Right-Most Position of a Last Progeny Modified Branching Random Walk

TL;DR

This work analyzes a last progeny modified branching random walk (LPM-BRW), introducing end-generation i.i.d. displacements drawn from a distribution μ and scaled by a parameter θ. By coupling the maximum displacement with a linear statistic via a smoothing transformation, the authors establish a phase diagram around a critical inverse temperature θ0 and derive precise centering constants and limiting laws for the maximum in three regimes: below, at, and above the boundary. In the boundary and below-boundary regimes, the centering includes logarithmic corrections with explicit coefficients, and the limit distributions resemble randomly shifted Gumbel laws, linked to derivative martingales; in the above-boundary regime, a 3/2 log n correction appears under additional non-lattice hypotheses and μ=δ1. The paper also proves Brunet–Derrida type point-process convergence for θ ≤ θ0, showing convergence to Poisson point processes with intensity e^{-x}dx (with decorations vanishing in certain regimes), highlighting universality in extremes of log-correlated structures and demonstrating a powerful coupling technique via the smoothing transformation.

Abstract

In this work, we consider a modification of the usual Branching Random Walk (BRW), where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the $n$-th generation, which may be different from the driving increment distribution. We call this process last progeny modified branching random walk (LPM-BRW). Depending on the value of a parameter, $θ$, we classify the model in three distinct cases, namely, the boundary case, below the boundary case, and above the boundary case. Under very minimal assumptions on the underlying point process of the increments, we show that at the boundary case, $θ=θ_0$, where $θ_0$ is a parameter value associated with the displacement point process, the maximum displacement converges to a limit after only an appropriate centering, which is of the form $c_1 n - c_2 \log n$. We give an explicit formula for the constants $c_1$ and $c_2$ and show that $c_1$ is exactly the same, while $c_2$ is $1/3$ of the corresponding constants of the usual BRW Aidekon (2013). We also characterize the limiting distribution. We further show that below the boundary, $θ< θ_0$, the logarithmic correction term is absent. For above the boundary case, $θ> θ_0$, the logarithmic correction term is exactly the same as that of the classical BRW. For $θ\leq θ_0$, we further derive Brunet-Derrida -type results of point process convergence of our LPM-BRW to a Poisson point process. Our proofs are based on a novel method of coupling the maximum displacement with a linear statistic associated with a more well-studied process in statistics, known as the smoothing transformation.

Figures (3)

• Figure 1: Last progeny modified branching random walk (LPM-BRW)
• Figure 2: Illustration of $\theta_0$
• Figure 3: Graph of $y = \nu(x)/x$

Theorems & Definitions (56)

• Remark 1.1
• Remark 2.1
• Remark 2.2
• Theorem 2.1
• Remark 2.3
• Theorem 2.2
• Remark 2.4
• Theorem 2.3
• Remark 2.5
• Remark 2.6