Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The structure of low-complexity Gibbs measures on product spaces   Let $K_1$, $\dots$, $K_n$ be bounded, complete, separable metric spaces. Let
$\lambda_i$ be a Borel probability measure on $K_i$ for each $i$. Let
$f:\prod_i K_i \to \mathbb{R}$ be a bounded and continuous potential function,
and let $$\mu(d \mathbf{x})\ \propto\ e^{f(\mathbf{x})}\lambda_1(d x_1)\cdots
\lambda_n(d x_n)$$ be the associated Gibbs distribution. At each point
$\mathbf{x} \in \prod_i K_i$, one can define a `discrete gradient' $\nabla
f(\mathbf{x},\,\cdot\,)$ by comparing the values of $f$ at all points which
differ from $\mathbf{x}$ in at most one coordinate. In case $\prod_i K_i =
\{-1,1\}^n \subset \mathbb{R}^n$, the discrete gradient $\nabla
f(\mathbf{x},\,\cdot\,)$ is naturally identified with a vector in
$\mathbb{R}^n$.
  This paper shows that a `low-complexity' assumption on $\nabla f$ implies
that $\mu$ can be approximated by a mixture of other measures, relatively few
in number, and most of them close to product measures in the sense of optimal
transport. This implies also an approximation to the partition function of $f$
in terms of product measures, along the lines of Chatterjee and Dembo's theory
of `nonlinear large deviations'.
  An important precedent for this work is a result of Eldan in the case
$\prod_i K_i = \{-1,1\}^n$. Eldan's assumption is that the discrete gradients
$\nabla f(\mathbf{x},\,\cdot\,)$ all lie in a subset of $\mathbb{R}^n$ that has
small Gaussian width. His proof is based on the careful construction of a
diffusion in $\mathbb{R}^n$ which starts at the origin and ends with the
desired distribution on the subset $\{-1,1\}^n$. Here our assumption is a more
naive covering-number bound on the set of gradients $\{\nabla
f(\mathbf{x},\,\cdot\,):\ \mathbf{x} \in \prod_i K_i\}$, and our proof relies
only on basic inequalities of information theory. As a result, it is shorter,
and applies to Gibbs measures on arbitrary product spaces.

13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int