes are freely available and provided by the authors. The BSD100 dataset is free to download for non-commercial research.}
 \citep{martin2001database}.  The evaluated algorithms include EDSR \citep{lim2017enhanced}, ESRGAN \citep{wang2018esrgan}, SinGAN \citep{shaham2019singan}, ZSSR \citep{shocher2018zero}, DIP \citep{ulyanov2018deep}, SRResNet variants which optimize MSE and VGG$_{2,2}$, SRGAN variants which optimize MSE, VGG$_{2,2}$ and VGG$_{5,4}$ in addition to an adversarial loss \citep{ledig2017photo}, ENet \citep{sajjadi2017enhancenet} (``PAT'' and ``E'' variants). Low resolution images were obtained by $ 4\times$ downsampling using a bicubic kernel.

In Figure \ref{fig:DG} we plot each method on the distortion-perception plane. Specifically, we consider natural (and reconstructed) images to be stationary random sources, and use $9 \times 9 $ patches  (totally $1.6\times 10^6$ patches) from the RGB images to empirically estimate the mean and covariance matrix for the ground-truth images, and for the reconstructions produced by each method. We then use the estimated Gelbrich distances \eqref{eq:Gelbrich_dist} between the patch distribution of each method and that of ground-truth images, as a perceptual quality index. Recall this is a lower bound on the Wasserstein distance.

We consider EDSR \cite{lim2017enhanced} to be the best MSE estimator $X^*$ since it achieves the lowest distortion among the evaluated methods. We therefore estimate the lower bound \eqref{eq:Thm2::LwrBound_G_Sx_S*} as
\[\hat D (P)=D_{\text{EDSR}}+\left[(P_{\text{EDSR}}-P)_+\right]^2,\]
where $D_{\text{EDSR}}$ is the MSE of EDSR, and $P_{\text{EDSR}}$ is the estimated Gelbrich distance between EDSR reconstructions and ground-truth images. Note the unoccupied region under the estimated curve in Figure~\ref{fig:DG}, which is indeed unattainable according to the theory.

We also present 11 estimators $\hat X_t$ which we construct by interpolation between EDSR and ESRGAN \cite{wang2018esrgan},
	$ \hat X_t = tX_{\text{EDSR}} + (1-t)X_{\text{ESRGAN}}$.
	 We observe (Figure \ref{fig:DG}) that estimators constructed using these two extreme points are closer to the optimal DP tradeoff than the evaluated methods. Also note that since ESRGAN does not attain $0$-perception index, we are practically able to use negative values $t<0$ to extrapolate better perception-quality estimators $\hat X_{-0.05}$ and $\hat X_{-0.1}$. In Figure \ref{fig:SrganVSintp} we present a visual comparison between SRGAN-VGG$_{2,2}$ \citep{ledig2017photo}
	and our interpolated estimator $\hat X_{0.12}$. Both achieve roughly the same RMSE distortion ($18.09$ for SRGAN, $18.15$ for $\hat X_{0.12}$), but our estimator achieves a lower perception index. Namely, by using interpolation, we manage to achieve improvement in perceptual quality, without degradation in distortion. The improvement in visual quality is also apparent in the figure.	Additional visual comparisons can be found in the Appendix.

	\begin{figure}[h]
	\begin{centering}
			\includegraphics[bb=10bp 0bp 460bp 307bp,clip,scale=0.85,width=0.80\linewidth]{plot/20210520_22_29_19.pdf}
			\par\end{centering}

		\centering{}\caption{\label{fig:DG} \textbf{Evaluation of SR algorithms}. We plot 12 algorithms (Blue) on the Distortion-Perception plane. Here we estimate perception using the Gelbrich distance between empirical means and covariances of the original data and reconstructed data. $\hat D(P)$ (Orange) is the estimated lower bound \eqref{eq:Thm2::LwrBound_G_Sx_S*} where we consider EDSR to be the global minimizer $X^*$. Note the unoccupied region under the estimated curve, which is unattainable.
			We also plot 11 estimators $\hat X_t$ (Green) created by an interpolation between EDSR and ESRGAN, using different relative weights $t$. Note that estimators constructed using these two extreme estimators are closer to the optimal DP curve than the compared methods.}
	\end{figure}

	\begin{figure}[]
	\begin{centering}
			\includegraphics[bb=12bp 200bp 1150bp 522bp,clip,scale=0.99,width=.95\linewidth]{plot/vgg22x12_horiz_largefont}
		\par\end{centering}

	\centering{}\caption{ \label{fig:SrganVSintp}
		\textbf{A visual comparison between estimators with approximately the same MSE}. Left: SRGAN-VGG$_{2,2}$. Right: $\hat X_{0.12}$, an interpolation between EDSR and ESRGAN using $t=0.12$. Observe the improvement in perceptual quality, without any significant degradation in distortion.}
	\end{figure}

\section{Conclusion}
In this paper we provide a full characterization of the distortion-perception tradeoff for the MSE distortion and the Wasserstein-$2$ perception index. We show that optimal estimators are obtained by interpolation between the minimum MSE estimator and an optimal perfect perception quality estimator. In the Gaussian case, we explicitly formulate these estimators. To the best of our knowledge, this is the first work to derive such closed-form expressions. Our work paves the way towards fully understanding the DP tradeoff under more general distortions and perceptual criteria, and bridging between fidelity and visual quality at test-time, without training different models.

\setcitestyle{numbers}


\section{Introduction}\label{section:Intro}

The study of extreme eigenvalue statistics in Random Matrix Theory (RMT) has attracted much attention during the last twenty years. In particular, the Tracy-Widom (TW) distributions \cite{TW94a,TW96} describing the largest eigenvalue $\lambda_{\max}$ (as well as the smallest one, $\lambda_{\min}$) in the classical Gaussian ensembles, orthogonal (GOE, $\beta =1$), unitary (GUE, $\beta= 2$) and symplectic (GSE, $\beta =4$) -- where $\beta$ is the Dyson index -- have become cornerstones of the theory of extreme value statistics of strongly correlated variables. Quite remarkably, it was shown that the TW distributions, denoted by ${\cal F}_{\beta}$ in the following, appear in a wide variety of problems \cite{Majumdar06}, a priori not directly related to RMT, ranging from the longest increasing sequence of random permutations of integers \cite{Baik99}, stochastic growth and related directed polymer models in the Kardar-Parisi-Zhang \cite{KPZ86} universality class \cite{Johansson00,Prahofer00,Sasamoto10,Calabrese10,Dotsenko10,Amir11} and sequence alignment problems \cite{Majumdar05} to non-intersecting interfaces and Brownian motions \cite{Nadal09,Forrester11,Liechty12} as well as in finance \cite{Biroli07}. While the TW distribution describes the typical fluctuations of $\lambda_{\max}$ and $\lambda_{\min}$, a large body of work has also been devoted to the study of large deviations of extreme eigenvalues in Gaussian ensembles \cite{Majumdar14}.

\begin{figure}[h!]
\begin{center}
\resizebox{120mm}{!}{\includegraphics{Wishart_soft_vs_hard.pdf}}
\caption{Plot of the Marchenko-Pastur distribution for $c<1$ ({\bf left}) and for $c=1$ ({\bf right}).}
\label{fig:Wishart_soft_vs_hard}
\end{center}
\end{figure}

Another interesting ensemble of random matrices, which we focus on in this paper, is the so-called Wishart-Laguerre ensemble -- here we focus on the case of complex matrices ($\beta = 2$). Let $X$ be a $M \times N$ rectangular matrix with i.i.d. complex Gaussian entries and $M-N=a  \geq 0$. The Wishart-Laguerre matrix $W$ is defined as $W=X\,^\dagger X$ which is thus a $N \times N$ Hermitian matrix, having $N$ real and positive eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_N$. The joint probability density function (PDF) of these $N$ eigenvalues is given by \cite{Mehta91,Forrester10}
\begin{eqnarray}\label{Pjoint}
P_{\rm joint}(\lambda_1,\lambda_2,\cdots,\lambda_N)=\frac{1}{Z_N} \prod_{i < j}^N (\lambda_i-\lambda_j)^2 \,\exp{\left(-\sum_{i=1}^N \lambda_i\right)} \prod_{i=1}^N\lambda_i^a,
\end{eqnarray}
where $\lambda_1$, $\lambda_2$ ... $\lambda_N$ are positive and where $Z_N$ is a normalization constant, depending on $a$. These Wishart-Laguerre matrices play an important role in statistics, in particular in principal component analysis, where the matrix $W$ is a covariance matrix. Hence in this case both $M$ and $N$, and thus $a = M-N$, are positive integers. However, this joint PDF in Eq. (\ref{Pjoint}) is well defined for any real value of $a \geq 0$, which, for some non-integer value of $a$, may have some physical applications. An interesting example is the case of non-intersecting Brownian excursions \cite{Tracy07}, i.e., $N$ non-colliding positive Brownian paths $x_i(t) \ge 0$ on the unit time interval $t\in[0,1]$ constrained to start and end at the origin $x_i(0)=x_i(1)=0$. The joint PDF of the positions of the $N$ walkers, at a given time $t$, can indeed be written as \cite{Schehr08}
\begin{eqnarray}\label{pdf_watermelons}
P_{\rm joint}(x_1,\cdots,x_N;t)=\frac{1}{z_N(t)} \prod_{i =1}^N x_i^{2} \prod_{i < j}^N (x_i^2-x_j^2)^2 \,\exp{\left(-\sum_{i=1}^N \frac{x_i^2}{\sigma^2(t)} \right)} \;,
\end{eqnarray}
where $\sigma^2(t) = 2t(1-t)$ and $z_N(t)$ is a normalization constant. From this expression (\ref{pdf_watermelons}) we obtain that the scaled variables
$x_i^2/\sigma^2(t)$ behave statistically like the eigenvalues of random matrices from the Wishart-Laguerre ensemble (\ref{Pjoint}) with a non-integer parameter $a=1/2$. Hence it is physically relevant to study the joint distribution in Eq.~(\ref{Pjoint}) for any real $a \geq 0$.

