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The frog model with death revisited

Samyah Ahmed, Matthew Junge

TL;DR

This work analyzes recurrence in the frog model with death and drift on the infinite $d$-ary tree by introducing a recursive frog model (RFM) that remains tractable under death. The key idea is to show that recurrence of $\mathrm{RFM}(d,q,p)$ implies recurrence of the original model $\mathrm{FM}(d,q,p)$, and to prove a positive-probability recurrence for a concrete parameter rectangle $p\in[0.4950,0.4975]$ and $q\in[0.9999998,1]$ for all $d\ge 2$. The authors establish nonmonotonicity in the drift parameter within this regime, highlighting nuanced phase structure when death is present, and supplement the analysis with explicit transience criteria and a random-walk excursion lemma in the appendix. The methodological contribution lies in leveraging first- and second-moment bounds within the RFM framework, aided by a geometric thinning factor $\mathfrak{q}$ and the Paley-Zygmund inequality, to derive positive recurrence while avoiding the complications that death imposes on self-similar approaches.

Abstract

We prove that the probability the frog model with death and drift on the $d$-ary tree is recurrent can be made positive and thus is not monotone in the drift parameter.

The frog model with death revisited

TL;DR

This work analyzes recurrence in the frog model with death and drift on the infinite -ary tree by introducing a recursive frog model (RFM) that remains tractable under death. The key idea is to show that recurrence of implies recurrence of the original model , and to prove a positive-probability recurrence for a concrete parameter rectangle and for all . The authors establish nonmonotonicity in the drift parameter within this regime, highlighting nuanced phase structure when death is present, and supplement the analysis with explicit transience criteria and a random-walk excursion lemma in the appendix. The methodological contribution lies in leveraging first- and second-moment bounds within the RFM framework, aided by a geometric thinning factor and the Paley-Zygmund inequality, to derive positive recurrence while avoiding the complications that death imposes on self-similar approaches.

Abstract

We prove that the probability the frog model with death and drift on the -ary tree is recurrent can be made positive and thus is not monotone in the drift parameter.
Paper Structure (10 sections, 8 theorems, 18 equations, 2 figures)

This paper contains 10 sections, 8 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model and results
  3. Proof Discussion
  4. Organization
  5. The recursive frog model
  6. Definition
  7. Properties
  8. Proof of \ref{['thm:main']}
  9. Proof of \ref{['prop:p']}
  10. Random walk excursion lengths

Key Result

Theorem 1

$\mathop{\mathrm{FM}}\nolimits(d,q,p) \text{ is recurrent}$ with positive probability for any $d \geq 2$ for all $(p,q) \in[0.4950, 0.4975] \times [0.9999998,1]$.

Figures (2)

  • Figure 1: The $(p,q)$-phase diagram for $\mathop{\mathrm{FM}}\nolimits(2,q,p)$ implied by our results. In the gray shaded regions the process is almost surely transient. The blue region represents the rectangle $[0.4950,0.4975] \times [0.9999998,1]$ (enlarged to make it visible at this scale) for which the process is recurrent with positive probability. The white region is not covered by our results.
  • Figure 2: The recursive structure of $\mathop{\mathrm{RFM}}\nolimits(2,q,p)$. Here, $V_{t}$ is the total number of visits to $\varnothing$ with sleeping frogs placed up to distance $t$ from the root. It can be expressed as a binomial thinning of the number of visits to $\varnothing'$ from each of its children $x$ and $y$. Conditional on $x$ and $y$ being visited, these quantities are i.i.d. and distributed like $V_{t-1}$.

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 2 more