In this talk, we will consider a modification of the usual Branching Random Walk (BRW), where we will give certain independent and identically distributed (i.i.d.) displacements/perturbations to all the particles at the generation $n$. We call this process last progeny modified branching random walk (LPM-BRW). Depending on the value of a parameter, $theta > 0$, which works as a “scale parameter’’ for the perturbations, we will 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 will show that in the boundary case, the maximum displacement converges to a limit after only an appropriate centering, which will be the form $c_1 n – c_2 log n$. We will give explicit formulas for the constants $c_1$ and $c_2$ and will show that $c_1$ is exactly the same, while $c_2$ is $1/3$ of the corresponding constants of the Classical BRW [Aídekon 2013]. We will also be able to characterize the limiting distribution as a randomly shifted Gamble distribution. We will further show that below the boundary the logarithmic correction term will be absent, while for above the boundary case, the logarithmic correction term is exactly same as that of the classical BRW . If time permits then we will also show that Brunet-Derrida-type results of point process convergence of our LPM -BRW to a Poisson point process also hold. Our proofs are based on a novel method of coupling the maximum displacement with a linear statistic associated with a well-studied process in statistics known as the smoothing transformation.
[This is a joint work with Partha Pratim Ghosh]