Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universality of cutoff for independent random walks on the circle conditioned not to intersect

Anna Ben-Hamou, Pierre Tarrago

TL;DR

This work proves a universal cutoff in the mixing of a broad class of non-intersecting Markov processes on the circle, modeled as conditioned multi-particle jumps with a common invariant measure. By representing the overall kernel as a Doob h-transform and exploiting commutativity, the authors derive precise spectral-gap estimates and analyze eigenvalues via complex-analytic and steepest-descent techniques, yielding a cutoff at t ~ (log n)/γ with γ tied to the induced jump distribution. The results are illustrated through three concrete models—constant kernel, conditioned ASEP, and periodic dimer models on hexagonal lattices—demonstrating universality and detailing how the cutoff time depends on the underlying jump statistics through γ. The methods provide sharp lower and upper bounds on mixing, reveal the role of the spectral decomposition in high-dimensional limits, and connect to hyperuniform/discrete Circular Unitary Ensemble phenomena. Practical implications include efficient sampling from the stationary distribution and insights into threshold behaviors in related interacting particle systems.

Abstract

In the present paper, we consider a class of Markov processes on the discrete circle which has been introduced by König, O'Connell and Roch. These processes describe movements of exchangeable interacting particles and are discrete analogues of the unitary Dyson Brownian motion: a random number of particles jump together either to the left or to the right, with trajectories conditioned to never intersect. We provide asymptotic mixing times for stochastic processes in this class as the number of particles goes to infinity, under a sub-Gaussian assumption on the random number of particles moving at each step. As a consequence, we prove that a cutoff phenomenon holds independently of the transition probabilities, subject only to the sub-Gaussian assumption and a minimal aperiodicity hypothesis. Finally, an application to dimer models on the hexagonal lattice is provided.

Universality of cutoff for independent random walks on the circle conditioned not to intersect

TL;DR

This work proves a universal cutoff in the mixing of a broad class of non-intersecting Markov processes on the circle, modeled as conditioned multi-particle jumps with a common invariant measure. By representing the overall kernel as a Doob h-transform and exploiting commutativity, the authors derive precise spectral-gap estimates and analyze eigenvalues via complex-analytic and steepest-descent techniques, yielding a cutoff at t ~ (log n)/γ with γ tied to the induced jump distribution. The results are illustrated through three concrete models—constant kernel, conditioned ASEP, and periodic dimer models on hexagonal lattices—demonstrating universality and detailing how the cutoff time depends on the underlying jump statistics through γ. The methods provide sharp lower and upper bounds on mixing, reveal the role of the spectral decomposition in high-dimensional limits, and connect to hyperuniform/discrete Circular Unitary Ensemble phenomena. Practical implications include efficient sampling from the stationary distribution and insights into threshold behaviors in related interacting particle systems.

Abstract

In the present paper, we consider a class of Markov processes on the discrete circle which has been introduced by König, O'Connell and Roch. These processes describe movements of exchangeable interacting particles and are discrete analogues of the unitary Dyson Brownian motion: a random number of particles jump together either to the left or to the right, with trajectories conditioned to never intersect. We provide asymptotic mixing times for stochastic processes in this class as the number of particles goes to infinity, under a sub-Gaussian assumption on the random number of particles moving at each step. As a consequence, we prove that a cutoff phenomenon holds independently of the transition probabilities, subject only to the sub-Gaussian assumption and a minimal aperiodicity hypothesis. Finally, an application to dimer models on the hexagonal lattice is provided.
Paper Structure (18 sections, 27 theorems, 304 equations, 2 figures)

This paper contains 18 sections, 27 theorems, 304 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and statement of the result
  3. Related work
  4. Structure of the paper and specific notations
  5. Application to simple models
  6. Constant Markov kernel
  7. Conditioned ASEP
  8. Periodic dimer model on the hexagonal lattice
  9. Spectral gap
  10. Lower bound in Theorem \ref{['thm:main']}
  11. Upper bound in Theorem \ref{['thm:main']}
  12. Bounds on $\lambda^{\ell}_J$
  13. Proof of the upper bound
  14. Estimates in the super-logarithmic regime
  15. Rewriting the eigenvalues as complex integrals
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\eta\in (0,1/2)$ and $\delta,K_g,K_a>0$. Any sequence of kernels belonging to $\mathcal{A}(\eta,\delta,K_g,K_a)$ satisfies

Figures (2)

  • Figure 1: Possible move corresponding to $A^{(3)}$ on $B_{5,10}$ and graph corresponding to the adjacency matrix $A^{(-1)}$ on $B_{2,4}$ (the dashed line marks the origin of the circle).
  • Figure 2: Typical representative elements in classes of $\mathcal{J}_1$, $\mathcal{J}_2$ and $\mathcal{J}_3$ for $B_{15,30}$. The last picture is the representative $J_{\omega}$ of a class $\omega\in\mathcal{J}_3$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • proof
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:spectral-gap']}
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • ...and 42 more