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Cutoff Phenomenon for Inhomogeneous Nonlinear Recombination in Arbitrary Finite Product Spaces

Junho Kim, Insuk Seo

TL;DR

This work extends the cutoff phenomenon for nonlinear recombination from the homogeneous two-spin case to general finite product spaces with inhomogeneous marginals under a uniform nondegeneracy condition. It introduces a tractable algebraic representation of the density fluctuation via an sitewise orthonormal polynomial basis and a comonotonic coupling to handle inhomogeneity, enabling sharp upper and lower bounds on the mixing time and a precise cutoff window at time $t = \log_2 n + O(1)$. The main contributions include a general cutoff result, asymptotic sharpness without relying on an explicit convergence profile, and a complete cutoff profile in the homogeneous setting, extending previously known two-spin results to arbitrary state spaces. The methods provide a robust framework for nonlinear Markov dynamics on product spaces and have potential implications for understanding mixing in population-genetics-inspired models with heterogeneous marginals. The work thereby enhances both theoretical understanding and quantitative tools for nonlinear convergence in high-dimensional probabilistic systems.

Abstract

In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space $S^n$ representing the state of spins on $n$ sites. Although its stationary distribution has a product structure, and its evolution is Markovian, the dynamics of the model is nonlinear. Consequently, the estimation of the mixing time becomes a highly non-trivial task. The special case with two spins and homogeneous stationary measure was considered in Caputo, Labbé, and Lacoin [The Annals of Applied Probability 35:1164-1197, 2025], where the cutoff phenomenon for the mixing behavior has been verified. In this article, we extend this result to the general case with finite spins and inhomogeneous stationary measure by developing a novel algebraic representation for the density fluctuation of the system with respect to its stationary state.

Cutoff Phenomenon for Inhomogeneous Nonlinear Recombination in Arbitrary Finite Product Spaces

TL;DR

This work extends the cutoff phenomenon for nonlinear recombination from the homogeneous two-spin case to general finite product spaces with inhomogeneous marginals under a uniform nondegeneracy condition. It introduces a tractable algebraic representation of the density fluctuation via an sitewise orthonormal polynomial basis and a comonotonic coupling to handle inhomogeneity, enabling sharp upper and lower bounds on the mixing time and a precise cutoff window at time . The main contributions include a general cutoff result, asymptotic sharpness without relying on an explicit convergence profile, and a complete cutoff profile in the homogeneous setting, extending previously known two-spin results to arbitrary state spaces. The methods provide a robust framework for nonlinear Markov dynamics on product spaces and have potential implications for understanding mixing in population-genetics-inspired models with heterogeneous marginals. The work thereby enhances both theoretical understanding and quantitative tools for nonlinear convergence in high-dimensional probabilistic systems.

Abstract

In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space representing the state of spins on sites. Although its stationary distribution has a product structure, and its evolution is Markovian, the dynamics of the model is nonlinear. Consequently, the estimation of the mixing time becomes a highly non-trivial task. The special case with two spins and homogeneous stationary measure was considered in Caputo, Labbé, and Lacoin [The Annals of Applied Probability 35:1164-1197, 2025], where the cutoff phenomenon for the mixing behavior has been verified. In this article, we extend this result to the general case with finite spins and inhomogeneous stationary measure by developing a novel algebraic representation for the density fluctuation of the system with respect to its stationary state.

Paper Structure

This paper contains 22 sections, 13 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Main Results
  3. The Model
  4. The Cutoff Phenomenon
  5. The Cutoff Profile for Monochromatic Initial States
  6. Key Tools
  7. Quenched Distribution
  8. Orthonormal Polynomial Basis
  9. Quenched Moments and Density Expansion
  10. Comonotonic Coupling and Its Correlation Properties
  11. Proof of Theorem \ref{['thm:upper_bound']}
  12. Proof of the Upper Bound
  13. Proof of the Asymptotic Sharpness
  14. Proof of Theorem \ref{['thm: discrete lb']}
  15. Proof of the Lower Bound
  16. ...and 7 more sections

Key Result

Lemma 2.1

The law of the random configuration $\sigma^*$ is $\mu_t$.

Figures (1)

  • Figure 1: The graphical representation of the distribution $\mu_t$ for the case $t=2$. Each internal node represents the collision product of its children's distributions, with the root yielding the final distribution $\mu_2$.

Theorems & Definitions (37)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3: Cutoff Phenomenon
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • proof : Proof of Theorem \ref{['thm:cutoff_phenomenon']}
  • Definition 2.8: Monochromatic Initial Distribution
  • Theorem 2.9
  • ...and 27 more