[Paper Review] A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation

Jinho Baik, Toufic Suidan|ArXiv.org|Dec 18, 2004

Random Matrices and Applications13 references4 citations

TL;DR

This paper establishes a GUE central limit theorem for i.i.d. random variables with finite fourth moments, proving that directed first and last passage site percolation in thin rectangles exhibit universal Tracy-Widom fluctuations. The key result shows that the rescaled passage times converge in distribution to the GUE Tracy-Widom law under specific scaling regimes, extending universality beyond Gaussian cases.

ABSTRACT

We prove a GUE central limit theorem for random variables with finite fourth moment. We apply this theorem to prove that the directed first and last passage percolation problems in thin rectangles exhibit universal fluctuations given by the Tracy-Widom law.

Motivation & Objective

• To establish a central limit theorem for general i.i.d. random variables with finite fourth moments, where the limiting distribution is the GUE Tracy-Widom law.
• To demonstrate universality of the Tracy-Widom distribution in directed first and last passage percolation on thin rectangles.
• To extend known results from Gaussian to general i.i.d. weights in percolation models, under controlled scaling of system size.
• To identify the precise scaling regime (k = o(N^α) with α < 3/14) under which the GUE limit holds for non-Gaussian weights.

Proposed method

• Derives a GUE central limit theorem by analyzing the largest eigenvalue of a random matrix ensemble constructed from partial sums of i.i.d. random variables.
• Uses a coupling argument between the percolation model and Brownian motion via the Dyson process and Brownian Gibbs line ensemble.
• Applies the Baryshnikov and Gravner-Gravner-Tracy-Widom theorems to link the last-passage time to the largest eigenvalue of GUE matrices.
• Employs moment bounds and exponential tail estimates to control the difference between the percolation time and the Brownian motion approximation.
• Uses a symmetrization argument and reflection principle to handle both first and last passage times.
• Applies a concentration inequality and integral bounds to show that the correction terms vanish under the scaling k = o(N^α) with α < 3/14 for non-Gaussian weights.

Experimental results

Research questions

• RQ1 Does the Tracy-Widom distribution universally describe the fluctuations of last passage percolation for general i.i.d. weights with finite fourth moments?
• RQ2 What is the maximal scaling regime (in terms of k relative to N) for which the GUE Tracy-Widom limit holds in non-Gaussian directed percolation?
• RQ3 Can a central limit theorem for the largest eigenvalue of random matrices be established under only finite fourth moment, not Gaussianity?
• RQ4 How do the fluctuations of first and last passage times relate to the GUE Tracy-Widom law in thin rectangles?

Key findings

• For i.i.d. random variables with mean 0, variance 1, and finite fourth moment, the rescaled last passage time converges in distribution to the GUE Tracy-Widom law when k = o(N^α) with α < 3/14.
• For i.i.d. Gaussian weights, the convergence holds under the larger regime k = o(N^α) with α < 3/7.
• The rescaled first passage time satisfies a similar limit law, with the cumulative distribution function 1 - F_GUE(-s).
• The proof relies on coupling the percolation model to Brownian motion and using the known distribution of the largest eigenvalue of GUE matrices.
• The technical restriction α < 3/14 is believed to be an artifact of the method, and universality is expected to hold without such constraints.
• The correction terms between the percolation time and the Brownian motion approximation vanish in probability under the stated scaling, ensuring convergence.