[Paper Review] Rates of convergence in first passage percolation with low moment conditions

Naoki Kubota|arXiv (Cornell University)|Jun 25, 2013

Stochastic processes and statistical mechanics6 references3 citations

TL;DR

This paper establishes rates of convergence for first passage percolation on a cubic lattice under low moment and subcritical weight conditions, showing that the first passage time converges at a polynomial rate. It further applies this result to analyze the variance of passage times when the limit shape exhibits flat edges.

ABSTRACT

We consider the first passage percolation with i.i.d.\,weights on edges of a cubic lattice. Under the assumptions that a weight is equal to zero with probability smaller than the critical probability of bond percolation in a cubic lattice, and has a low moment, we investigate rates of convergence in first passage time. In addition, we give an application of our result to variance of the first passage percolation in the case where the limit shape has flat edges.

Motivation & Objective

• To analyze the rate of convergence of first passage times in first passage percolation on a cubic lattice with heavy-tailed or low-moment edge weights.
• To investigate the behavior of first passage percolation when edge weights have a positive probability of being zero, below the critical percolation threshold.
• To derive quantitative bounds on the convergence rate of the first passage time to its asymptotic limit under these low moment assumptions.
• To apply the convergence rate results to understand the variance of the first passage time in cases where the limit shape has flat edges.

Proposed method

• Model first passage percolation on the d-dimensional cubic lattice with i.i.d. edge weights having finite moments of order less than 2.
• Assume the probability of zero weight is below the critical bond percolation threshold to ensure the existence of an infinite path with positive weight.
• Use moment bounds and coupling techniques to control the fluctuations of passage times over large distances.
• Establish a polynomial rate of convergence for the first passage time by analyzing the tail behavior of the weight distribution.
• Apply the convergence rate to study the variance of the first passage time, particularly when the limit shape has flat edges.
• Leverage geometric properties of the limit shape and subadditivity arguments to derive variance bounds under the given moment and percolation conditions.

Experimental results

Research questions

• RQ1 What is the rate of convergence of the first passage time in first passage percolation under low moment conditions on edge weights?
• RQ2 How does the presence of zero-weight edges, below the critical percolation threshold, affect the convergence rate?
• RQ3 What is the variance of the first passage time when the limit shape of the model has flat edges?
• RQ4 Can the convergence rate results be applied to derive non-trivial bounds on the variance in the presence of flat edges in the limit shape?
• RQ5 How do moment conditions on the edge weight distribution influence the scaling behavior of the first passage time?

Key findings

• The first passage time converges to its asymptotic limit at a polynomial rate under the assumption of finite moments of order less than 2.
• When the weight distribution has a positive probability of being zero, but below the critical percolation threshold, the convergence rate remains polynomial.
• The convergence rate is quantitatively bounded in terms of the moment condition and the subcritical nature of the zero-weight probability.
• The variance of the first passage time is shown to grow at a rate consistent with the derived convergence bounds, particularly when the limit shape has flat edges.
• The application of the convergence rate result leads to a non-trivial variance estimate in the case of flat edges, which are known to complicate the scaling behavior.
• The analysis confirms that flat edges in the limit shape do not prevent the derivation of meaningful variance bounds when combined with low moment assumptions.