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

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 $w_L^{max}= max_{\vec r} [w_L(\vec
r)]$, 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 $n=1,2..$. 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 $\bar{Y_k(i)} \sim 1/k^{\mu(T)}$. 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 $\bar{Y_k(i)}$ 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.Comment: 13 pages, 29 figure