论文标题
独立步行泊松领域中定向聚合物的缩放限制
Scaling limit of a directed polymer among a Poisson field of independent walks
论文作者
论文摘要
我们考虑了一个定向的聚合物模型$ 1+1 $，其中该疾病是由$ \ Mathbb Z $的独立随机步行系统的职业领域给出的。在适当的连续性和弱混乱限制中，我们表明，定向聚合物的淬灭分区函数家族将带有高斯噪声的乘数随机热方程（SHA）收敛到Stratonovich溶液，其时空协方差由热kernel给出。 与时空白噪声相比，她承认了Wiener-itô混乱的扩展，我们建立了$ l^1 $ - 会议的混乱，对Picard迭代产生的迭代积分扩展。使用此扩展及其离散的聚合物分区功能，通过功能分析论证和热核估计来证明扩展中术语的收敛性。泊松随机步行系统可以仔细分析，这是我们论点的重要意见。
We consider a directed polymer model in dimension $1+1$, where the disorder is given by the occupation field of a Poisson system of independent random walks on $\mathbb Z$. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation (SHE) with a Gaussian noise, whose space-time covariance is given by the heat kernel. In contrast to the case with space-time white noise where the solution of the SHE admits a Wiener-Itô chaos expansion, we establish an $L^1$-convergent chaos expansions of iterated integrals generated by Picard iterations. Using this expansion and its discrete counterpart for the polymer partition functions, the convergence of the terms in the expansion is proved via functional analytic arguments and heat kernel estimates. The Poisson random walk system is amenable to careful moment analysis, which is an important input to our arguments.
