We consider a modification of the usual Branching Random Walk, where we give certain i.i.d. displacements to all the particles at the n-th generation. We call this process a last progeny modified branching random walk (LPM-BRW). For θ, a scaling-parameter for the extra displacements, we see that when θ = θ0, the maximum displacement converges to a limit only after a centering c1n−c2logn, where c1 is exactly the same and c2 is 1/3 of the corresponding constants of the usual BRW. We also characterize the limiting distribution. When θ < θ0, the logarithmic correction disappears. For θ > θ0, we have only a partial result, which indicates that the logarithmic correction matches with that of the classical BRW. For θ ≤ θ0, we further derive Brunet-Derrida-type results of point process convergence of LPM-BRW to a Poisson point process. Similar results have been obtained also for the time-inhomogeneous setting.
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Meeting ID: 869 3584 1349