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

Abstract

We consider the low-temperature $T<T_c$ disorder-dominated phase of the directed polymer in a random potentiel in dimension 1+1 (where $T_c=\infty$) and 1+3 (where $T_c<\infty$). To characterize the localization properties of the polymer of length $L$, we analyse the statistics of the weights $w_L(\vec{r})$ of the last monomer as follows. We numerically compute the probability distributions $P_1(w)$ of the maximal weight $w_L^{max}=ma{x}_{\vec{r}}[w_L(\vec{r})]$, the probability distribution $\Pi (Y_2)$ of the parameter $Y_2(L)={\sum }_{\vec{r}}{w}_L^2(\vec{r})$ as well as the average values of the higher order moments $Y_k(L)={\sum }_{\vec{r}}{w}_L^k(\vec{r})$. We find that there exists a temperature $T_{gap}<T_c$ such that (i) for $T<T_{gap}$, the distributions $P_1(w)$ and $\Pi (Y_2)$ present the characteristic Derrida-Flyvbjerg singularities at $w=1/n$ and $Y_2=1/n$ for $n=1,\mathrm{2..}$. In particular, there exists a temperature-dependent exponent $\mu (T)$ that governs the main singularities $P_1(w)\sim (1-w{)}^{\mu (T)-1}$ and $\Pi (Y_2)\sim (1-Y_2{)}^{\mu (T)-1}$ as well as the power-law decay of the moments $\overline{Y_k(i)}\sim 1/k^{\mu (T)}$. The exponent $\mu (T)$ grows from the value $\mu (T=0)=0$ up to $\mu (T_{gap})\sim 2$. (ii) for $T_{gap}<T<T_c$, the distribution $P_1(w)$ vanishes at some value $w_0(T)<1$, and accordingly the moments $\overline{Y_k(i)}$ 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.

13 pages, 29 figures

References in corpus (3)

• Exact scaling functions for one-dimensional stationary KPZ growth
• Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)
• Freezing transition of the random bond RNA model: statistical properties of the pairing weights