A first important characteristic associated to this ensemble (\ref{Pjoint}) is the mean density of eigenvalues, $\rho(\lambda,N)$, defined by
\begin{eqnarray}
\rho(\lambda,N)=\frac{1}{N} \sum_{i=1}^{N} \overline{\delta(\lambda_i-\lambda)},
\end{eqnarray}
where the overline denotes an average over the different realizations of the random variables $\lambda_i$'s according to the joint PDF in Eq.  (\ref{Pjoint}).
In the large $N$ limit, it is well known that $\rho(\lambda,N)$ is given by the Marchenko-Pastur (MP) distribution (see Fig. \ref{fig:Wishart_soft_vs_hard})
\begin{eqnarray}\label{density_MP}
\rho(\lambda,N) &\underset{N \to \infty}{\to}& \frac{1}{N}\rho_{\rm MP}\left( \frac{\lambda}{N}\right),\\
\rho_{\rm MP}\left(x\right)&=&\frac{\sqrt{(x-x_-)(x_+-x)}}{2 \pi x} ,
\end{eqnarray}
where $x_\pm = (c^{-1/2} \pm 1)^2$ are the right and left edges of the support, with $c=N/M \le 1$. The case where $c<1$ corresponds to the case where $a \sim {\cal O}(N)$. Here, we mainly focus on the case $c=1$ which corresponds instead to the case where $a$ is finite, while both $N$ and $M$ are large. In this case, which we will mainly focus on in this paper, the MP distribution takes the particular form (see the left panel of Fig.~\ref{fig:Wishart_soft_vs_hard})
\begin{eqnarray}\label{eq:MP_afinite}
\rho_{\rm MP}\left(x\right)&=&\frac{1}{2\pi}\sqrt{\frac{4-x}{x}} \;.
\end{eqnarray}
At the right edge, near $x_+ = 4$, $\rho_{\rm MP}\left(x\right)$ vanishes as a square-root, $\rho_{\rm MP}\left(x\right) \propto \sqrt{4 - x}$ and therefore the fluctuations near this {\it soft} edge are governed, for large $N$, by the Airy kernel. In particular, the distribution of the largest eigenvalue $\lambda_{\max}$, appropriately shifted and scaled, converges, when $N \to \infty$, to the TW distribution ${\cal F}_2$ mentioned above (the same as for GUE) \cite{Johansson00,Johnstone01}. It is now well known that this distribution can be expressed in terms of a special solution of a Painlev\'e II equation. While this connection to Painlev\'e transcendents was initially obtained by Tracy and Widom using Fredholm operator techniques, Nadal and Majumdar \cite{Nadal11} provided, more recently, a derivation of this result (for $\beta =2$) using semi-classical orthogonal polynomials (OPs), see also Ref. \cite{Chen05}. This method is at the heart of the present paper.

On the other hand, at the left edge, $x_- = 0$, $\rho_{\rm MP}\left(x\right)$ has a square-root singularity, $\rho_{\rm MP}\left(x\right) \propto 1/\sqrt{x}$ (see the left panel of Fig. \ref{fig:Wishart_soft_vs_hard}). What about the fluctuations of the smallest eigenvalue $\lambda_{\min}$ in this case? One can estimate the typical scale of $\lambda_{\min}$, for large $N$, by considering that there is typically one eigenvalue in the interval $[0,\lambda_{\min}]$, i.e.,
\begin{eqnarray}
N \int_0^{\lambda_{\min}} \rho\left(\lambda,N\right) \, {\rm d} \lambda \sim {\cal O}(1)\;,
\end{eqnarray}
which implies that $\lambda_{\min} \sim {\cal O}(1/N)$. Within this scale ${\cal O}(1/N)$, the fluctuations are governed by the Bessel kernel \cite{For93,Ver93}. Furthermore, it has been shown that the distribution of $N \lambda_{\min}$
converges to a limiting form which (i) is different from the standard TW distribution ${\cal F}_2$ and depends continuously on the exponent $a$ in (\ref{Pjoint}) and (ii) can be written in terms of a special solution of a Painlev\'e III equation~\cite{Tracy94}. If one introduces $F_N(t)={\Pr}(\lambda_{\min} \ge t)$ then, one has indeed
\begin{eqnarray}\label{eq:result_TW}
\underset{N \to \infty}{\lim}F_N\!\left(\frac{x}{N}\right) = F_{\infty}(x),\hspace{0.2cm} F_{\infty}(x)=\exp\left( \int_0^x \frac{f(u)}{u} {\rm d}u\right) \;,
\end{eqnarray}
where $f(x)$ is the unique solution of a Painlev\'e III~\cite{Tracy94}:
\begin{eqnarray}\label{PIII}
(xf'')^2+4f'(1+f')(xf'-f)=(a f')^2 \;,
\end{eqnarray}
satisfying
\begin{eqnarray}\label{asympt_PIII}
f(x) \sim  - \frac{x^{a+1}}{\Gamma(a+1) \Gamma(a+2)}  \;, \;  {\rm as} \; x \to 0 \;.
\end{eqnarray}
This result (\ref{eq:result_TW}) was shown by Tracy and Widom using Fredholm operator techniques~\cite{Tracy94}. Note that for integer values of $a$, the limiting distribution $F_{\infty}(x)$ can be written as an $a \times a$ determinant whose entries are expressed in terms of Bessel functions (see Eq. (\ref{fdetBessel}) below) -- a result which can be obtained by clever manipulations of determinants~\cite{Forrester94}. In particular for $a=0$ the result is extremely simple as $F_N(x) = \exp{(-N\,x)}$ for all $N$, implying $f(x) = -x$, which is obviously solution of Eq. (\ref{PIII}) with the boundary condition~(\ref{asympt_PIII}).

The limiting distribution of $\lambda_{\min}$ for complex Wishart matrices, in the limit $N \to \infty$, is thus well known (\ref{eq:result_TW}) -- we refer the reader to Ref. \cite{akemann} to a recent work on the smallest eigenvalue for real Wishart matrices in the hard edge limit. What about the finite $N$ corrections to this asymptotic form? Such a question is quite natural for practical applications of extreme value statistics (EVS), where one always deals with finite samples -- here matrices of finite size. This issue was recently revisited for EVS of independent and identically distributed random variables using a renormalization group approach \cite{Gyorgyi10}. For EVS of strongly correlated variables, there are actually few cases where these corrections have been worked out explicitly, including random walks \cite{Schehr06}, the {\it largest} eigenvalue of random matrices belonging to various ensembles \cite{karoui2006,johnstone2008,johnstonema2011,Ma12}, non-intersecting Brownian motions \cite{Schehr13} or (Poissonized) random matchings \cite{Baik13}. For {\it real} Wishart matrices, the first corrections to the limiting distribution of the {\it smallest} eigenvalue in the soft edge limit were studied in Ref. \cite{Ma12} where it was shown that corrections to the limiting distribution of $\lambda_{\min}$ and $\lambda_{\max}$ are quite different, although the limiting distributions for both observables are actually the same, namely the TW distribution for GOE, ${\cal F}_1$. What about the corrections to the limiting distribution $F_{\infty}$ in Eq. (\ref{eq:result_TW}) of $\lambda_{\min}$ for complex Wishart matrices in the hard edge limit? This question was recently raised by Edelman, Guionnet et P\'ech\'e \cite{Edelman14} in their study of finite size covariance matrices with non-Gaussian entries. Based on the large $N$ expansion of the exact formulas obtained in Ref.~\cite{Forrester94} for small integer values of $a$, they conjectured the following form of the first $1/N$-correction
\begin{eqnarray}\label{conjecture_finite_N}
F_N\left(\frac{x}{N}\right) = F_\infty(x) + \frac{a}{2N} x \, F'_\infty(x) +  o\!\left(\frac{1}{N}\right) \;.
\end{eqnarray}
Note that this first $1/N$ correction in Eq. (\ref{conjecture_finite_N}) can be interpreted as a correction to the width, i.e.,
\begin{eqnarray}\label{eq:correction}
F_N\left(\frac{x}{N}\right) = F_{\infty}\left(x\left(1 + \frac{a}{2\,N}\right)\right) +  o\!\left(\frac{1}{N}\right) \;.
\end{eqnarray}
It is interesting to notice \cite{Baik13} that for most of the cases which have been studied in RMT~\cite{karoui2006,johnstone2008,johnstonema2011}, it was actually found that the first order correction to the limiting distribution of extreme eigenvalue actually corresponds to a correction of the scaling variable, as in Eq.~(\ref{eq:correction}). One exception concerns the smallest eigenvalue of real Wishart matrices in the soft-edge limit, where the first correction has a more complicated structure~\cite{Ma12}.

