Abstract
This work is a continuation of [6], in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. Our main finding here is that most of the particles in an extreme level set come from only a small fraction of the clusters, which are atypically large.
Highlights
Introduction and results1.1 Setup and state of the artThis work is a continuation of [6], in which the fine structure of the extreme values of branching Brownian motion (BBM) was studied
This work is a continuation of [6], in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum
Let us first recall the definition of BBM and some of the state-of-the-art concerning its extreme value statistics
Summary
This work is a continuation of [6], in which the fine structure of the extreme values of branching Brownian motion (BBM) was studied. Other extreme values of h can be studied simultaneously by considering the extremal process associated with it To describe the latter, given t ≥ 0, x ∈ Lt and r > 0, we let Ct,r(x) denote the cluster of relative heights of particles in Lt, which are at genealogical distance at most r from x. The asymptotic growth of the number of all extreme values, which is arguably the more interesting quantity, is not a straightforward consequence of (1.6) This is because the limiting process E is a superposition of i.i.d. clusters C, and the law ν of the latter will determine the number of points inside any given set in the overall process. This was combined with (1.5) and (1.6) to derive (Theorem 1.1 in [6])
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