Extreme value statistics in a continuous time branching process: a pedagogical primer

TL;DR

This paper analyzes a continuous time branching process using an 'agitated random walk' model to derive exact extreme value statistics across different phases, revealing distinct behaviors in subcritical, critical, and supercritical regimes.

Contribution

It introduces an exact mapping of the branching process to an 'agitated random walk' model, enabling explicit calculations of the maximal population size distribution in all phases.

Findings

Exact distribution of maximum population size derived for all phases.

Distinct asymptotic behaviors identified in subcritical, critical, and supercritical regimes.

Analytical results confirmed by numerical simulations.

Abstract

We study a continuous time branching process where an individual splits into two daughters with rate b and dies with rate a, starting from a single individual at t=0. We show that the model can be mapped exactly to a random walk problem where the population size N(t) performs a random walk on a positive semi-infinite lattice. The hopping rate of this random walker out of a site labelled n is proportional to n, i.e., the walker gets more and more `agitated' as it moves further and further away from the origin--we call this an `agitated random walk' (ARW). We demonstrate that this random walk problem is particularly suitable to obtain exact explicit results on the extreme value statistics, namely, on the distribution of the maximal population size M(t)= \max_{0\tau\le t}[N(\tau)] up to time t. This extreme value distribution displays markedly different behaviors in the three phases: (i) subcritical (b<a) (ii) critical (b=a) and (iii) supercritical (b>a). In the subcritical and critical phases , Q(L,t) becomes independent of time t for large t and the stationary distribution Q(L, \infty) decays to zero with increasing L, respectively exponentially (subcritical) and algebraically (critical). For finite but large t, the distribution at the critical point exhibits a scaling form Q(L,t)\sim f_c(L/{at})/L^2 where the scaling function f_c(z) has a nontrivial shape that we compute analytically. In the supercritical phase, the distribution Q(L,t) has a `fluid' part that becomes independent of t for large t and a `condensate' part (a delta peak centered at e^{(b-a)t}) which gets disconnected from the `fluid' part and moves rapidly to \infty as time increases. We also verify our analytical predictions via numerical simulations finding excellent agreement.