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Exponential moments of truncated branching random walk martingales

Heng Ma, Pascal Maillard

TL;DR

The paper investigates exponential tails for truncated branching random walk martingales. By combining mgf-based methods (for the truncated additive martingale) with spine decompositions and perpetuity-type analysis (for the truncated derivative martingale in the critical case), it establishes exponential decay in the tails of both truncated limits under explicit moment conditions. This contrasts with polynomial tails in the untruncated setting and supports a deeper understanding of large-deviation behavior under killing, with relevance to first-passage percolation on sparse random graphs. The results also introduce novel use of spine techniques for additive functionals with exponential tails, suggesting further applications to absorbed BRW quantities and related perpetuity problems.

Abstract

For a branching random walk that drifts to infinity, consider its Malthusian martingale, i.e.~the additive martingale with parameter $θ$ being the smallest root of the characteristic equation. When particles are killed below the origin, we show that the limit of this martingale admits an exponential tail, contrary to the case without killing, where the tail is polynomial. In the critical case, where the characteristic equation has a single root, the same holds for the (truncated) derivative martingale, as we show. This study is motivated by recent work on first passage percolation on Erdős-Rényi graphs.

Exponential moments of truncated branching random walk martingales

TL;DR

The paper investigates exponential tails for truncated branching random walk martingales. By combining mgf-based methods (for the truncated additive martingale) with spine decompositions and perpetuity-type analysis (for the truncated derivative martingale in the critical case), it establishes exponential decay in the tails of both truncated limits under explicit moment conditions. This contrasts with polynomial tails in the untruncated setting and supports a deeper understanding of large-deviation behavior under killing, with relevance to first-passage percolation on sparse random graphs. The results also introduce novel use of spine techniques for additive functionals with exponential tails, suggesting further applications to absorbed BRW quantities and related perpetuity problems.

Abstract

For a branching random walk that drifts to infinity, consider its Malthusian martingale, i.e.~the additive martingale with parameter being the smallest root of the characteristic equation. When particles are killed below the origin, we show that the limit of this martingale admits an exponential tail, contrary to the case without killing, where the tail is polynomial. In the critical case, where the characteristic equation has a single root, the same holds for the (truncated) derivative martingale, as we show. This study is motivated by recent work on first passage percolation on Erdős-Rényi graphs.
Paper Structure (9 sections, 6 theorems, 88 equations)

This paper contains 9 sections, 6 theorems, 88 equations.

Table of Contents

  1. Main Results
  2. Background and related work
  3. Proof ideas
  4. Outlook and open problems
  5. Overview of paper
  6. Truncated Additive Martingales
  7. Truncated Derivative Martingales
  8. Definitions and preliminaries
  9. Proof of Theorem \ref{['thm-tail-D-x']}

Key Result

Theorem 1.1

Suppose $\Phi'(1)<0$ and set Let $\gamma\in (1,\kappa)$. If $W_1 = \sum_{|u|=1} e^{-V(u)}$ admits a finite exponential moment, then there are constants $C,c>0$ such that for all $x>0$, $y\ge1$, and $n\in\mathbb{N}\cup\{\infty\}$.

Theorems & Definitions (13)

  • Theorem 1.1: Tail of truncated additive martingale
  • Theorem 1.2: Tail of truncated derivative martingale
  • Conjecture 1.3
  • Lemma 2.1
  • proof : Proof of Theorem \ref{['thm-tail-W-x']} assuming Lemma \ref{['lem-moments-W-x']}
  • proof : Proof of Lemma \ref{['lem-moments-W-x']}
  • Proposition 3.1: BK04
  • Lemma 3.2
  • proof : Proof of theorem \ref{['thm-tail-D-x']} assuming Lemma \ref{['lem-exp-mon-D-x']}
  • Lemma 3.3
  • ...and 3 more