 The main goal of this paper is to provide an explicit computation of this first correction in the hard edge limit and we will show that it has indeed the conjectured form given above in Eq.~(\ref{conjecture_finite_N}). To perform this computation, we will use a method relying on semi-classical OPs, in the spirit of Refs. \cite{Nadal11} and \cite{Chen05,Basor09}. As we will see, our method does not only allow us to compute explicitly the first $1/N$ corrections but provides also a rather straightforward derivation of the expression for the limiting distribution $F_{\infty}(x)$ in terms of the solution of a Painlev\'e III equation, without using Fredholm operators theory but relying instead only on the recurrence relations associated to the (semi-classical) OPs system. Finally, we will also study the first finite $N$ corrections to the limiting distribution of $\lambda_{\min}$ at the soft edge.

Note that after the results obtained in the present paper were presented in a conference \cite{conf_montevideo}, another independent proof of the conjecture in Eq. (\ref{conjecture_finite_N}) was achieved in Ref. \cite{Bornemann15}, using operator theoretic techniques. More recently, yet another independent proof of this conjecture was given in Ref. \cite{najim2015}.

\section{Summary of main results and outline of the paper}\label{section:Summary}
The distribution of the smallest eigenvalue $\lambda_{\min}=\underset{1 \le i \le N}{\min}\lambda_i$ is given by
\begin{equation}\label{def_FN}
F_N(t)={\rm Prob}(\lambda_{\min} \ge t)=\int_t^{\infty} \!{\rm d}\lambda_1 \int_t^{\infty} \!{\rm d}\lambda_2 ... \int_t^{\infty} \!{\rm d}\lambda_N P_{\rm joint}(\lambda_1,\lambda_2,\cdots,\lambda_N) \;.
\end{equation}
In this paper, we will compute $F_N(t)$ using semi-classical OPs $\{\pi_k(\lambda)\}_{k \in \mathbb{N}}$ which are polynomials of the variable $\lambda$ while $t$ and $a$ are parameters (for the sake of clarity in the notations, this dependence is omitted here):
\begin{eqnarray}\label{eq:def_OP}
\left\{
\begin{array}{l}
  \braket{\pi_k}{\pi_{k'}}=\int_t^{\infty} e^{-\lambda} \lambda^a \pi_k(\lambda) \pi_{k'}(\lambda) {\rm d}\lambda = h_k \delta_{k,k'}, \\
  \\
  \pi_k(\lambda)=\lambda^k + \zeta_k \lambda^{k-1} + ...
\end{array}
\right.
\end{eqnarray}
The cumulative distribution $F_N(t)$ can be expressed, using standard manipulations, in terms of the norms $h_k$'s as
\begin{eqnarray}
F_N(t) = \frac{N!}{Z_N} \prod_{k=0}^{N-1} h_k \;.
\end{eqnarray}
As we will see, the norms $h_k$'s can be computed from the three term recurrence relation satisfied by the OPs
\begin{eqnarray}
\lambda \pi_k=\pi_{k+1}+S_k \pi_k +R_k \pi_{k-1} \;,
\end{eqnarray}
from which we deduce the following important relations:
\begin{eqnarray}\label{important_eq}
&&R_k = \frac{h_k}{h_{k-1}}, \\
&&S_k = - t \partial_t \log h_k + 2k + a + 1 \;,\\
&& \zeta_k = - \sum_{i=0}^{k-1} S_i \;.
\end{eqnarray}
Note that the starting point of our analysis is very similar to the one of Basor and Chen in Ref. \cite{Basor09} but the analysis of the recursion relations is different. In particular, we do not make use of ladder operators techniques, which are heavily used in Ref. \cite{Basor09}. In addition, we provide an asymptotic analysis of this OP system (\ref{eq:def_OP}) for large $N$, beyond the leading order.

In section \ref{section:OP}, we will study the variables $S_k$, $R_k$, $h_k$ and $\zeta_k$. In particular, we will show that $S_k$ and $R_k$
satisfy a coupled set of equations, named Schlesinger equations in the literature on OPs
\begin{eqnarray}\label{schlesinger_intro}
\left\{
\begin{array}{ccl}
S_k-R_{k+1}+R_{k} &=& t\partial_t S_k,\\
\\
2-S_{k+1}+S_{k} &=& t \dfrac{\partial_t R_{k+1}}{R_{k+1}} \;.
\end{array}
\right.
\end{eqnarray}
This system of equations, together with the initial condition given in Eq. (\ref{CI}) below, determines uniquely the values of $R_k$ and $S_k$ for all values of $k$, as they can be computed by induction. It is however quite difficult to analyze the large $N$ behavior of $R_N$ and $S_N$ using only this set of equations (\ref{schlesinger_intro}). To circumvent this difficulty, it is customary to use another set of relations, called the Laguerre-Freud equations, which we derive here using the method based on Tur\'an determinants, as developed in Ref. \cite{Belmehdi94}. They read
\begin{eqnarray}\label{LF_intro}
\left\{
\begin{array}{rcl}
R_{k+2}-R_k&=&S_{k+1}(2k+4+a+t-S_{k+1})-S_k(2k+a+t-S_k)-2t,\\
\\
S_{k+1}(S_{k+1}-t) &=& R_{k+1}(2k+1+a+t-S_{k+1}-S_k) \\
&-&R_{k+2}(2k+5+a+t-S_{k+2}-S_{k+1}).
\end{array}
\right.
\end{eqnarray}
Some details of this derivation have been relegated in \ref{appendix:Turan}.

In section \ref{section:PV}, by manipulating these two sets of equations (\ref{schlesinger_intro}) and (\ref{LF_intro}), we show that
$F_N(t)$, for finite $N$, is related to a special solution of a Painlev\'e V equation [see Eqs. (\ref{def:Hn}, \ref{Painleve5})], thus recovering a previous result of Tracy and Widom. Section \ref{section:PIII} is devoted to the hard edge scaling limit:
\begin{eqnarray}\label{def_hard_edge}
N \to \infty , \, t \to 0,\,  x=N\, t \in \mathbb{R}^+ \;\;{\rm fixed} \;.
\end{eqnarray}
It is known that the limiting distribution $F_{\infty}(x)$ can be obtained by analyzing the large $N$ limit of this Painlev\'e V equation in Eqs. (\ref{def:Hn}, \ref{Painleve5}), obtained for finite $N$, leading to (\ref{eq:result_TW}) and (\ref{PIII}). As we show in section 5, the finite $N$ corrections are then easily obtained from the Schlesinger equation (\ref{schlesinger_intro}), from which we obtain Eq. (\ref{conjecture_finite_N}). Finally, section \ref{section:PII} is devoted to the soft edge scaling limit, when $a \sim {\cal O}(N)$, for $N$ large.

\section{Semi-classical Orthogonal Polynomials}\label{section:OP}

To study this OP system (\ref{eq:def_OP}) it is useful \cite{Nadal11} to introduce a deformation parameter $\alpha$ and study the following OP system
\begin{eqnarray}\label{eq:OP_deformed}
\left\{
\begin{array}{l}
  \braket{\pi_k}{\pi_{k'}}=\int_t^{\infty} e^{-\alpha\lambda} \lambda^a \pi_k(\lambda) \pi_{k'}(\lambda) {\rm d}\lambda = h_k \delta_{k,k'}, \\
  \\
  \pi_k(\lambda)=\lambda^k + \ldots \;,
\end{array}
\right.
\end{eqnarray}
such that the norms $h_k$'s are defined by
\begin{eqnarray}
h_k &=& \braket{\pi_k}{\pi_{k}}. \label{def_hn}
\end{eqnarray}
As we show here, some useful relations can be obtained by varying $\alpha$. Eventually, we will of course set $\alpha =1$ (\ref{eq:def_OP}). The first polynomials can be computed from (\ref{eq:OP_deformed}) to obtain
\begin{eqnarray}\label{first_polynomials0}
\pi_0(\lambda)&=&1\,,\\
\pi_1(\lambda)&=&\lambda-\frac{1+a}{\alpha}-\frac{e^{-t\alpha}t(t\alpha)^a}{\Gamma(1+a,t\alpha)}\,,\label{first_polynomials1}
\end{eqnarray}
where $\Gamma(\nu,x) = \int_x^\infty y^{\nu-1} e^{-y} {\rm d}y$ is the incomplete gamma function. Obviously, the expression of the OPs $\pi_k$ becomes more and more complicated as $k$ grows.
The polynomials $\pi_k$ being OPs, they satisfy a three-term recursion relation, which can be obtained as follows. As
$\lambda \pi_k$ is a polynomial of degree $k+1$, we can expand it on the basis of these OPs. Because $\braket{\pi_{k-2-i}}{\lambda \pi_k}= 0 $ if $i \ge 0$, we can write the three term recurrence relation \cite{Szego75}:
\begin{eqnarray}\label{3term_rec}
\lambda \pi_k=\pi_{k+1}+S_k \pi_k +R_k \pi_{k-1},
\end{eqnarray}
where, by definition
\begin{eqnarray}
S_k h_k &=& \braket{\pi_k}{\lambda\pi_{k}},\\
R_k h_{k-1} &=& \braket{\pi_{k-1}}{\lambda\pi_{k}}.
\end{eqnarray}
From (\ref{first_polynomials0}) and (\ref{first_polynomials1}), we can compute the first terms
\begin{eqnarray}\label{CI0}
\left\{
\begin{array}{ccl}
h_0 &=& \int_t^{\infty} e^{-\alpha\lambda} \lambda^a {\rm d}\lambda = \dfrac{\Gamma(1+a,\alpha t)}{\alpha^{a+1}},\\
\\
S_0 &=& -\partial_\alpha \log{h_0}=\dfrac{1+a}{\alpha}+\dfrac{e^{-t\alpha}t(t\alpha)^a}{\Gamma(1+a,t\alpha)},\\
\\
R_0 &=& 0,\\
\\
\zeta_0&=& 0.
\end{array}
\right.
\end{eqnarray}

