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arXiv 0912.2875cond-mat.dis-nn
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无序系统中典型涨落与大偏差的匹配:应用于SK自旋玻璃模型中基态能量的统计

Matching between typical fluctuations and large deviations in disordered systems : application to the statistics of the ground state energy in the SK spin-glass model

Cecile Monthus, Thomas Garel

摘要:

对于无序系统中的全局可观量统计,我们讨论了典型涨落与大偏差之间的匹配。我们专注于统计的基态能量$E_0$在两种类型的无序模型:(i)为定向聚合物的长度$N$在一个二维介质中,其中存在许多精确的结果(ii)为谢灵顿-柯克帕特里克自旋玻璃模型的$N$自旋,其中各种可能性已被提出。这里我们强调,除了定义波动指数$ω_f$的无序平均$E_0^{av}(N)$和标准差$ ΔE_0(N) \sim N^{ω_f}$的行为之外,研究重新标度的变量$u= \frac{E_0(N)-E_0^{av}(N)}{ΔE_0(N)}$的全概率分布$Π(u)$是非常有指导意义的:(a)在数字上,向$Π(u)$的收敛通常非常迅速,因此相当小尺寸但具有高统计量的数据允许测量定义为$\ln Π(u \to \pm \infty) \sim - | u |^{η_{\pm}}$的双尾指数$η_{\pm}$。在一般情况下,$1< η_{\pm} < +\infty$,这导致显式的非平凡条款的渐近行为的时刻$\bar{Z_N^n}$的配分函数时,组合$[| n | N^{ω_f}]$变得大(b)简单的罕见事件参数通常可以找到获得明确的关系$η_{\pm}$和$ω_f$。这些罕见的事件通常对应于广义形式$R(w_{\pm} = \frac{E_0(N)-E_0^{av}(N)}{N^{κ_{\pm}}}) \sim e^{- N^{ρ_{\pm}} {\cal R}_{\pm}(w_{\pm})}$的“异常”大偏差性质("通常“大偏差形式对应于$κ_{\pm}=1=ρ_{\pm}$)。

英文摘要:

For the statistics of global observables in disordered systems, we discuss the matching between typical fluctuations and large deviations. We focus on the statistics of the ground state energy $E_0$ in two types of disordered models : (i) for the directed polymer of length $N$ in a two-dimensional medium, where many exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass model of $N$ spins, where various possibilities have been proposed. Here we stress that, besides the behavior of the disorder-average $E_0^{av}(N)$ and of the standard deviation $ ΔE_0(N) \sim N^{ω_f}$ that defines the fluctuation exponent $ω_f$, it is very instructive to study the full probability distribution $Π(u)$ of the rescaled variable $u= \frac{E_0(N)-E_0^{av}(N)}{ΔE_0(N)}$ : (a) numerically, the convergence towards $Π(u)$ is usually very rapid, so that data on rather small sizes but with high statistics allow to measure the two tails exponents $η_{\pm}$ defined as $\ln Π(u \to \pm \infty) \sim - | u |^{η_{\pm}}$. In the generic case $1< η_{\pm} < +\infty$, this leads to explicit non-trivial terms in the asymptotic behaviors of the moments $\bar{Z_N^n}$ of the partition function when the combination $[| n | N^{ω_f}]$ becomes large (b) simple rare events arguments can usually be found to obtain explicit relations between $η_{\pm}$ and $ω_f$. These rare events usually correspond to 'anomalous' large deviation properties of the generalized form $R(w_{\pm} = \frac{E_0(N)-E_0^{av}(N)}{N^{κ_{\pm}}}) \sim e^{- N^{ρ_{\pm}} {\cal R}_{\pm}(w_{\pm})}$ (the 'usual' large deviations formalism corresponds to $κ_{\pm}=1=ρ_{\pm}$).

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