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Cutoff for generalised Bernoulli-Laplace urn models

Ritesh Goenka, Jonathan Hermon, Dominik Schmid

TL;DR

The paper introduces a generalised Bernoulli–Laplace urn model with $d$ urns and $m$ colours, where at each step one ball from every urn is moved according to a random permutation from $\mu$ on $S_d$. It proves a sharp cutoff for the mixing time as $n\to\infty$ when the single-ball chain on $[d]$ is irreducible, with a precise scaling $t_{\text{mix}}^{(n)}(\varepsilon) \sim mn\, t_{\text{mix, single}}^{(n)}(1/\sqrt{n})$; the result extends to the labeled model and yields partial cutoff for a card-shuffling variant. The methodology blends hitting-time concentration to a centre of high stationary mass with spectral-profile and Dirichlet-form techniques, plus comparison arguments that transfer centre-scale estimates to the full chain. Additional contributions include a balanced variant, a rigorous equivalence with the labeled model, explicit examples (cyclic shift and mean-field), and an application to multi-stack random-to-random card shuffles, furnishing concrete mixing-time bounds in terms of $\gamma$ and $q$. Overall, the work provides a robust framework for establishing cutoff in interacting particle systems with centre concentration and product-chain-like mixing behavior, applicable to a broad class of multi-urn, multi-colour dynamics.

Abstract

We introduce a multi-colour multi-urn generalisation of the Bernoulli-Laplace urn model, consisting of $d$ urns, $m$ colours, and $dmn$ balls, with $dn$ balls of each colour and $mn$ balls in each urn. At each step, one ball is drawn uniformly at random from each urn, and the chosen balls are redistributed among the urns based on a permutation drawn from a distribution $μ$ on the symmetric group $S_d$. We study the mixing time of this Markov chain for fixed $m$, $d$, and $μ$, as $n \rightarrow \infty$. We show that there is cutoff whenever the chain on $[d]$ corresponding to the evolution of a single ball is irreducible, and that the same holds for a labeled version of the model. As an application, we also obtain partial results on cutoff for a card shuffling version of the model in which the cards are labeled and their ordering within each stack matters.

Cutoff for generalised Bernoulli-Laplace urn models

TL;DR

The paper introduces a generalised Bernoulli–Laplace urn model with urns and colours, where at each step one ball from every urn is moved according to a random permutation from on . It proves a sharp cutoff for the mixing time as when the single-ball chain on is irreducible, with a precise scaling ; the result extends to the labeled model and yields partial cutoff for a card-shuffling variant. The methodology blends hitting-time concentration to a centre of high stationary mass with spectral-profile and Dirichlet-form techniques, plus comparison arguments that transfer centre-scale estimates to the full chain. Additional contributions include a balanced variant, a rigorous equivalence with the labeled model, explicit examples (cyclic shift and mean-field), and an application to multi-stack random-to-random card shuffles, furnishing concrete mixing-time bounds in terms of and . Overall, the work provides a robust framework for establishing cutoff in interacting particle systems with centre concentration and product-chain-like mixing behavior, applicable to a broad class of multi-urn, multi-colour dynamics.

Abstract

We introduce a multi-colour multi-urn generalisation of the Bernoulli-Laplace urn model, consisting of urns, colours, and balls, with balls of each colour and balls in each urn. At each step, one ball is drawn uniformly at random from each urn, and the chosen balls are redistributed among the urns based on a permutation drawn from a distribution on the symmetric group . We study the mixing time of this Markov chain for fixed , , and , as . We show that there is cutoff whenever the chain on corresponding to the evolution of a single ball is irreducible, and that the same holds for a labeled version of the model. As an application, we also obtain partial results on cutoff for a card shuffling version of the model in which the cards are labeled and their ordering within each stack matters.

Paper Structure

This paper contains 39 sections, 43 theorems, 338 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The generalised Bernoulli--Laplace model
  3. The balanced model
  4. The labeled model
  5. Examples
  6. Related work on the Bernoulli--Laplace urn model
  7. Related models and their inter-relationships
  8. Application to card shuffling
  9. Proof strategy
  10. Coupling approach
  11. Open problems
  12. Outline of the paper
  13. Notation
  14. Preliminaries on Markov chains
  15. Properties of the generalised Bernoulli--Laplace chain
  16. ...and 24 more sections

Key Result

Lemma 1.2

Let $d \geqslant 2$. Consider a sequence of continuous time rate $1$ single ball chains where the $n$th chain has parameters $(d, n, \mu_n)$. Suppose that the Cheeger constant of these chains satisfies $\Phi_{\ast}^{(n)} = \Omega(1)$. Then, we have Further, if $\mu_n$ is symmetric for each $n \in \mathbb{N}$, then the condition $\Phi_{\ast}^{(n)} = \Omega(1)$ is no longer required. In fact, in th

Figures (3)

  • Figure 1: A step in the $(3, 3, 5, \mu_0)$ generalised Bernoulli--Laplace chain, where $\mu_0$ is the Dirac measure on the $3$-cycle $\readlist\thecycle{1,2,3}(\foreachitem\i\in\thecycle{\i})$ inside the symmetric group $S_3$.
  • Figure 2: The inter-relationships between the four chains we define. Arrows denote projection with the text corresponding to the operation. Forgetting order and labels correspond to treating the slots in each urn and balls with the same colour as indistinguishable, respectively.
  • Figure 3: Illustration of the construction in Lemma \ref{['lem:SausageIntersection']}. Let $\zeta$ be such that the canonical path $\Gamma_{\eta, \zeta}$ contains $e_+$. Then $\zeta$ is contained inside the cone $\mathcal{C}$, and hence the tube $\mathcal{T}$, which is in turn contained inside the union of the dotted balls.

Theorems & Definitions (94)

  • Remark 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Example 1: Bernoulli--Laplace with a cyclic shift
  • Example 2: Mean-field Bernoulli--Laplace
  • Theorem 1.7
  • Conjecture 1.10
  • ...and 84 more