On the genealogy of branching random walks and of directed polymers
1 Collège de France - 11 Place Marcelin Berthelot, 75005 Paris, France
2 LPS, École Normale Supérieure - 24 rue Lhomond, 75005 Paris, France
3 School of Physics and Astronomy, University of Edinburgh - James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
Received: 22 July 2016
Accepted: 31 August 2016
It is well known that the mean-field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite-size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.
PACS: 02.50.-r – Probability theory, stochastic processes, and statistics / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 65.60.+a – Thermal properties of amorphous solids and glasses: heat capacity, thermal expansion, etc.
© EPLA, 2016