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Cutoff for congestion dynamics and related generalized exclusion processes

Ryokichi Tanaka

TL;DR

The paper analyzes congestion-type Markov chains with $n$ players and $Q$ resources under capacity $\kappa$, in the regime $n=\lfloor \rho \kappa Q\rfloor$, $\rho\in(0,1/2]$. By constructing a load-profile chain and a mean-field load-matrix framework, the authors prove sharp cutoff phenomena: the labeled Glauber dynamics mixes in time $T_{mix}\sim \tfrac{1}{2}n\log n$, while the unlabeled process (a special case of sampling from log $M$-concave distributions) exhibits the cutoff at $T_{mix}\sim \tfrac{1}{2}(1-\rho)n\log n$. They relate the unlabeled dynamics to generalized exclusion processes on complete graphs, provide a time-change interpretation relative to exclusion processes, and show that uniform sampling on some $M$-convex sets need not have a cutoff. The results advance understanding of abrupt convergence in discrete-convex sampling problems and offer a self-contained approach via coupling and mean-field techniques, with an appendix illustrating a non-cutoff counterexample in the $M$-convex setting.

Abstract

We consider congestion dynamics with $n$ players and $Q$ resources under the constraint that the number of each resource is $κ$ and that $n<κQ$ in the regime that $n$ and $κ$ diverge but $Q$ is fixed with $n=\lfloor{ρκQ\rfloor}$ for a fixed constant $ρ\in (0, 1/2]$. We show that the Glauber dynamics and its unlabeled version exhibit cutoff at time $(1/2)n \log n$ and $(1/2)(1-ρ)n\log n$ in total variation respectively. The unlabeled version is a special case of natural Markov chains for sampling from log M-concave distributions. We also show that a family of Markov chains for uniform sampling on M-convex sets does not necessarily exhibit cutoff.

Cutoff for congestion dynamics and related generalized exclusion processes

TL;DR

The paper analyzes congestion-type Markov chains with players and resources under capacity , in the regime , . By constructing a load-profile chain and a mean-field load-matrix framework, the authors prove sharp cutoff phenomena: the labeled Glauber dynamics mixes in time , while the unlabeled process (a special case of sampling from log -concave distributions) exhibits the cutoff at . They relate the unlabeled dynamics to generalized exclusion processes on complete graphs, provide a time-change interpretation relative to exclusion processes, and show that uniform sampling on some -convex sets need not have a cutoff. The results advance understanding of abrupt convergence in discrete-convex sampling problems and offer a self-contained approach via coupling and mean-field techniques, with an appendix illustrating a non-cutoff counterexample in the -convex setting.

Abstract

We consider congestion dynamics with players and resources under the constraint that the number of each resource is and that in the regime that and diverge but is fixed with for a fixed constant . We show that the Glauber dynamics and its unlabeled version exhibit cutoff at time and in total variation respectively. The unlabeled version is a special case of natural Markov chains for sampling from log M-concave distributions. We also show that a family of Markov chains for uniform sampling on M-convex sets does not necessarily exhibit cutoff.

Paper Structure

This paper contains 8 sections, 16 theorems, 210 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and related results
  3. More on the setting
  4. Outlines of the proofs
  5. Organization
  6. The load profile chain
  7. The congestion dynamics
  8. Uniform sampling on M-convex sets

Key Result

Theorem 1.1

Let $Q$ be a fixed integer at least two, $\rho$ be a fixed real in $(0, 1/2]$. For a positive integer $\kappa$, let $n=\left\lfloor \rho \kappa Q \right\rfloor$. For the Markov chain $\{\sigma(t)\}_{t\in {\mathbb Z}_+}$ on ${\mathcal{S}}_n$, it holds that for each $\varepsilon \in (0, 1)$, there exi

Figures (2)

  • Figure 1: An illustration of a sequence $(\bm{s}_i)_{i \in [N]}$ of elements in $(S\sqcup \overline S)^2$, where the first coordinates are aligned on the upper row and the second coordinates are aligned on the lower row for $\bm{s}_i=(v_i, v'_i)$ and $i \in [N]$.
  • Figure 2: An illustration of sequences: $(\bm{u}_i)_{i \in [n]}$ and $(\bm{\widetilde{u}}_i)_{i \in [n]}$ are aligned on the upper row and on the lower row respectively (above), and similarly $(\bm{v}_i)_{i \in [N-n]}$ and $(\bm{\widetilde{v}}_i)_{i \in [N-n]}$ are aligned (below).

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • Lemma 3.2
  • ...and 7 more