2020/03/09 by Le, Anh Dung, Mueller, Alfred H., Munier, Stéphane
#FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2003.04362
We implement a discretization of the one-dimensional branching Brownian motion in the form of a Monte Carlo event generator, designed to efficiently produce ensembles of realizations in which the rightmost lead particle at the final time T is constrained to have a position X larger than some predefined value Xmin. The latter may be chosen arbitrarily far from the expectation value of X, and the evolution time after which observables on the particle density near the lead particle are measured may be as large as T∼ 104. We then calculate numerically the probability distribution pn(Δx) of the number n of particles in the interval [X-Δx,X] as a function of Δx. When Xmin is significantly smaller than the expectation value of the position of the rightmost lead particle, i.e. when X is effectively unconstrained, we check that both the mean and the typical values of n grow exponentially with Δx, up to a linear prefactor and to finite-T corrections. When Xmin is picked far ahead of the latter but within a region extending over a size of order √(T) to its right, the mean value of the particle number still grows exponentially with Δx, but its typical value is lower by a multiplicative factor consistent with e^-ζΔx2/3, where ζ is a number of order unity. These numerical results bring strong support to recent analytical calculations and conjectures in the infinite-time limit.