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

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    We consider the low-temperature T<TcT<T_c disorder-dominated phase of the directed polymer in a random potentiel in dimension 1+1 (where Tc=T_c=\infty) and 1+3 (where Tc<T_c<\infty). To characterize the localization properties of the polymer of length LL, we analyse the statistics of the weights wL(r)w_L(\vec r) of the last monomer as follows. We numerically compute the probability distributions P1(w)P_1(w) of the maximal weight wLmax=maxr[wL(r)]w_L^{max}= max_{\vec r} [w_L(\vec r)], the probability distribution Π(Y2)\Pi(Y_2) of the parameter Y2(L)=rwL2(r)Y_2(L)= \sum_{\vec r} w_L^2(\vec r) as well as the average values of the higher order moments Yk(L)=rwLk(r)Y_k(L)= \sum_{\vec r} w_L^k(\vec r). We find that there exists a temperature Tgap<TcT_{gap}<T_c such that (i) for T<TgapT<T_{gap}, the distributions P1(w)P_1(w) and Π(Y2)\Pi(Y_2) present the characteristic Derrida-Flyvbjerg singularities at w=1/nw=1/n and Y2=1/nY_2=1/n for n=1,2..n=1,2... In particular, there exists a temperature-dependent exponent μ(T)\mu(T) that governs the main singularities P1(w)(1w)μ(T)1P_1(w) \sim (1-w)^{\mu(T)-1} and Π(Y2)(1Y2)μ(T)1\Pi(Y_2) \sim (1-Y_2)^{\mu(T)-1} as well as the power-law decay of the moments Yk(i)ˉ1/kμ(T) \bar{Y_k(i)} \sim 1/k^{\mu(T)}. The exponent μ(T)\mu(T) grows from the value μ(T=0)=0\mu(T=0)=0 up to μ(Tgap)2\mu(T_{gap}) \sim 2. (ii) for Tgap<T<TcT_{gap}<T<T_c, the distribution P1(w)P_1(w) vanishes at some value w0(T)<1w_0(T)<1, and accordingly the moments Yk(i)ˉ\bar{Y_k(i)} decay exponentially as (w0(T))k(w_0(T))^k in kk. The histograms of spatial correlations also display Derrida-Flyvbjerg singularities for T<TgapT<T_{gap}. Both below and above TgapT_{gap}, the study of typical and averaged correlations is in full agreement with the droplet scaling theory.Comment: 13 pages, 29 figure