Exponential moments of truncated branching random walk martingales

2025/04/29 by Heng Ma, Ma, Heng, Pascal Maillard +1 · 1 citation
Mathematics · Physics and Astronomy · #60G55 #92D25 #FOS: Mathematics #Primary 60J80 #Probability (math.PR) #Random Matrices and Applications #Secondary 60G70 #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2504.20963

openalex publication_date 2025/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• On the derivative martingale in a branching random walk
• The tail distribution of the Derivative martingale and the global minimum of the branching random walk
• Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift
• A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk
• On distributional properties of perpetuities
• Invariance principles for random walks conditioned to stay positive

Cited by

• Minimum-Weight Path in a Sparse Erdős--Rényi Graph with Signed Weights

Related

• The tail distribution of the Derivative martingale and the global minimum of the branching random walk
• A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk
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• On the derivative martingale in a branching random walk
• A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks