arXiv 1202.43982012-02math.PRcond-mat.stat-mech

$1+1$维定向高分子的中间无序区

The intermediate disorder regime for directed polymers in dimension $1+1$

Tom Alberts, Konstantin Khanin, Jeremy Quastel

备注：: Published in at http://dx.doi.org/10.1214/13-AOP858 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Probability 2014, Vol. 42, No. 3, 1212-1256
DOI：: 10.1214/13-AOP858

摘要：

我们引入了一个新的无序制度的定向聚合物在尺寸$1+1$之间的弱和强的无序制度。我们称之为中间无序状态。当聚合物长度$n$趋于无穷大时，通过将逆温度参数$β$缩放为零来访问它。缩放的自然选择是$β_n:=βn^{-1/4}$。我们表明，在这种缩放下的聚合物措施有以前看不见的行为。虽然聚合物终点的波动指数和对数分拆函数与简单随机游走（$ζ=1/2,χ=0$）的波动指数相同，但波动本身是不同的。这些波动仍然受到随机环境的影响，并且没有聚合物测量的自平均。特别地，聚合物端点的广义过程的随机分布在律上（在空间的扩散标度下）收敛到实数线上的随机绝对连续测量。该测度的随机性是从平稳过程$A_β$继承的，该平稳过程具有最近发现的交叉分布作为其单点边缘，对于大的$β$，其成为GUE Tracy-Widom分布。我们还证明了存在一个限制律的四参数场的聚合物转移概率，可以描述的随机热方程。特别是，在这个弱噪声限制，我们得到的点到点的自由能波动的GUE Tracy-Widom分布的收敛。我们强调，所获得的标度行为是普遍的，并不依赖于法律的混乱。

英文摘要：

We introduce a new disorder regime for directed polymers in dimension $1+1$ that sits between the weak and strong disorder regimes. We call it the intermediate disorder regime. It is accessed by scaling the inverse temperature parameter $β$ to zero as the polymer length $n$ tends to infinity. The natural choice of scaling is $β_n:=βn^{-1/4}$. We show that the polymer measure under this scaling has previously unseen behavior. While the fluctuation exponents of the polymer endpoint and the log partition function are identical to those for simple random walk ($ζ=1/2,χ=0$), the fluctuations themselves are different. These fluctuations are still influenced by the random environment, and there is no self-averaging of the polymer measure. In particular, the random distribution of the polymer endpoint converges in law (under a diffusive scaling of space) to a random absolutely continuous measure on the real line. The randomness of the measure is inherited from a stationary process $A_β$ that has the recently discovered crossover distributions as its one-point marginals, which for large $β$ become the GUE Tracy-Widom distribution. We also prove existence of a limiting law for the four-parameter field of polymer transition probabilities that can be described by the stochastic heat equation. In particular, in this weak noise limit, we obtain the convergence of the point-to-point free energy fluctuations to the GUE Tracy-Widom distribution. We emphasize that the scaling behaviour obtained is universal and does not depend on the law of the disorder.

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英文摘要：

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