[Paper Review] Hydrodynamic limit and cutoff for the biased adjacent walk on the simplex

Cyril Labbé, Enguérand Petit|arXiv (Cornell University)|Mar 8, 2022

Random Matrices and Applications4 citations

TL;DR

This paper establishes a cutoff phenomenon for the mixing time of a biased adjacent random walk on the N-simplex, where particles are resampled according to asymmetric beta distributions. Using a hydrodynamic limit derived from a non-linear Hamilton-Jacobi equation with degenerate boundary conditions, the authors prove that when the asymmetry parameter λ is small but vanishes slower than log N / N, the mixing time scales as 4N / λ, confirming a sharp cutoff with precise prefactor in the asymptotic regime as N → ∞.

ABSTRACT

We investigate the asymptotic in $N$ of the mixing times of a Markov dynamics on $N-1$ ordered particles in an interval. This dynamics consists in resampling at independent Poisson times each particle according to a probability measure on the segment formed by its nearest neighbours. In the setting where the resampling probability measures are symmetric, the asymptotic of the mixing times were obtained and a cutoff phenomenon holds. In the present work, we focus on an asymmetric version of the model and we establish a cutoff phenomenon. An important part of our analysis consists in the derivation of a hydrodynamic limit, which is given by a non-linear Hamilton-Jacobi equation with degenerate boundary conditions.

Motivation & Objective

• . To establish the cutoff phenomenon for the mixing time of a biased adjacent random walk on the N-simplex with asymmetric resampling.
• . To derive the hydrodynamic limit of the process, showing convergence to a non-linear Hamilton-Jecobi equation with degenerate boundary conditions.
• . To determine the precise asymptotic scaling of the mixing time in the regime where the asymmetry parameter λ → 0 but λ ≫ log N / N.
• . To analyze the invariant measure and its large-scale behavior under asymmetric beta resampling, contrasting with the symmetric case.

Proposed method

• . Use of a continuous-time Markov process on ordered particle configurations in [0, N], with resampling at Poisson times based on asymmetric beta(αk, αk+1) laws.
• . Derive a hydrodynamic limit by scaling space and time, leading to a non-linear Hamilton-Jacobi equation with degenerate boundary conditions.
• . Construct sub- and super-solutions to the discrete Hamilton-Jacobi system to control the evolution of the expected maximum particle position.
• . Employ comparison principles for viscosity solutions to bound the solution of the discrete system and establish convergence to the hydrodynamic limit.
• . Use concentration estimates on the invariant measure and the hydrodynamic limit to derive sharp lower bounds on the mixing time.
• . Apply a perturbation analysis near the origin and use Taylor expansions to control the behavior of the solution in the presence of asymmetry.

Experimental results

Research questions

• RQ1 . Does a cutoff phenomenon occur in the mixing time for the biased adjacent walk on the simplex when the resampling distribution is asymmetric?
• RQ2 . What is the precise asymptotic scaling of the mixing time when the asymmetry parameter λ → 0 but λ ≫ log N / N?
• RQ3 . How does the hydrodynamic limit of the process behave, and what PDE does it converge to in the scaling limit?
• RQ4 . What is the role of the degenerate boundary conditions in the hydrodynamic equation, and how do they affect the solution's behavior?
• RQ5 . How does the invariant measure and the typical particle profile differ from the symmetric case under asymmetric resampling?

Key findings

• . A cutoff phenomenon is established for the biased adjacent walk on the simplex when the asymmetry parameter λ satisfies λ → 0 and λ ≫ log N / N.
• . The mixing time scales asymptotically as t^N_mix(ε) ∼ 4N / λ as N → ∞, with a sharp prefactor.
• . The hydrodynamic limit is described by a non-linear Hamilton-Jacobi equation with degenerate boundary conditions, arising from the scaling limit of the discrete dynamics.
• . The solution to the hydrodynamic equation captures the front propagation of the maximum particle position, with a sharp transition at time t = x.
• . The invariant measure remains explicit under asymmetric beta resampling, but the typical height function is skewed toward the left, unlike the symmetric case.
• . The analysis relies on constructing sub- and super-solutions to the discrete Hamilton-Jacobi system, with careful control of error terms depending on λ and N.