Directed polymer in a random medium of dimension 1+1 and 1+3: weights statistics in the low temperature phase

作者：Monthus Cecile^*; Garel Thomas

来源：Journal of Statistical Mechanics: Theory and Experiment , 2007, P03011.

DOI：10.1088/1742-5468/2007/P03011

• disordered systems (theory)

We consider the low temperature T < T-c disorder-dominated phase of the directed polymer in a random potential in dimension 1+1 (where T-c = infinity) and 1 + 3 ( where Tc < 8). To characterize the localization properties of the polymer of length L, we analyse the statistics of the weights w(L)(empty set) of the last monomer as follows. We numerically compute the probability distributions P-1(w) of the maximal weight w(L)(max) = max empty set[w(L)(empty set)], the probability distribution II(Y-2) of the parameter Y-2( L) = Sigma empty set w(L)(2)(empty set) as well as the average values of the higher-order moments Y-k(L) = Sigma empty set w(L)(k)(empty set). We find that there exists a temperature T-gap < T-c such that (i) for T < Tgap, the distributions P-1(w) and.( Y2) present the characteristic Derrida-Flyvbjerg singularities at w = 1/n and Y-2 = 1/n for n = 1, 2... In particular, there exists a temperature-dependent exponent mu(T) that governs the main singularities P-1( w) similar to (1 - w)(mu(T)-1) and II(Y-2) similar to (1 - Y-2)(mu(T)-1) as well as the power-law decay of the moments Y-k(i) similar to 1/k(mu(T)). The exponent mu(T) grows from the value mu(T = 0) = 0 up to mu(T-gap) similar to 2. (ii) For T-gap < T < T-c, the distribution P-1(w) vanishes at some value w(0)(T) < 1, and accordingly the moments Y-k(i) decay exponentially as (w(0)(T))(k) in k. The histograms of spatial correlations also display Derrida-Flyvbjerg singularities for T < T-gap. Both below and above T-gap, the study of typical and averaged correlations is in full agreement with the droplet scaling theory.

• 出版日期2007
• 单位中国地震局