Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics

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In this paper, we view fluctuating fronts made of particles on a one-dimensional lattice as an extreme value problem. The idea is to denote the configuration for a single front realization at time t by the set of co-ordinates {kᵢ(t)}≡[k₁(t),k₂(t),...,kₙ₍ₜ₎(t)] of the constituent particles, where N(t) is the total number of particles in that realization at time t. When {kᵢ(t)} are arranged in the ascending order of magnitudes, the instantaneous front position can be denoted by the location of the rightmost particle, i.e., by the extremal value kf(t)=max[k₁(t),k₂(t),...,kₙ₍ₜ₎(t)]. Due to interparticle interactions, {kᵢ(t)} at two different times for a single front realization are naturally not independent of each other, and thus the probability distribution Pₖf(t) [based on an ensemble of such front realizations] describes extreme value statistics for a set of correlated random variables. In view of the fact that exact results for correlated extreme value statistics are rather rare, here we show that for a fermionic front model in a reaction-diffusion system, Pₖf(t) is Gaussian. In a bosonic front model however, we observe small deviations from the Gaussian.