{
  "id": "cond-mat/0702131",
  "version": "v1",
  "published": "2007-02-06T09:07:55.000Z",
  "updated": "2007-02-06T09:07:55.000Z",
  "title": "Directed polymer in a random medium of dimension 1+1 and 1+3: weights statistics in the low-temperature phase",
  "authors": [
    "Cecile Monthus",
    "Thomas Garel"
  ],
  "comment": "13 pages, 29 figures",
  "journal": "J. Stat. Mech. (2007) P03011",
  "doi": "10.1088/1742-5468/2007/03/P03011",
  "categories": [
    "cond-mat.dis-nn",
    "math.PR"
  ],
  "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 $w_L^{max}= max_{\\vec r} [w_L(\\vec r)]$, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-02-06T09:07:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "directed polymer",
      "random medium",
      "weights statistics",
      "low-temperature phase",
      "probability distribution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2007,
      "month": "Mar",
      "volume": 2007,
      "number": 3,
      "pages": 3011
    },
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007JSMTE..03...11M"
    }
  }
}