Maxima of branching random walks with piecewise constant variance

Abstract

This article extends the results of Fang and Zeitouni [Electron. J. Probab. 17 (2012a) 18] 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 O P (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 [Electron. J. Probab. 20 (2015b) 40] when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang and Zeitouni [Electron. J. Probab. 17 (2012a) 18] 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.

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Sub-leading order and exponential tails

“…In the discrete analogue model of (variable-speed) BBM, the (time-inhomogeneous) branching random walk (BRW) on the Galton Watson tree, there are results on the first and second order correction by Fang and Zeitouni [28], Mallein [45] and Ouimet [49]. A notable difference in the context of (timeinhomogeneous) BRW is that one does not need to assume that increments are Gaussian (see [45]).…”

Extremes of the two-dimensional Gaussian free field with scale-dependent variance

“…In this paper, we consider a family of Gaussian fields constructed from the GFF {φ v } v∈V N on the square box V N {0, 1, ..., N } 2 . These Gaussian fields are the analogues, in the context of the GFF, of the time-inhomogeneous branching random walks studied in Bovier and Kurkova (2004); Fang and Zeitouni (2012a); Bovier and Hartung (2014); Ouimet (2015). We study the maxima and the number of high points of this family of Gaussian fields as N → ∞.…”

“…Similar results for non-Gaussian IBRWs and more general variances are proved in Mallein (2015), though not at the level of convergence of the law. In Ouimet (2015), the second order of the maximum for the Gaussian IBRW with a finite number of variances is shown by generalizing the approach of Fang and Zeitouni (2012a) and the tightness follows from Fang (2012).…”

Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

“…Fyodorov and Bouchaud (2008b); Cao et al (2016); • The branching random walk in time-inhomogeneous environment, see e.g. Fang and Zeitouni (2012a); Mallein (2015a,b); Ouimet (2017); • The variable speed branching Brownian motion, see e.g. Bovier andHartung (2014, 2015); Fang and Zeitouni (2012b); Maillard and Zeitouni (2016).…”