134 research outputs found
Tightness for Maxima of Generalized Branching Random Walks
We study generalized branching random walks, which allow time dependence and
local dependence between siblings. Under appropriate tail assumptions, we prove
the tightness of , where is the maxima
distribution at time and is the median of . The main
component in the argument is a proof of exponential decay of the right tail
.Comment: 18 pages, no figur
Branching Random Walks in Time Inhomogeneous Environments
We study the maximal displacement of branching random walks in a class of
time inhomogeneous environments. Specifically, binary branching random walks
with Gaussian increments will be considered, where the variances of the
increments change over time macroscopically. We find the asymptotics of the
maximum up to an (stochastically bounded) error, and focus on the
following phenomena: the profile of the variance matters, both to the leading
(velocity) term and to the logarithmic correction term, and the latter exhibits
a phase transition
Extremes of the two-dimensional Gaussian free field with scale-dependent variance
In this paper, we study a random field constructed from the two-dimensional
Gaussian free field (GFF) by modifying the variance along the scales in the
neighborhood of each point. The construction can be seen as a local martingale
transform and is akin to the time-inhomogeneous branching random walk. In the
case where the variance takes finitely many values, we compute the first order
of the maximum and the log-number of high points. These quantities were
obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when
the variance is constant on all scales. The proof relies on a truncated second
moment method proposed by Kistler (2015), which streamlines the proof of the
previous results. We also discuss possible extensions of the construction to
the continuous GFF.Comment: 30 pages, 4 figures. The argument in Lemma 3.1 and 3.4 was revised.
Lemma A.4, A.5 and A.6 were added for this reason. Other typos were corrected
throughout the article. The proof of Lemma A.1 and A.3 was simplifie
Maxima of branching random walks with piecewise constant variance
This article extends the results of Fang & Zeitouni (2012a) on branching
random walks (BRWs) with Gaussian increments in time inhomogeneous
environments. We treat the case where the variance of the increments changes a
finite number of times at different scales in [0,1] under a slight restriction.
We find the asymptotics of the maximum up to an OP(1) error and show how the
profile of the variance influences the leading order and the logarithmic
correction term. A more general result was independently obtained by Mallein
(2015b) when the law of the increments is not necessarily Gaussian. However,
the proof we present here generalizes the approach of Fang & Zeitouni (2012a)
instead of using the spinal decomposition of the BRW. As such, the proof is
easier to understand and more robust in the presence of an approximate
branching structure.Comment: 28 pages, 4 figure
Slowdown for time inhomogeneous branching Brownian motion
We consider the maximal displacement of one dimensional branching Brownian
motion with (macroscopically) time varying profiles. For monotone decreasing
variances, we show that the correction from linear displacement is not
logarithmic but rather proportional to . We conjecture that this is
the worse case correction possible
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