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Register a new userLarge Deviations for the Right-Most Position of a Last Progeny Modified Branching Random Walk
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Partha Pratim Ghosh
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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. This model was first introduced by Bandyopadhyay and Ghosh (2021) and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Under very minimal assumptions, we 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 H{o}felsauer (2018).
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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, $theta$, 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 $theta$ takes a particular value $theta_0$, 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. We also characterize the limiting distribution. We further show that below the boundary (that is, when $theta < theta_0$), the logarithmic correction term is absent. For above the boundary case (that is, when $theta > theta_0$), 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 $theta leq theta_0$, we further derive Brunet--Derrida-type results of point process convergence of our LPM-BRW to a decorated Poisson point process. Our proofs are based on a novel method of coupling the maximum displacement with a linear statistics associated with a more well-studied process in statistics, known as the smoothing transformation.
We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cramers condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e.~both tails decay as $e^{-|t|^r}$ for some $rin( 0,1)$. It is known that in this case, $M_n$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_n$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $tilde M_n$, and we show a large deviation principle for the laws of $n^{-1/r}tilde M_n$ as well.
We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $mathbb{Z}^d$, in dimension $dge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in cite{AS}.
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