Overlap of Brownian trajectories
Mikhail Tamm, Physics Dept., Moscow State University
I will present our recent results on the mean number of common sites visited by several random walk trajectories of a given length starting at a given distance from each other. I will start with the formulation of the problem and show the general solution in the Laplace space which is viable for any Markovian random walks in any geometry; then proceed to consideration of the Brownian motion in two specific cases: that of several trajectories starting from a common center, and that of two trajectories starting at a given distance. For the first case, I will show the existence of three distinct regimes depending on the dimensionality of space and the number of trajectories, and construct the corresponding morphology diagram. For the second case, I will show that in 1D and 3D it is possible to do the reverse Laplace transform and get the explicit expressions for the scaling functions of interest, which turn out to be in perfect agreement with computer simulation results.
The work is done in collaboration with S. Majumdar (LPTMS), A. Ilyina (Moscow State University), and D. Grebenkov (Ecole Polytechnique).