Sharp approximation for density dependent Markov chains

TL;DR

This paper develops a sharp approximation method for density-dependent Markov chains, enabling precise analysis of exit probabilities and finite size effects, with applications to randomized algorithms and coding theory.

Contribution

It introduces a coupling technique to approximate Markov chains with Brownian motion, providing detailed asymptotic expansions and error bounds for exit probabilities.

Findings

Accurate asymptotic expansion of exit probabilities for density-dependent Markov chains.

Sharp characterization of correction terms using differential equations.

Application to analyze phase transitions in randomized algorithms and coding schemes.

Abstract

Consider a sequence (indexed by n) of Markov chains Z^n in R^d characterized by transition kernels that approximately (in n) depend only on the rescaled state n^{-1} Z^n. Subject to a smoothness condition, such a family can be closely coupled on short time intervals to a Brownian motion with quadratic drift. This construction is used to determine the first two terms in the asymptotic (in n) expansion of the probability that the rescaled chain exits a convex polytope. The constant term and the first correction of size n^{-1/6} admit sharp characterization by solutions to associated differential equations and an absolute constant. The error is smaller than O(n^{-b}) for any b < 1/4. These results are directly applied to the analysis of randomized algorithms at phase transitions. In particular, the `peeling' algorithm in large random hypergraphs, or equivalently the iterative decoding scheme for low-density parity-check codes over the binary erasure channel is studied to determine the finite size scaling behavior for irregular hypergraph ensembles.