On the genealogy of branching random walks and of directed polymers

Abstract

PACS 02.50.-r -Probability theory, stochastic processes and statistics PACS 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion PACS 65.60.+a -Thermal properties of amorphous solids and glasses: heat capacity, thermal expansion, etc.Abstract -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.Introduction. -The study of branching Brownian motion and of branching random walks is central in the theory of probability [1-4] and appears in several physical contexts [5][6][7][8][9][10]. Here we focus on its revelance in the mean field theory of directed polymers in a random medium [11][12][13][14]. This mean field version is an example of a disordered system which exhibits replica symmetry breaking in its low temperature phase, [15][16][17]. In the models considered here only one step of replica symmetry breaking is required in the thermodynamic limit [18]. Our motivation here is to determine how this broken symmetry is affected by finite size fluctuations. In the present work we will show that the one step replica symmetry breaking is smoothed in a universal way for which one can obtain an explicit analytic expression.

Rapidity Gaps and Ancestry

“…We also expect it to be correct (up to the overall numerical factor) for related quantities, such as the rapidity distribution of the common ancestor of all dipoles larger than some given (large enough) size 1/Q MV . 6 Equation (5) was found by assuming that the common ancestor was an unusually large object generated around the rapidityỹ 0 = Y −y 0 in the evolution of the onium [7]. This is exactly the same mechanism as in the case of the diffraction problem (see Sec.…”

Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions

“…The leading large-M behavior of such a generating function is then obtained as the Legendre transform of the large deviation function given in eq. (8). The result is [14] (see Fig.…”

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“…More generally, size-dependent corrections of logREM are difficult to get access to analytically, especially using replicas; however, for simple versions of REM there is a recent progress using alternative approach [61].…”