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A model of discrete interacting updates

Denis Denisov, Seva Shneer, Vitali Wachtel

TL;DR

The article studies a discrete interacting update model of N counters updated by random pair interactions, proving positive recurrence of the distances between ordered counters and defining the stationary speed V(N) as the average number of updated counters per step. It develops a configuration-based framework and Lyapunov drift techniques to derive both asymptotic and non-asymptotic bounds on V(N), confirming tight bounds of order 1 and identifying a conjectured limit V(∞) around 1.24. The authors provide a detailed analysis for N=4, outline a general approach via test functions and level-merging/redistribution, and present a rich set of open problems including mean-field limits and traveling-wave interpretations for the tail behavior. Collectively, the work advances understanding of simple but intricate interacting-update systems and highlights challenging directions for exact limiting behavior and field-theoretic analogies.

Abstract

We consider $N$ counters taking integer values which are subject to the following dynamics. At every time, a pair of distinct counters is chosen uniformly at random and their states are updated according to the following rule. If the states are different, then the smaller one is increased by $1$, while if the states are the same, both of them are increased by $1$. We show that, for a fixed $N$, the distances between consecutive ordered counters form a positive recurrent Markov chain and there exists the speed $V(N)$ defined as the average number of counters updated per time step in the stationary regime. We provide non-trivial upper and lower bounds for $V(N)$ as $N\to \infty$. Despite the simple formulation of the problem, its analysis seems to be highly complicated. We also provide a list of open problems and discuss various methods one may want to use, and obstacles one encounters.

A model of discrete interacting updates

TL;DR

The article studies a discrete interacting update model of N counters updated by random pair interactions, proving positive recurrence of the distances between ordered counters and defining the stationary speed V(N) as the average number of updated counters per step. It develops a configuration-based framework and Lyapunov drift techniques to derive both asymptotic and non-asymptotic bounds on V(N), confirming tight bounds of order 1 and identifying a conjectured limit V(∞) around 1.24. The authors provide a detailed analysis for N=4, outline a general approach via test functions and level-merging/redistribution, and present a rich set of open problems including mean-field limits and traveling-wave interpretations for the tail behavior. Collectively, the work advances understanding of simple but intricate interacting-update systems and highlights challenging directions for exact limiting behavior and field-theoretic analogies.

Abstract

We consider counters taking integer values which are subject to the following dynamics. At every time, a pair of distinct counters is chosen uniformly at random and their states are updated according to the following rule. If the states are different, then the smaller one is increased by , while if the states are the same, both of them are increased by . We show that, for a fixed , the distances between consecutive ordered counters form a positive recurrent Markov chain and there exists the speed defined as the average number of counters updated per time step in the stationary regime. We provide non-trivial upper and lower bounds for as . Despite the simple formulation of the problem, its analysis seems to be highly complicated. We also provide a list of open problems and discuss various methods one may want to use, and obstacles one encounters.

Paper Structure

This paper contains 23 sections, 4 theorems, 135 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main results and discussions
  3. Main results
  4. Case $N=4$
  5. Description of our approach
  6. Open problems
  7. Preliminary computations
  8. Computation of $\mathbb{E}[\Delta X_{k} ]$
  9. Computation of values of test functions for each configuration
  10. Proof of Theorem \ref{['thm:positive.recurrent']}
  11. Proof with quadratic Lyapunov function
  12. Proof with exponential Lyapunov function
  13. Monotonicity property of configurations
  14. Merging levels $k-1$ and $k$ for $k\ge 2$
  15. Redistribution of counters between levels $k-1$ and $k$ when $k\ge 2$
  16. ...and 8 more sections

Key Result

Theorem 1

For each $N\ge 3$, $X$ is a positive recurrent Markov chain with stationary distribution $\pi$.

Figures (2)

  • Figure 1: Plots of $\phi_1,\phi_3,\phi_5,\phi_7, \phi_9$ and $\phi_{11}$
  • Figure 2: Fitting of $f$ by a parabola for $N=16$.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Lemma 5
  • proof