Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tridiagonal Models for Dyson Brownian Motion   In this paper, we consider tridiagonal matrices the eigenvalues of which
evolve according to $\beta$-Dyson Brownian motion. This is the stochastic
gradient flow on $\mathbb{R}^n$ given by, for all $1 \leq i \leq n,$ \[
  d\lambda_{i,t} = \sqrt{\frac{2}{\beta}}dZ_{i,t} - \biggl(
\frac{V'(\lambda_i)}{2} - \sum_{j: j \neq i} \frac{1}{\lambda_i - \lambda_j}
\biggr)\,dt \] where $V$ is a constraining potential and $\left\{ Z_{i,t}
\right\}_1^n$ are independent standard Brownian motions. This flow is
stationary with respect to the distribution \[
  \rho^{\beta}_N(\lambda) = \frac{1}{Z^{\beta}_N} e^{-\frac{\beta}{2}
  \left( -\sum_{1 \leq i \neq j \leq N} \log|\lambda_i - \lambda_j| +
\sum_{i=1}^N V(\lambda_i) \right) }. \] The particular choice of $V(t)=2t^2$
leads to an eigenvalue distribution constrained to lie roughly in
$(-\sqrt{n},\sqrt{n}).$ We study evolution of the entries of one choice of
tridiagonal flow for this $V$ in the $n\to \infty$ limit.
  On the way to describing the evolution of the tridiagonal matrices we give
the derivative of the Lanczos tridiagonalization algorithm under perturbation.

16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess
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The categories of this paper are: supr-con
9
The categories of this paper are: nlin
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The categories of this paper are: adap-org