\subsection{Schlesinger equations}\label{section:Schlesinger}

In this section, we derive a couple of recursion relations called the Schlesinger equations that couple $R_k$ and $S_k$. We first write
\begin{eqnarray}
R_k h_{k-1} = \braket{\pi_{k-1}}{\lambda\pi_{k}}= \braket{\lambda \pi_{k-1}}{\pi_k} \;.
\end{eqnarray}
Therefore, using Eq. (\ref{3term_rec}) with the substitution $k \to k-1$, we have $\braket{\lambda \pi_{k-1}}{\pi_k} = \braket{\pi_k}{\pi_k} = h_k$ and finally we obtain the standard relation
\begin{eqnarray}\label{R_en_h}
R_k = \frac{h_k}{h_{k-1}}.
\end{eqnarray}
On the other hand, using the definition of the scalar product in (\ref{eq:OP_deformed}), we have
\begin{eqnarray}\label{Snhn}
S_k h_k = \braket{\pi_k}{\lambda\pi_{k}} = \int_t^{\infty} e^{-\alpha\lambda} \lambda^a \, \lambda \, \pi_{k}^2(\lambda)\, {\rm d}\lambda = -\partial_\alpha \braket{\pi_k}{\pi_{k}} = -\partial_\alpha h_k,
\end{eqnarray}
from which we deduce the relation between $S_k$ and $h_k$
\begin{eqnarray}\label{S_en_h}
S_k = -\partial_\alpha \log{h_k}.
\end{eqnarray}
By combining Eq. (\ref{S_en_h}) and Eq. (\ref{R_en_h}), we obtain straightforwardly
\begin{eqnarray}\label{1_recu}
S_{k+1}-S_{k} = -\partial_\alpha \log{\frac{h_{k+1}}{h_{k}}} =  -\partial_\alpha \log{R_{k+1}}.
\end{eqnarray}
We now study the coefficient $\zeta_k$ of the term of degree $k-1$ in the polynomial $\pi_k$ [see Eq. (\ref{eq:def_OP})]:
\begin{eqnarray}\label{def_zeta}
\pi_k(\lambda)=\lambda^k + \zeta_k \lambda^{k-1}+...
\end{eqnarray}
Taking the derivative of this equation with respect to (w.r.t.) $\alpha$ we obtain
\begin{eqnarray}\label{eq:deriv_zeta}
\partial_\alpha\pi_k(\lambda)=\partial_\alpha \zeta_k \lambda^{k-1}+ \ldots \;.
\end{eqnarray}
Multiplying both sides of Eq. (\ref{eq:deriv_zeta}) by $\lambda$ and projecting on $\pi_k$ yields
\begin{eqnarray}\label{eq:deriv_zeta2}
\partial_\alpha \zeta_k h_k = \braket{\lambda \partial_\alpha \pi_k}{ \pi_{k}}.
\end{eqnarray}
Besides, by looking at the term of degree $k$, i.e. $\propto \lambda^k$, in Eq. (\ref{3term_rec}) we find a recursion relation between $S_k$ and $\zeta_k$
\begin{eqnarray}\label{zeta_S}
\zeta_k = \zeta_{k+1}+S_k \;,
\end{eqnarray}
which can be solved for $\zeta_k$, using the initial condition (\ref{CI0}), to get
\begin{eqnarray}\label{zeta_k_S}
\zeta_k = -\sum_{i=0}^{k-1} S_i.
\end{eqnarray}
Furthermore, by differentiating Eq. (\ref{Snhn}) w.r.t. to $\alpha$ we have
\begin{eqnarray}
\partial_\alpha (S_k h_k) &=& - \int_t^\infty e^{-\alpha \lambda} \lambda^\alpha \, \lambda^2 \, \pi_k^2(\lambda) \, {\rm d}\lambda + 2 \int_t^\infty e^{-\alpha \lambda} \lambda^\alpha \, \lambda \,\partial_\alpha \pi_k(\lambda) \pi_k(\lambda) \, {\rm d }\lambda \nonumber \\
&=& -\braket{\lambda \pi_k}{\lambda\pi_{k}} + 2 \braket{\lambda \partial_\alpha \pi_k}{\pi_{k}} \;,
\end{eqnarray}
where, in the second line, we have simply used the definition of the scalar product in (\ref{eq:def_OP}). Using the three-term recurrence relation (\ref{3term_rec}) to rewrite the first term and Eq. (\ref{eq:deriv_zeta2}) to rewrite the second one, we have
\begin{eqnarray}\label{deriv_prod}
\partial_\alpha (S_{k} h_k)=  -h_{k+1} -S^2_k h_k -R_k^2 h_{k-1} +2 \partial_\alpha \zeta_k h_k \;.
\end{eqnarray}
We can also write
\begin{eqnarray}\label{id_prod}
\partial_\alpha (S_{k} h_k)= (\partial_\alpha S_{k}) h_k+S_k (\partial_\alpha h_k) \;.
\end{eqnarray}
By replacing the left hand side of Eq. (\ref{deriv_prod}) by Eq. (\ref{id_prod}), dividing the resulting equation by $h_k$ and using
Eqs. (\ref{R_en_h}) and (\ref{S_en_h}), we obtain a last recursion relation between $R_k$, $S_k$ and $\zeta_k$
\begin{eqnarray}\label{2_recu}
R_{k+1} +R_k = -\partial_\alpha S_{k} +2 \partial_\alpha \zeta_k \;.
\end{eqnarray}
Combining Eq. (\ref{zeta_S}) and (\ref{2_recu}), we have
\begin{eqnarray}
R_{k+1}+R_{k} = \partial_\alpha \zeta_{k+1}+ \partial_\alpha \zeta_k.
\end{eqnarray}
And finally, with the initial condition $R_0 = \zeta_0 = 0$ in Eq. (\ref{CI0}) we obtain
\begin{eqnarray}\label{eq:rel_zeta_R}
\partial_\alpha \zeta_k = R_k \;.
\end{eqnarray}
Therefore substituting this relation in Eq. (\ref{2_recu}) we finally obtain a closed system of two coupled recursion relations between $S_k$ and $R_k$
\begin{eqnarray}\label{recu_derivalpha}
\left\{
\begin{array}{ccl}
R_{k+1}-R_{k} &=& -\partial_\alpha S_k,\\
\\
S_{k+1}-S_{k} &=& -\partial_\alpha \log{R_{k+1}} \;,
\end{array}
\right.
\end{eqnarray}
where the second equation was previously obtained in Eq. (\ref{1_recu}).

Our goal now is to relate $\alpha$-derivatives to $t$-derivatives, i.e., find a relation between $\partial_\alpha h_k$ and $\partial_t h_k$. Differentiating Eq. (\ref{def_hn}) w.r.t. $t$, using that $\braket{\partial_t \pi_k}{\pi_{k}}=0$ as $\partial_t \pi_k$ is a polynomial of degree $k-1$ (\ref{eq:def_OP}), we have
\begin{eqnarray}\label{dot_hn}
\partial_t h_k=-e^{-\alpha t}t^{a} \pi_k^2(t).
\end{eqnarray}
We now start from the expression of $\partial_\alpha h_k$ given in Eq. (\ref{Snhn}) and use integration by parts to obtain
\begin{eqnarray}
-\partial_\alpha h_k&=& \int_t^\infty{\rm d}\lambda \pi_k^2(\lambda)e^{-\alpha\lambda}\lambda^{a+1}, \nonumber \\
&=&\big[-\frac{1}{\alpha} e^{-\alpha\lambda}\lambda^{a+1} \pi_k^2(\lambda)\big]^{\lambda=\infty}_{\lambda=t}\\
&+&\frac{1}{\alpha}\int_t^\infty{\rm d}\lambda e^{-\alpha\lambda}\lambda^{a} \left((a+1) \pi_k^2(\lambda) +  2 \lambda
 \pi_k(\lambda) \partial_\lambda\pi_k(\lambda)\right),\nonumber \\
 &=&\frac{1}{\alpha} (-t\partial_t h_k+(a+1)h_k+2\braket{ \pi_k}{\lambda\partial_\lambda\pi_{k}}) \;,
\end{eqnarray}
where we have used Eq. (\ref{dot_hn}). We can easily calculate the last scalar product
\begin{eqnarray}
\braket{ \pi_k}{\lambda\partial_\lambda\pi_{k}}=\braket{\pi_k}{(k \lambda^k + ...)}=k h_k \;,
\end{eqnarray}
to obtain the desired relation between $\partial_\alpha h_k$ and $\partial_t h_k$
\begin{eqnarray}\label{deriv_alpha_t_h}
\alpha \partial_\alpha h_k=t\partial_t h_k-(2k+a+1)h_k.
\end{eqnarray}
With this relation (\ref{deriv_alpha_t_h}), it is then straightforward to relate $\partial_\alpha R_k$ and $\partial_\alpha S_k$ to $\partial_t R_k$ and $\partial_t S_k$, using  (\ref{R_en_h}) and (\ref{S_en_h}). Note that from now on, we set $\alpha =1$
\begin{eqnarray}\label{S_en_h_sans_alpha}
S_k = -t \partial_t \log{h_k}+(2k+a+1).
\end{eqnarray}
We finally obtain, using Eq. (\ref{recu_derivalpha}), the so-called Schlesinger equations \cite{Magnus95}
\begin{eqnarray}
\left\{
\begin{array}{ccl}\label{recurrence_dotS}
S_k-R_{k+1}+R_{k} &=& t\partial_t S_k,\\
\\
2-S_{k+1}+S_{k} &=& t \partial_t \log R_{k+1}.
\end{array}
\right.
\end{eqnarray}
Note that, for $\alpha =1$, the initial condition (\ref{CI0}) reads
\begin{eqnarray}\label{CI}
\left\{
\begin{array}{ccl}
h_0 &=& \int_t^{\infty} e^{-\lambda} \lambda^a{\rm d}\lambda = \Gamma(1+a,t),\\
\\
S_0 &=& -t \partial_t \log{h_0} +a+1 = \frac{e^{-t}t^{a+1}}{\Gamma(1+a,t)}+a+1,\\
\\
R_0 &=& 0,\\
\\
\zeta_0&=& 0.
\end{array}
\right.
\end{eqnarray}
Using the Schlesinger equations (\ref{recurrence_dotS}) and this initial condition (\ref{CI}), we can compute step by step all the terms for arbitrary $k$.

