[Paper Review] On the largest-eigenvalue process for generalized Wishart random matrices

A. B. Dieker, Jon Warren|arXiv (Cornell University)|Dec 8, 2008

Random Matrices and Applications11 references17 citations

TL;DR

This paper establishes a distributional equivalence between the largest eigenvalue process of a generalized Wishart random matrix and a last-passage percolation process via a change-of-measure argument. The key result confirms a conjecture by Borodin and Péché (2008), showing that under a general class of parameters, the largest eigenvalue of the matrix process has the same law as the maximal directed path weight in a last-passage percolation model with exponentially distributed weights.

ABSTRACT

Using a change-of-measure argument, we prove an equality in law between the process of largest eigenvalues in a generalized Wishart random-matrix process and a last-passage percolation process. This equality in law was conjectured by Borodin and Peche.

Motivation & Objective

• To establish a distributional equivalence between the largest eigenvalue process of a generalized Wishart random matrix and a last-passage percolation process.
• To resolve a conjecture by Borodin and Péché (2008) regarding the equality in law between these two stochastic processes.
• To extend known results from the standard Wishart case (i.e., i.i.d. Gaussian entries) to a broader class of matrix ensembles with structured variances.
• To provide a rigorous probabilistic framework using change-of-measure techniques and connections to the RSK correspondence and Gelfand-Tsetlin patterns.
• To demonstrate the convergence of discrete RSK dynamics with geometric inputs to continuous percolation processes in the exponential limit.

Proposed method

• Construct a generalized Wishart matrix process $\{M(n): n \geq 0\}$ via independent complex Gaussian entries with variances inversely proportional to $\pi_i + \hat{\pi}_j$.
• Define a last-passage percolation process $Y(N,n)$ as the maximum weight over up-right paths in a directed grid with i.i.d. exponential weights of rate $\pi_i + \hat{\pi}_j$.
• Apply a change-of-measure argument to relate the law of the eigenvalue process $\text{sp}(M(n))_1$ to the percolation process $Y(N,n)$ under the same probability measure $P^{\pi,\hat{\pi}}$.
• Use the RSK correspondence to link the dynamics of GT patterns from geometric inputs to the eigenvalue evolution of the Wishart matrix.
• Leverage the continuity of the RSK algorithm and weak convergence to show that the rescaled discrete GT patterns converge to a continuous process with the same law as $Y(N,n)$.
• Establish that the bottom row of the limiting GT pattern process follows an inhomogeneous Markov chain with transition kernels $Q^{\pi,\hat{\pi}}_{n-1,n}$, matching the eigenvalue dynamics of $M(n)$.

Experimental results

Research questions

• RQ1 Does the largest eigenvalue process of a generalized Wishart matrix have the same distribution as a last-passage percolation process under general parameter settings?
• RQ2 Can the conjecture by Borodin and Péché (2008) regarding the equality in law between the largest eigenvalue and last-passage percolation be extended beyond fixed $n$ to the full process level?
• RQ3 How do the transition mechanisms of the eigenvalue process and the percolation process align under a change-of-measure framework?
• RQ4 What is the role of the RSK correspondence and GT patterns in connecting matrix eigenvalue dynamics to percolation models?
• RQ5 Can the convergence of discrete RSK dynamics with geometric inputs to continuous percolation processes be rigorously established in the exponential limit?

Key findings

• The largest eigenvalue process $\{\text{sp}(M(n))_1: n \geq 1\}$ and the last-passage percolation process $\{Y(N,n): n \geq 1\}$ have identical finite-dimensional distributions under $P^{\pi,\hat{\pi}}$ for any strictly positive $\pi$ and nonnegative $\hat{\pi}$.
• The result confirms the conjecture of Borodin and Péché (2008) that the laws of the largest eigenvalue and the last-passage percolation time coincide not only marginally but also as stochastic processes.
• The change-of-measure technique successfully links the Wishart matrix process to the percolation model, providing a novel method for proving such equalities in law.
• The limiting process of rescaled RSK dynamics with geometric inputs converges weakly to a continuous process whose top row matches the law of $Y(N,n)$ under $P^{\pi,\hat{\pi}}$.
• The bottom row of the limiting GT pattern process evolves as an inhomogeneous Markov chain with transition kernels $Q^{\pi,\hat{\pi}}_{n-1,n}$, which are shown to match the eigenvalue dynamics of $M(n)$.
• The result generalizes previous findings in the standard Wishart case (e.g., Defosseux, 2008; Forrester and Rains, 2006), extending them to non-i.i.d. variance structures.