[Paper Review] Subgaussian rates of convergence of means in directed first passage percolation

Kenneth S. Alexander|arXiv (Cornell University)|Jan 7, 2011

Random Matrices and Applications9 references6 citations

TL;DR

This paper establishes subgaussian convergence rates for the expected passage time in directed first passage percolation on the integer lattice, showing that the difference between the expected passage time $\mathbb{E}a_{0n}$ and the asymptotic linear growth $n\mu$ is bounded by $O\left(\frac{n^{1/2}\log\log n}{(\log n)^{1/2}}\right)$ under nearly gamma-distributed passage times. The result improves upon prior bounds by leveraging refined exponential tail estimates and path decomposition techniques in a clean path framework.

ABSTRACT

We consider directed first passage percolation on the integer lattice, with time constant $μ$ and passage time $a_{0n}$ from the origin to $(n,0,...,0)$. It is shown that under certain conditions on the passage time distribution, $Ea_{0n} - nμ= O(n^{1/2}(\log\log n)/\log n)$.

Motivation & Objective

• To establish tighter bounds on the convergence of expected passage times to the time constant in directed first passage percolation.
• To improve upon existing $O(n^{1/2}\log n)$ bounds for $\mathbb{E}a_{0n} - n\mu$ by achieving subgaussian-type decay.
• To extend the use of exponential tail bounds from fluctuation analysis to bias estimation (i.e., deviation from the time constant).
• To demonstrate that under nearly gamma-distributed passage time distributions, the convergence rate is $O\left(\frac{n^{1/2}\log\log n}{(\log n)^{1/2}}\right)$, which is subgaussian.

Proposed method

• Utilizes the nearly gamma condition on passage time distributions to ensure compatibility with log-Sobolev inequalities and exponential tail bounds.
• Applies refined exponential tail estimates from Benaïm and Rossignol (2008) on the scale $(n/\log n)^{1/2}$ to control fluctuations.
• Employs a clean path decomposition strategy to express the passage time from origin to $(n,0,\dots,0)$ as a sum of a bounded number of increments.
• Uses symmetry and reflection techniques (via $\zeta$, $\xi^j$, and $\eta$) to construct symmetrized paths and bound the number of fast increments.
• Applies concentration inequalities to bound the probability that the sum of passage times on a path falls below a threshold, leading to a contradiction if the bias is too large.
• Combines path counting with exponential decay estimates to show that the probability of a path being too short is less than 1, implying the bias must be bounded.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence of $\mathbb{E}a_{0n} - n\mu$ in directed first passage percolation under general passage time distributions?
• RQ2 Can subgaussian-type bounds be achieved for the bias $\mathbb{E}a_{0n} - n\mu$ using exponential tail estimates from fluctuation theory?
• RQ3 How does the nearly gamma condition on the passage time distribution enable tighter control over the convergence rate of the expected passage time?
• RQ4 Can path decomposition and symmetry arguments be used to bound the number of slow increments in a geodesic, leading to improved bias estimates?

Key findings

• The paper establishes the bound $\mathbb{E}a_{0n} - n\mu = O\left(\frac{n^{1/2}\log\log n}{(\log n)^{1/2}}\right)$ for directed first passage percolation with nearly gamma-distributed passage times.
• This bound is subgaussian in nature, improving upon the prior $O(n^{1/2}\log n)$ result from subadditivity and Kesten's exponential bounds.
• The result is derived by showing that the probability of a path being significantly shorter than its expected length is less than 1, implying the bias cannot exceed the derived rate.
• The proof relies on decomposing the geodesic into a bounded number of increments and using symmetry to construct paths with controlled passage time sums.
• The nearly gamma condition ensures that the passage time distribution supports the necessary log-Sobolev and tail-estimate machinery used in the analysis.
• The method avoids reliance on exact solvability techniques, making it applicable to a broad class of distributions beyond geometric or discrete cases.