We end up this section by providing a useful relation between $\zeta_k$ [see Eq. (\ref{def_zeta})] and the cumulative distribution of the smallest eigenvalue $F_N(t)={\rm Prob}(\underset{1 \le i \le N}{\min}\lambda_i \ge t)$. One has
\begin{equation}\label{F_k_fonction_de_hn}
F_N(t)=\int_t^{\infty} \!{\rm d}\lambda_1 \int_t^{\infty} \!{\rm d}\lambda_2 ... \int_t^{\infty} \!{\rm d}\lambda_N \, P_{\rm joint}(\lambda_1,\lambda_2,\cdots,\lambda_N) = \frac{N!}{Z_N} \prod_{k=0}^{N-1} h_k,
\end{equation}
where the last equality is obtained by using the classical tricks of replacing the Vandermonde determinant by the determinant built from the OPs $\pi_k$'s (\ref{eq:def_OP}) and then use the Cauchy-Binet formula \cite{Mehta91,Forrester10}. Using Eq. (\ref{zeta_k_S}) with (\ref{S_en_h_sans_alpha}) as well as (\ref{deriv_alpha_t_h}) , we can write
\begin{eqnarray}
\zeta_N=-N(N+a)+t\partial_t\log\left(\prod_{k=0}^{N-1} h_k \right).
\end{eqnarray}
And therefore, from the expression of $F_N(t)$ given in Eq. (\ref{F_k_fonction_de_hn}), we have
\begin{eqnarray}\label{lien_zeta_F}
\zeta_N=-N(N+a)+t\partial_t\log \left(F_N(t)\right) \;.
\end{eqnarray}
This expression thus provides a link between $\zeta_N$, which is the first non trivial coefficient of the OP's [see Eq.  (\ref{def_zeta})] and the cumulative distribution of the smallest eigenvalue.

\subsection{Laguerre-Freud equations}\label{section:LF}

In this section, we derive another set of recursion relations between the coefficients $R_k$'s and $S_k$'s, the so-called Laguerre-Freud equations, following the procedure used by Belmehdi and Ronveaux in \cite{Belmehdi94}. To derive these equations, we start by searching two functions $\Psi$ and $\Phi$, which are polynomials in $\lambda$, satisfying for any polynomial~$p$
\begin{eqnarray}\label{def_phi_psi}
\braket{\Psi}{p}=\braket{\Phi}{p'} \;,
\end{eqnarray}
where the polynomials $\Phi$ and $\Psi$ may depend explicitly on the parameters $t$ and $a$ and $p'(\lambda)\equiv \partial_\lambda p(\lambda) $. Denoting by $w(\lambda)=e^{-\lambda} \lambda^a$ the weight associate to the scalar product in Eq. (\ref{eq:def_OP}) and using an integration by parts, this relation (\ref{def_phi_psi}) is satisfied provided that
\begin{eqnarray}
\Psi w+ (\Phi w)' = 0 \hspace{0.2cm} {\rm and} \hspace{0.2cm} \Phi(\lambda=t)=0.
\end{eqnarray}
We find that the simplest non trivial solution to this equation is given by
\begin{eqnarray}
\left\{
\begin{array}{ccl}
\Psi(\lambda)&=&\lambda^2-(2+a+t)\lambda+t(1+a),\\
\\
\Phi(\lambda)&=&\lambda^2-t \lambda \;.
\end{array}
\right.
\end{eqnarray}
Then, using this property (\ref{def_phi_psi}) for $p=\pi_k^2$ and for $p=\pi_k\pi_{k+1}$, we can write after expansion, using the three-term
recursion relation (\ref{3term_rec})
\begin{eqnarray}
\left\{
\begin{array}{ccl}
I_{2,k}-(2+a+t)I_{1,k}+t(1+a)I_{0,k}&=& 2(J_{2,k}-tJ_{1,k}),\\
\\
K_{2,k}-(2+a+t)K_{1,k}+t(1+a)K_{0,k}&=& L_{2,k}-tL_{1,k},
\end{array}
\right.
\end{eqnarray}
where we have introduced the four family of integrals
\begin{eqnarray}
\left\{
\begin{array}{ccl}
I_{m,k} &=& \braket{\lambda^m}{\pi_{k}^2},\\
\\
J_{m,k} &=& \braket{\lambda^m}{\pi_{k}\pi'_{k}},\\
\\
K_{m,k} &=& \braket{\lambda^m}{\pi_{k}\pi_{k+1}},\\
\\
L_{m,k} &=& \braket{\lambda^m}{(\pi_{k+1}\pi_k)'}.
\end{array}
\right.
\end{eqnarray}
These integrals are all calculated (for $m=0, 1$ and $2$) in \ref{appendix:Turan}.
Using these expressions together with Eq. (\ref{R_en_h}), we find the two relations
\begin{eqnarray}
\left\{
\begin{array}{rcl}\label{LF01}
&&R_{k+1}+R_k+S_k(S_k-a-t-2k-2)+t(2k+1+a) = 2 \,\displaystyle{\sum_{i=0}^{k-1} }S_i, \\
\\
&&(S_{k+1}+S_k-3-a-t-2k)R_{k+1} = 2 \,\displaystyle{\sum_{i=1}^k R_i+\sum_{i=0}^k S_i^2 - t \sum_{i=0}^k S_i}.
\end{array}
\right.
\end{eqnarray}
We can rewrite these equations by subtracting rank $k$ to the rank $k+1$ and find the two Laguerre-Freud recurrence equations (which are here of order 2):
\begin{eqnarray}\label{laguerre-freud}
\left\{
\begin{array}{rcl}
R_{k+2}-R_k&=&S_{k+1}(2k+4+a+t-S_{k+1})-S_k(2k+a+t-S_k)-2t,\\
\\
S_{k+1}(S_{k+1}-t) &=& R_{k+1}(2k+1+a+t-S_{k+1}-S_k)\\
&-&R_{k+2}(2k+5+a+t-S_{k+2}-S_{k+1}).
\end{array}
\right.
\end{eqnarray}
As we show below these two sets of equations (\ref{recurrence_dotS}) and (\ref{laguerre-freud}) allow us to (i) derive the connection to the Painlev\'e equation and (ii) perform the asymptotic analysis of these coefficients for large $N$.

