[Paper Review] Large deviations for some corner growth models with inhomogeneity

Elnur Emrah, Christopher Janjigian|arXiv (Cornell University)|Sep 8, 2015

Random Matrices and Applications17 references14 citations

TL;DR

This paper establishes large deviation principles for inhomogeneous corner growth models with exponentially distributed weights whose means depend on random row and column parameters. It derives tractable variational representations for right-tail rate functions in both quenched and annealed settings and provides expansions near the shape function consistent with KPZ-type fluctuations.

ABSTRACT

We study an inhomogeneous generalization of the classical corner growth in which the weights are exponentially distributed with random parameters. Our main interest is in the quenched and annealed large deviation properties of the last passage times. We derive variational representations of the rate functions for right tail large deviations. The quenched rate function can be computed explicitly for certain choices of the parameter distributions. We present a mechanism for rate n left tail annealed large deviations. In the quenched model these deviations have rate strictly greater than n. The annealed right tail rate function is connected to the quenched rate function through a variational problem involving relative entropy. We identify the speed at which the rate functions decay to zero near the shape function.

Motivation & Objective

• To analyze large deviation properties of last-passage times in a corner growth model with inhomogeneous exponential weights.
• To derive tractable variational representations for the right-tail large deviation rate functions in quenched and annealed settings.
• To estimate left-tail large deviations and analyze fluctuations near the shape function.
• To connect the rate functions to KPZ universality by computing expansions near the deterministic shape function.
• To extend known results from i.i.d. models to inhomogeneous settings with random row and column parameters.

Proposed method

• Model weights as independent exponential random variables with mean $(a_i + b_j)^{-1}$, where $\mathbf{a}$ and $\mathbf{b}$ are i.i.d. sequences bounded away from zero.
• Define quenched and annealed measures by conditioning on or averaging over the random parameters $\mathbf{a}$ and $\mathbf{b}$.
• Use subadditive ergodic theory to establish almost sure convergence of $n^{-1}G(\lfloor ns\rfloor, \lfloor nt\rfloor)$ to a deterministic limit.
• Derive variational representations for the right-tail rate functions via Legendre-Fenchel transforms of cumulant generating functions.
• Apply Hölder’s inequality and monotonicity arguments to prove convexity and homogeneity of the rate functions in $(s,t)$.
• Use scaling and approximation arguments to extend results from rational to real $s,t > 0$.

Experimental results

Research questions

• RQ1 What is the large deviation rate function for last-passage times in the inhomogeneous corner growth model under quenched and annealed measures?
• RQ2 How do the right-tail rate functions behave near the shape function, and do they exhibit KPZ-type fluctuation scaling?
• RQ3 What are the asymptotic behaviors of the cumulant generating functions $\mathbf{L}_{s,t}(\lambda)$ and $\mathbb{L}_{s,t}(\lambda)$?
• RQ4 How do the rate functions differ between quenched and annealed settings, and what role does the inhomogeneity play?
• RQ5 Can the left-tail large deviations be estimated, and how do they compare to the right-tail behavior?

Key findings

• The right-tail large deviation rate function in the quenched setting is given by a variational formula involving the cumulant generating function of the weights.
• In the annealed setting, the rate function is also variational and can be expressed as an expectation over the random parameters $\mathbf{a}$ and $\mathbf{b}$.
• Expansions of the rate functions near the shape function are consistent with $n^{1/3}$-scale fluctuations, indicating KPZ-type behavior.
• The cumulant generating functions $\mathbf{L}_{s,t}(\lambda)$ and $\mathbb{L}_{s,t}(\lambda)$ converge almost surely and are convex and nondecreasing in $\lambda$.
• The functions $\lambda \mapsto \lambda \mathbf{L}_{s,t}(\lambda)$ and $\lambda \mapsto \lambda \mathbb{L}_{s,t}(\lambda)$ are homogeneous and concave in $(s,t)$.
• The large deviation principle holds with speed $n$, and the rate functions are characterized via subadditive ergodic theory and scaling limits.