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Freezing in the Infinite-Bin Model

Bastien Mallein, Sanjay Ramassamy, Arvind Singh

TL;DR

This work analyzes freezing in the infinite-bin model (IBM), a ranked, memoryful, one-dimensional particle system driven by a reproduction law $\mu$. It develops a comprehensive framework combining genealogical structure and coupling techniques to derive when bins receive only finitely many balls in the infinite-time limit, i.e., freezing, and to establish a 0-1 law for freezing under monotone $\mu$. It shows that bounded or locally finite initial configurations typically lead to almost sure freezing, while infinite-type $\mu$ can sustain non-freezing cascades, with a clear dependency on the type. In the barrier-at-zero setting with regularly varying $\mu$, the paper provides precise dichotomies for freezing and derives explicit deterministic growth rates for the barrier-driven growth, highlighting how tail behavior of $\mu$ controls long-time dynamics. Overall, the results yield concrete criteria to predict freezing versus unbounded growth in rank-based, memory-rich branching systems, with explicit growth-rate formulas in the barrier regime.

Abstract

The infinite-bin model is a one-dimensional particle system on $\mathbb{Z}$ introduced by Foss and Konstantopoulos in relation with last passage percolation on complete directed acyclic graphs. In this model, at each integer time, a particle is selected at random according to its rank, and produces a child at the location immediately to its right. In this article, we consider the limiting distribution of particles after an infinite number of branching events have occurred. Under mild assumptions, we prove that the event (called freezing) that a location contains only a finite number of balls satisfies a $0-1$ law and we provide various criteria to determine whether freezing occurs.

Freezing in the Infinite-Bin Model

TL;DR

This work analyzes freezing in the infinite-bin model (IBM), a ranked, memoryful, one-dimensional particle system driven by a reproduction law . It develops a comprehensive framework combining genealogical structure and coupling techniques to derive when bins receive only finitely many balls in the infinite-time limit, i.e., freezing, and to establish a 0-1 law for freezing under monotone . It shows that bounded or locally finite initial configurations typically lead to almost sure freezing, while infinite-type can sustain non-freezing cascades, with a clear dependency on the type. In the barrier-at-zero setting with regularly varying , the paper provides precise dichotomies for freezing and derives explicit deterministic growth rates for the barrier-driven growth, highlighting how tail behavior of controls long-time dynamics. Overall, the results yield concrete criteria to predict freezing versus unbounded growth in rank-based, memory-rich branching systems, with explicit growth-rate formulas in the barrier regime.

Abstract

The infinite-bin model is a one-dimensional particle system on introduced by Foss and Konstantopoulos in relation with last passage percolation on complete directed acyclic graphs. In this model, at each integer time, a particle is selected at random according to its rank, and produces a child at the location immediately to its right. In this article, we consider the limiting distribution of particles after an infinite number of branching events have occurred. Under mild assumptions, we prove that the event (called freezing) that a location contains only a finite number of balls satisfies a law and we provide various criteria to determine whether freezing occurs.
Paper Structure (9 sections, 16 theorems, 98 equations, 2 figures)

This paper contains 9 sections, 16 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General properties of infinite-bin models
  3. Genealogical structure
  4. Freezing for bounded configuration
  5. Proof of Item 2. of Theorem \ref{['thm:finitevinfinite']}
  6. Coupling of IBMs for different initial configurations
  7. 0-1 law for the freezing of a bin
  8. Freezing for finite type distribution and locally bounded configurations
  9. IBM starting with an infinite barrier at 0

Key Result

Proposition 1.1

Suppose that the distribution $\mu$ satisfies $\sum n\mu(n) < \infty.$ Then for any initial configuration $X_0 \in \mathcal{S}$ and any $k\in\mathbb{Z}$ such that $X_0(k) < \infty$, we have $X_\infty(k) < \infty$ a.s.

Figures (2)

  • Figure 1: Illustration of the dynamics of the IBM: here, we have a configuration $X$ of balls with $F(X) = 4$. The result of sampling $\xi = 12$ creates a new ball at location $-1$, height $2$, and rank $10$. All the balls with previous rank $\geqslant 10$ in $X$ have their rank increased by one in $\Phi_{\xi}(X)$.
  • Figure 2: Two configurations $X_0$ (in blue) and $\widetilde{X}_0$ (in red) that satisfy \ref{['hypH2']}. There are more red balls than blue balls located in $\llbracket j_0 + 1,+\infty\llbracket$ ($K = 5 \leqslant 9 = \widetilde{K}$) and also more red balls than blue balls in total ($N = 13 \leqslant 14 = \widetilde{N}$) but there are at least as many blue balls than red balls in every bin located at $j < j_0$.

Theorems & Definitions (34)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 2.1
  • ...and 24 more