\section{Painlev\'e V equation for finite $N$}\label{section:PV}

In this section, we proceed to the derivation of the Painlev\'e V equation, following the method of Ref. \cite{Basor09}. First, it is useful to introduce the quantities~\cite{Forrester07}
\begin{eqnarray}
\left\{
\begin{array}{rcl}\label{def_omega_txt}
\theta_k&=&2k+1+a-S_k,\\
\\
\omega_k&=&-R_k-\zeta_k.
\end{array}
\right.
\end{eqnarray}
Manipulating the Laguerre-Freud equations (\ref{laguerre-freud}) we can prove (see \ref{appendix:LF})
\begin{eqnarray}\label{LF02_bonne_limite}
\omega_k^2 -\theta_k (\theta_k-a-2k+t) \omega_k -\theta_k (kt(k+a)+(\theta_k+t)\zeta_k) = 0 \;,
\end{eqnarray}
which is a simple algebraic relation (and not a recursion relation) between the different variables $\omega_k, \theta_k$ and $\zeta_k$.
Using the previous relations with the index $k=N$, we find from Eq. (\ref{lien_zeta_F}),
\begin{eqnarray}\label{def:Hn}
H_N=t \partial_t \log(F_N(t)) =N(N+a)+ \zeta_N.
\end{eqnarray}
Summing up the first Schlesinger equation (\ref{recurrence_dotS}) from $k=0$ to $k=N-1$, we find
\begin{eqnarray}
\sum_{k=0}^{N-1} S_k + \sum_{k=0}^{N-1}(R_k-R_{k-1})= t \partial_t \sum_{k=0}^{N-1} S_k
\end{eqnarray}
which can be simplified using (\ref{zeta_k_S}), (\ref{CI0}) and (\ref{def_omega_txt}) to yield
\begin{eqnarray}\label{zeta:omegan}
-t \partial_t \zeta_N = -R_N-\zeta_N =\omega_N .
\end{eqnarray}
On the other hand, using (\ref{def:Hn}) and (\ref{zeta:omegan}), we have
\begin{eqnarray}\label{Hn_to_Rn}
t \partial_t H_N = - \omega_N = \zeta_N+R_N =  H_N -N(N+a)+R_N \;.
\end{eqnarray}
Taking a derivative of this equation w.r.t. $t$, we find
\begin{eqnarray}\label{partialRn:H''}
\partial_t R_N = t \,\partial_t^2 H_N.
\end{eqnarray}
From the second Schlesinger equation (\ref{recurrence_dotS}), we find
\begin{eqnarray}
t \partial_t R_N &=& R_N (2-S_N-S_{N-1}) = R_N (\theta_N-\theta_{N-1}) \;,
\end{eqnarray}
where we have used (\ref{def_omega_txt}). Using the relation derived in \ref{appendix:LF} in Eq. (\ref{theta_km}), we can rewrite this recursion equation as a simple algebraic relation
\begin{eqnarray}\label{partialR_k:omegaRnthetan}
t \partial_t R_N &=& R_N \theta_N-\frac{\omega_N^2}{\theta_N} \;.
\end{eqnarray}
Therefore, combining (\ref{partialRn:H''}) and (\ref{partialR_k:omegaRnthetan}), we find
\begin{eqnarray}\label{H'':omegaRnthetan}
t^2 \partial_t^2 H_N &=& R_N \theta_N-\frac{\omega_N^2}{\theta_N} \;.
\end{eqnarray}
On the other hand, from our Eq. (\ref{LF02_bonne_limite}) in which we inject the definition of $\omega_N$ (\ref{def_omega_txt}) to eliminate $\zeta_N$
\begin{eqnarray}\label{LF_modif_kozeta}
N(N+a)t-(2N+a)\omega_N-t R_N &=& R_N \theta_N+\frac{\omega_N^2}{\theta_N}.
\end{eqnarray}
Summing and subtracting the last two equations (\ref{H'':omegaRnthetan}) and (\ref{LF_modif_kozeta}), we obtain
\begin{eqnarray}
2\frac{\omega_N^2}{\theta_N} &=& N(N+a)t-(2N+a)\omega_N-t R_N -t^2 \partial_t^2 H_N, \label{multi1}\\
2 R_N \theta_N &=&  N(N+a)t-(2N+a)\omega_N-t R_N +t^2 \partial_t^2 H_N. \label{multi2}
\end{eqnarray}
Multiplying these two equations (\ref{multi1}) and (\ref{multi2}) together, we find
\begin{eqnarray}
4 R_N \omega_N^2 &=& \left(N(N+a)t-(2N+a)\omega_N-t R_N \right)^2- \left(t^2 \partial_t^2 H_N\right)^2.
\end{eqnarray}
Finally, by using (\ref{Hn_to_Rn}), we eliminate $\omega_N$ and $R_N$ (by expressing them in terms of $H_N$ and $\partial_t H_N$) and find the equation
\begin{equation}\label{Painleve5}
\left(t \partial_t^2 H_N\right)^2 = 4\left( \partial_t H_N \right)^2 \left( H_N -N(N+a) -t \partial_t H_N \right) + \left((2N+a-t)\partial_t H_N +H_N\right)^2 \;,
\end{equation}
which is a Painlev\'e V equation in the Jimbo-Miwa-Okamoto $\sigma$ form \cite{Jimbo81,Okamoto82}. Note that this equation coincides exactly with the equation first found by Tracy and Widom in \cite{Tracy94b}.

\section{Large $N$ asymptotic limit at the hard edge: Painlev\'e III and first correction}\label{section:PIII}

In this section, we study the behavior of the quantities $h_N$, $R_N$, $S_N$ and $\zeta_N$ in the hard edge limit (\ref{def_hard_edge}), which, in the language of OPs, corresponds to a double scaling limit. Of course, as we are eventually interested in the study of the cumulative distribution of the smallest eigenvalue $F_N(t)$, we could perform this asymptotic analysis directly on the Painlev\'e V equation (\ref{Painleve5}), as done by Tracy and Widom in their original study of GUE \cite{TW94a}. But it turns out to be much more convenient, especially to extract the $1/N$ corrections, to perform this asymptotic analysis on the Schlesinger (\ref{recurrence_dotS}) and Laguerre-Freud (\ref{laguerre-freud}) equations.

