Universal Order and Gap Statistics of Critical Branching Brownian Motion

作者：Ramola Kabir^*; Majumdar Satya N; Schehr Gregory

来源：Physical Review Letters, 2014, 112(21): 210602.

DOI：10.1103/PhysRevLett.112.210602

We study the order statistics of one-dimensional branching Brownian motion in which particles either diffuse (with diffusion constant D), die (with rate d), or split into two particles (with rate b). At the critical point b = d, which we focus on, we show that at large time t the particles are collectively bunched together. We find indeed that there are two length scales in the system: (i) the diffusive length scale similar to root Dt, which controls the collective fluctuations of the whole bunch, and (ii) the length scale of the gap between the bunched particles similar to root D/b. We compute the probability distribution P) over tilde (g(k), t vertical bar n) of the kth gap g(k) = x(k) - x(k+1) between the kth and (k + 1)th particles given that the system contains exactly n > k particles at time t. We show that at large t, it converges to a stationary distribution (P) over tilde (g(k), t -> infinity vertical bar n) = p(g(k)vertical bar n) with an algebraic tail p(g(k)vertical bar n) similar to 8(D/b)g(k)(-3), for g(k) >> 1, independent of k and n. We verify our predictions with Monte Carlo simulations.

• 出版日期2014-5-29