A Last Progeny Modified Branching Random Walk

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 a 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, when θ takes a particular value θ0, the maximum displacement converges to a limit after only an appropriate centering, which is of the form c1n − c2logn. We give an explicit formula for the constants c1 and c2 and show that c1 is exactly the same, while c2 is 1/3 of the corresponding constants of the usual BRW. We also characterize the limiting distribution. We further show that below the boundary (that is, when θ < θ0), the logarithmic correction term is absent. For above the boundary case (that is, when θ > θ0), we have only a partial result, which indicates a possible existence of the logarithmic correction in the centering with exactly the same constant as that of the classical BRW. For θ ≤ θ0, we further derive Brunet--Derrida-type results of point process convergence of our LPM-BRW to a decorated Poisson point process. Similar results have been obtained also for the time inhomogeneous setting. Surprisingly, the limits in this case depend only on the increments in the initial fraction of time. Under very minimal assumptions, we also derive the large deviation principle (LDP) for the right-most position of a particle in generation n. As a byproduct, we also complete the LDP for the classical model, which complements the earlier work by Gantert and Höfelsauer. 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.