To understand the structure of the coefficients $h_N$, $R_N$, $S_N$ and $\zeta_N$ in this double scaling limit (\ref{def_hard_edge}), it is useful to study their behavior for small $t$ (keeping $N$ fixed). For $t=0$, the OPs $\pi_k$'s in Eq. (\ref{eq:def_OP}) can be expressed in terms of the generalized Laguerre polynomials
\begin{eqnarray}\label{Laguerre:def}
L_{k}^{(a)}(x)= \frac{\Gamma(a+k+1)}{k!} \sum_{i=0}^k \binom{k}{i} \frac{(-x)^i}{\Gamma(a+i+1)}\,.
\end{eqnarray}
Hence, thanks to standard properties of Laguerre polynomials, we easily obtain these coefficients, for $t=0$,
using (\ref{def_hn}), (\ref{R_en_h}) and (\ref{S_en_h_sans_alpha}) as:
\begin{eqnarray}
\left\{
\begin{array}{ccl}\label{polym0}
\left.  \pi_k(\lambda) \right|_{t=0}  &=& L_k^{(a)}(\lambda) k! (-1)^k,\\
\\
\left.  h_k \right|_{t=0} &=&\Gamma(k+a+1)k!,\\
\\
\left.  R_k \right|_{t=0} &=&k(k+a),\\
\\
\left.  S_k \right|_{t=0} &=&2k+a+1,\\
\\
\left.  \zeta_k \right|_{t=0} &=&-k(k+a).
\end{array}
\right.
\end{eqnarray}
We now take the index $k=N$. When $t$ is closed to 0, we can use Eq. (\ref{dot_hn}) and find
\begin{equation}\label{Eq:dot_h}
\partial_t h_N = -t^a e^{- t}\pi_N(t)^2 = -t^a  L_N^{(a)}(0)^2 N!^2 +o(t^a)= -t^a \frac{[\Gamma(N+a+1)]^2}{[\Gamma(a+1)]^2} +o(t^a) \;.
\end{equation}
Using (\ref{polym0}), we can integrate and find the first correction
\begin{eqnarray}
h_N=\Gamma(N+a+1)N!-\frac{t^{a+1}[\Gamma(N+a+1)]^2}{\Gamma(a+2) \Gamma(a+1)}+o(t^{a+1}).
\end{eqnarray}
We are interested in the scaling behavior when $N$ goes to infinity, $t$ goes to $0$ with $x=Nt$ finite (\ref{def_hard_edge}). In this double scaling regime, the expansion above reads
\begin{eqnarray}\label{hn_small_x}
h_N &=&\Gamma(N+a+1)N!\left[1-\frac{1}{N} \left(\frac{x^{a+1}}{\Gamma(a+2)\Gamma(a+1)}+ o(x^{a+1})\right) + o\left(\frac{1}{N}\right)\right].
\end{eqnarray}
This suggests the ansatz
\begin{eqnarray}\label{ansatz}
h_N=\Gamma(N+a+1)N!\left[1+\frac{1}{N}f(x= N\,t) + o\left(\frac{1}{N}\right)\right],
\end{eqnarray}
where the function $f$ has thus the small $x$ expansion, read from Eq.~(\ref{hn_small_x})
\begin{eqnarray}\label{f_dvpt0}
f(x) &=& -\frac{x^{a+1}}{\Gamma(a+2)\Gamma(a+1)} + o(x^{a+1}).
\end{eqnarray}
We can introduce the ansatz (\ref{ansatz}) in the relations (\ref{R_en_h}) and (\ref{S_en_h_sans_alpha}) which give $R_N$ and $S_N$ in terms of the function $f$. We obtain
\begin{eqnarray}\label{ansatz_R_S}
\left\{
\begin{array}{ccl}
R_N&=&N(N+a)+x f'(x) -f(x) + o\left(1\right),\\
\\
S_N&=&2N+a+1-\dfrac{1}{N}x f'(x) + o\left(\dfrac{1}{N}\right).
\end{array}
\right.
\end{eqnarray}
The first non trivial terms in Eq. (\ref{ansatz_R_S}) $xf'(x) - f(x)$ for $R_N$ and $-(1/N) x \,f'(x)$ for $S_N$ are necessary to compute the leading order of the cumulative distribution, $F_\infty(x)$. To compute the first $1/N$ correction to the limiting distribution, we need to expand $R_N$ and $S_N$ in Eq. (\ref{ansatz_R_S}) to the next order in $1/N$. One actually expects the following expansion
\begin{eqnarray}\label{R_S_expansion}
\left\{
\begin{array}{ccl}
R_N&=&N(N+a)+\displaystyle \sum_{i=0}^j r_i(x)N^{-i} + o\left(N^{-j}\right),\\
\\
S_N&=&2N+a+1+\displaystyle \sum_{i=1}^j s_i(x)N^{-i} + o\left(N^{-j}\right) \;,
\end{array}
\right.
\end{eqnarray}
where the first term in this expansion are given in (\ref{ansatz_R_S}), i.e., $r_0(x) = xf'(x)-f(x)$ and $s_1(x) = - xf'(x)$. Besides, using the exact relation $\partial_t h_N = -t^a e^{- t}\pi_N(t)^2$ [see Eq. (\ref{Eq:dot_h})], one can show that $r_i(x) = o(x)$ as well as $s_i(x)=o(x)$ when $x \to 0$. One can check, in principle, the validity of this expansion (\ref{R_S_expansion}) order by order in powers of $1/N$ by injecting it in the Schlesinger equations (\ref{recurrence_dotS}). Proving it rigorously to arbitrary order $j$ is however a hard task. However, here, we only need this asymptotic expansion up to order ${\cal O}(1/N^2)$, which can be obtained explicitly as follows. To compute the second correction, we truncate the expansion (\ref{R_S_expansion}) up to the $N^{-2}$ order and write
\begin{eqnarray}\label{R_S_expansion2}
\left\{
\begin{array}{ccl}
R_N&=&N(N+a)+x f'(x)-f(x)+\dfrac{r_1(x)}{N}+\dfrac{r_2(x)}{N^2}+o(N^{-2})\,,
\\
\\
S_N&=&2N+a+1-\dfrac{x f'(x)}{N}+\dfrac{s_2(x)}{N^2}+o(N^{-2})\,.
\end{array}
\right.
\end{eqnarray}
In the hard edge limit $N \to \infty$, $t \to 0$, keeping $x=N t$ fixed, we can obtain the expansion of $R_{N+1}$ and $S_{N+1}$ at the same order by replacing $N$ by $N+1$ and using $(N+1)t = x(1 + 1/N)$ in (\ref{R_S_expansion2})
\begin{eqnarray}\label{R_S_expansion}
\left\{
\begin{array}{ccl}
R_{N+1}&=&(N+1)(N+1+a)+x f'(x)-f(x)+\frac{1}{N}(r_1(x)+x^2 f''(x))\\
&&\!\!\!+\frac{1}{N^2}(r_2(x)+x r_1'(x)-r_1(x)+\frac{1}{2}x^2 f''(x)+\frac{1}{2}x^3 f'''(x))+o(N^{-2})\,,
\\
\\
S_{N+1}&=&2N+a+3-\frac{1}{N}x f'(x)+\frac{1}{N^2}(s_2(x)-xf'(x)-x^2f''(x))+o(N^{-2})\,.
\end{array}
\right.
\end{eqnarray}
By injecting these two expansions into the Schlesinger equation (\ref{recurrence_dotS}), we find (using $t \partial_t = x \partial_x$) at the first non trivial order (${\cal O}(N^{-2})$ for the first Schlesinger equation and at ${\cal O}(N^{-3})$ for the second)
\begin{eqnarray}\label{R_S_expansion2}
\left\{
\begin{array}{ccl}
s_2(x)-xs'_2(x)+r_1(x)-xr_1'(x)&=&\frac{1}{2}(x^2 f''(x)+x^3 f'''(x))\,,\\
\\
2s_2(x)-xs'_2(x)-xr_1'(x)&=&-a x^2 f''(x)+\frac{1}{2}x^3 f'''(x))\,,
\end{array}
\right.
\end{eqnarray}
which can be solved as follows. First, by subtracting the first equation of (\ref{R_S_expansion2}) to the second one, one obtains
\begin{eqnarray}\label{rel_s2_r1}
s_2(x) = r_1(x) - \left(a+\frac{1}{2}\right)x^2 f''(x) \;.
\end{eqnarray}
By injecting this relation (\ref{rel_s2_r1}) in the first equation of (\ref{R_S_expansion2}), one finds that $r_1$ satisfies the following equation
\begin{eqnarray}\label{eq_r1_only}
r_1(x) - x r_1'(x) = - \frac{a}{2} \left(x^2 f''(x) + x^3 f'''(x)\right) \;,
\end{eqnarray}
which can be solved, using that $r_1(x) = o(x)$ as $x \to 0$, yielding
\begin{eqnarray}\label{expr_r1}
r_1(x) = \frac{a}{2} x^2 f''(x) \;.
\end{eqnarray}
Consequently, from Eq. (\ref{rel_s2_r1}) one has
\begin{eqnarray}\label{expr_s2}
s_2(x) = - \frac{a+1}{2} x^2 f''(x) \;.
\end{eqnarray}
We finally find the two first terms of the expansion of $R_N$ and $S_N$, for large $N$, as
\begin{eqnarray}\label{expansion_second_order}
\left\{
\begin{array}{rcl}
R_N&=&N(N+a) + xf'(x)-f(x) + \dfrac{a}{2N}x^2 f''(x)+o\left(\dfrac{1}{N}\right),\\
\\
S_N&=&2N+a+1 - \dfrac{1}{N}xf'(x)-\dfrac{a+1}{2N^2} x^2 f''(x)+o\left(\dfrac{1}{N^2}\right).
\end{array}
\right.
\end{eqnarray}
From the expansion of $S_N$ in Eq. (\ref{expansion_second_order}), we compute $\zeta_N$ in the double scaling limit (\ref{def_hard_edge}), using Eq.~(\ref{zeta_S}), up to the second non-trivial order for large $N$
\begin{eqnarray} \label{ansatz_zeta}
\zeta_N&=&-(N + a) N + f(x) + \frac{a}{2N} x f'(x) + o\left(\frac{1}{N}\right) \;.
\end{eqnarray}
Finally, from the expression of $\zeta_N$ we obtain the large $N$ expansion of $F_N(t)$ in the hard edge limit (\ref{def_hard_edge}), using
Eq. (\ref{lien_zeta_F}), which is given by
\begin{eqnarray}\label{Dvpt_F0}
x\partial_x\log\left(F_N\left(\frac{x}{N}\right) \right) = f(x) + \frac{a}{2N} x f'(x) + o\left(\frac{1}{N}\right).
\end{eqnarray}
Using the initial condition $F_N(0)=1$ and (\ref{f_dvpt0}) we can rewrite this equation as
\begin{eqnarray}\label{Dvpt_F}
F_N\left(\frac{x}{N}\right) = \exp{ \left( \int_0^x \frac{f(u)}{u} {\rm d}u + \frac{a}{2N} f(x) + o\left(\frac{1}{N}\right) \right) }.
\end{eqnarray}
Expanding Eq. (\ref{Dvpt_F}) up to first order in $1/N$, one obtains finally
\begin{eqnarray}\label{Finfty2}
F_N\left(\frac{x}{N}\right) = F_\infty(x) + \frac{a}{2N} x F'_\infty(x) +  o\!\left(\frac{1}{N}\right) \;,
\end{eqnarray}
where $F_{\infty}(x)$ is given by
\begin{eqnarray}\label{Finfty(x)}
\underset{N \to \infty}{\lim}F_N\left(\frac{x}{N}\right) = F_\infty(x)\,,\hspace{0.7cm} F_\infty(x)=\exp\left( \int_0^x \frac{f(u)}{u} {\rm d}u\right) \;.
\end{eqnarray}
Hence we easily obtain the functional form of the $1/N$ correction as given in Eq.~(\ref{Finfty2}), fully consistent with the conjecture in (\ref{conjecture_finite_N}) made in \cite{Edelman14}. However, at this stage, we still need to find the equation satisfied by the function $f$, which enters in the definition of $F_{\infty}(x)$ in Eq. (\ref{Finfty(x)}). To obtain this equation, we expand
$\theta_N$ and $\omega_N$ in Eq.~(\ref{def_omega_txt}), which is easily done from the expansions of $R_N$, $S_N$ and $\zeta_N$ in (\ref{ansatz_R_S}) and (\ref{ansatz_zeta}) to yield
\begin{eqnarray}\label{dvpt_omega_theta}
\left\{
\begin{array}{rcl}
\theta_N&=& \dfrac{1}{N}xf'(x)+\dfrac{a+1}{2N^2} x^2 f''(x)+o\left(\dfrac{1}{N^2}\right),\\
\\
\omega_N&=& - xf'(x) - \dfrac{a}{2N}(x^2 f''(x)+x f'(x)) + o\left(\dfrac{1}{N}\right).
\end{array}
\right.
\end{eqnarray}
Finally, by injecting these expansions (\ref{ansatz_zeta}) and (\ref{dvpt_omega_theta}) in (\ref{LF02_bonne_limite}) with $t=x/N$, we obtain, by canceling the first term, of order ${\cal O}(N^{-2})$, in Eq. (\ref{LF02_bonne_limite}) that $f$ satisfies a Painlev\'e III equation
\begin{eqnarray}
(xf'')^2+4f'(1+f')(xf'-f)=(a f')^2, \label{PainleveIII}
\end{eqnarray}
with the small argument behavior in Eq. (\ref{f_dvpt0}). This result coincides with the one obtained, by a quite different method, by Tracy and Widom \cite{Tracy94} (note the correspondence $\sigma(s)=-f(s/4)$, where $\sigma(s)$ is the notation used in \cite{Tracy94}). Note that for integer values of $a$, $f(x)$ can be written explicitly in terms Bessel functions \cite{Forrester07} [see Eq. (\ref{fdetBessel})]. These results in Eqs. (\ref{Finfty2}), (\ref{Finfty(x)}) and (\ref{PainleveIII}) yield the results announced in the introduction in Eqs. (\ref{eq:result_TW}), (\ref{PIII}) and (\ref{conjecture_finite_N}). Finally, in \ref{appendix:Numeric} we present a comparison between numerical simulations of Wishart matrices of size $N=50$ and the asymptotic formula in Eq. (\ref{Finfty2}) describing the first $1/N$ correction.

\section{Large $N$ asymptotic limit at the soft edge: Painlev\'e II and first correction}\label{section:PII}

We now turn to the analysis of the PDF of the smallest eigenvalue $\lambda_{\min}$ in the case where $a \sim {\cal O}(N)$, and we set $a = \alpha \, N$. In this case, the density of eigenvalues has a single support on $[N x_-, N x_+]$ [see Eq. (\ref{density_MP})], with $x_{\pm} = (\sqrt{1+\alpha} \pm 1)^2$ with a soft edge at both extremities (see Fig. \ref{fig:Wishart_soft_vs_hard}). Therefore, one expects that $\lambda_{\min}$ will be close to $N x_-$, while its fluctuations, of order ${\cal O}(N^{1/3})$, are governed by the Tracy-Widom distribution for $\beta = 2$. In the soft edge limit, the large $N$ analysis of $F_N(t)$ in Eq. (\ref{def_FN}) is more conveniently done directly on the Painlev\'e V equation (\ref{Painleve5}) \cite{Baker98}. Following this route, one can indeed show \cite{Baker98} that, for large $N$
\begin{eqnarray}\label{baker_forrester}
\lambda_{\min} = N x_- - \frac{N^{1/3}}{m}\chi + {o}(N^{1/3})\;,
\end{eqnarray}
where $m$ is given by \cite{Baker98}
\begin{eqnarray}\label{def_m}
m=\frac{(1+\alpha)^{1/6}}{(\sqrt{1+\alpha}-1)^{4/3}} \;,
\end{eqnarray}
and where ${\chi}$ is distributed according to the Tracy-Widom distribution ${\cal F}_2$, i.e. $\Pr[\chi \leq s] = {\cal F}_2(s)$ where ${\cal F}_2(s)$ is given by \cite{TW94a}
\begin{eqnarray}\label{TW2}
{\cal F}_2(s) = \exp{\left(-\int_s^\infty (x-s) q^2(x) \, {\rm d}x \right)} \;.
\end{eqnarray}
Here $q(x)$ is the so-called Hatings-McLeod solution of the Painlev\'e II equation
\begin{eqnarray}\label{PII}
q''(x) = x\,q(x) + 2 q^3(x) \;, {\rm with} \; q(x) \sim {\rm Ai}(x) \;, \; {\rm for}\; x \to \infty \;,
\end{eqnarray}
where ${\rm Ai}(x)$ is the Airy function. The result in Eq. (\ref{baker_forrester}) can be equivalently written as
\begin{eqnarray}\label{tilde_f0}
F_N(t) = \tilde f_0\left(m \frac{N\, x_- -t}{N^{1/3}}\right) + {o}(1) \;,
\end{eqnarray}
which implies
\begin{eqnarray}\label{HN_0}
H_N(t) = t \partial_t \log F_N(t) &=& -(m x_-) \tilde h_0(x) N^{2/3} + o(N^{2/3}) \;, \; x = m \frac{N\, x_- -t}{N^{1/3}}.
\end{eqnarray}
with $\tilde h_0(x)=\tilde f_0'(x)/f_0(x)$ and where $\tilde f_0 = {\cal F}_2$, and where the tilde refers to the soft edge scaling limit. By injecting this form (\ref{HN_0}) into the Painlev\'e V equation satisfied by $H_N(t)$ (\ref{Painleve5}) one finds that $\tilde h_0$ satisfies the following equation
\begin{eqnarray}\label{PII_bis}
\left(\tilde h_0''(x)\right)^2+ 4 \tilde h'_0(x) \left[\left(\tilde h'_0(x)\right)^2 - x \tilde h'_0(x) + \tilde h_0(x) \right]=0 \;.
\end{eqnarray}
Using the Painlev\'e II equation (\ref{PII}), one can indeed check that $\tilde h_0(x) = \int_x^\infty q^2(u) {\rm d}u$ is solution of this equation (\ref{PII_bis}). Note that to check this, it is useful to use the identity
\begin{eqnarray}\label{identity}
\tilde h_0(x) = \int_x^\infty q^2(u) {\rm d} u = (q'(x))^2 - (q(x))^4 - x (q(x))^2 \;.
\end{eqnarray}

What is the first correction to the limiting form in Eq. (\ref{tilde_f0}) ? Unfortunately, in the soft edge limit, it turns out the Schlesinger equations  (\ref{recurrence_dotS}) do not allow to determine easily this first correction -- while they were very helpful in the hard edge scaling limit. An alternative way to compute this first correction is to analyze directly the Painlev\'e V equation in (\ref{Painleve5}). By inspection of this equation (\ref{Painleve5}), we conjecture that
the first correction to Eq. (\ref{tilde_f0}) is of the form
\begin{eqnarray}\label{HN_kext}
H_N(t) = -(m x_-) \left(\tilde h_0(x) N^{2/3} + \tilde h_1(x) N^{1/3}\right) + {o}(N^{1/3}) \;.
\end{eqnarray}
By inserting this expansion (\ref{HN_kext}) in Eq. (\ref{Painleve5}) we obtain that $\tilde h_1$ satisfies the following linear differential equation
\begin{eqnarray}\label{eq_h1}
2 \tilde h_1 \tilde h'_0 +2 (\tilde h_0 + h'_0 (3 \tilde h'_0 -2 x)) \tilde h'_1 + \tilde h''_0 \tilde h''_1= 0 \;,
\end{eqnarray}
where $\tilde h_0(x)$ is given in Eq. (\ref{identity}). Of course, we know that $\tilde h_1(x) \to 0$ when $x \to +\infty$ (\ref{HN_kext}). In addition, from Eq. (\ref{eq_h1}) one can show, using Eq.~(\ref{identity}) and the large argument behavior in (\ref{PII}) $q(x) \sim {\rm Ai}(x) \sim x^{-1/4} e^{-(2/3)x^{3/2}}/(2 \sqrt{\pi})$
that $\tilde h_1(x) \sim A \, x^{-1/2} e^{-(4/3) x^{3/2}}$. But, of course, the equation (\ref{eq_h1}) being linear, the amplitude $A$ can not be determined from this analysis. One way to determine it would be to analyze the OP system (\ref{eq:def_OP}) in the limit when $t$ is far from the left edge, i.e., for $(N x_- -t)\gg N^{1/3}$ (corresponding to the left large deviation tail of $\lambda_{\min}$~\cite{Katzav10}), and then match this result with the typical regime, for $|N x_-- t| \sim {\cal O}(N^{1/3})$. This program was carried out in detail in a similar albeit different context, involving discrete OPs in Ref. \cite{Schehr13}. Since we are interested in the first correction, this actually requires a very precise (and tedious) analysis of this regime $(N x_- -t)\gg N^{1/3}$ which goes beyond the scope of the present paper. Hence our result for the first correction $\tilde h_1(x)$ in Eq. (\ref{eq_h1}) does determine this function only up to a constant. We have not found any simple solution to this equation (\ref{eq_h1}), which could indicate that the corrections to scaling in this case are actually more complicated, as found recently in the case of real Wishart matrices \cite{Ma12}.

\section{Conclusion}\label{section:Conclusion}

To conclude, we have provided a direct computation of the cumulative distribution $F_N(t)$ of the smallest eigenvalue of complex Wishart random matrices (\ref{Pjoint}), for arbitrary parameter $a \geq 0$. This was done by studying a set of semi-classical orthogonal polynomials as defined in Eq. (\ref{eq:def_OP}) for which we derived (i) the Schlesinger (\ref{schlesinger_intro}) and (ii) the Laguerre-Freud (\ref{LF_intro}) equations. By combining these equations, we showed that $F_N(t)$ can be expressed in terms of a solution of a Painlev\'e V equation (\ref{Painleve5}), thus recovering the result of Tracy and Widom \cite{Tracy94b} using a quite different method. In the large $N$ limit, $F_N(t)$, properly shifted and scaled, converges to a limiting distribution $F_{\infty}(x)$ which can be expressed in terms of a solution of a Painlev\'e III equation (\ref{PainleveIII}) in the hard edge limit (corresponding to $a ={\cal O}(1)$) and of a Painlev\'e II equation (\ref{PII_bis}) in the soft edge limit (corresponding to $a = {\cal O}(N)$). Furthermore, we have computed explicitly the first correction to the limiting distribution when $N \to \infty$. In the hard edge case (\ref{conjecture_finite_N}), we confirmed a conjecture by Edelman, Guionnet et P\'ech\'e in Ref. \cite{Edelman14}. In this case, the first correction can be simply understood as a correction to the scale of the fluctuations of $\lambda_{\min}$, see Eq. (\ref{eq:correction}). On the other hand, in the soft edge limit, we found that this correction is a solution of a second order linear differential equation with varying coefficients (\ref{eq_h1}). Solving this equation remains a challenging open problem, which could suggest that the first correction does not correspond to a simple shift or rescaling of the scaling variable, as found for the hard edge (\ref{eq:correction}). This could be reminiscent of the result found for real Wishart matrices \cite{Ma12} and certainly deserves further investigations.

\section*{Acknowledgments}

We would like to thank Y. Chen, P. J. Forrester and N. S. Witte for useful correspondence and discussions.