AI 中文总结
研究具有幂律尾和空间相关噪声的1 + 1维有向聚合物，通过截断比较论证和多项式混沌不变性原理，确定中间无序区域对数配分函数极限涨落及临界尾指数，给出不同条件下的标度极限结果。
AI 中文摘要
我们研究了由幂律尾变量生成的空间相关随机环境中的(1 + 1)维有向聚合物：\(\omega(i,x)=\sum_{y\in\mathbb Z}\psi_{y - x}\xi(i,y)\)，\(\psi_y\sim \lambda_r |y|^{-r}\)，\(r\in(1/2,1)\)，其中变量\(\xi(i,y)\)是独立同分布的，且具有指数为\(\alpha>2\)的正则变化右尾。环境的空间协方差具有赫斯特参数\(H=\frac{3}{2}-r\in(1/2,1)\)的长程衰减。我们确定了中间无序区域中对数配分函数的极限涨落，并表明临界尾指数为\(\alpha_c=\frac{3}{H}=\frac{6}{3 - 2r}\)。当\(\alpha>\alpha_c\)时，模型具有与相应高斯空间相关聚合物相同的标度极限；当\(2<\alpha\leq\alpha_c\)时，在特定尺度下对数配分函数仍满足高斯涨落。主要方法是适用于长程移动平均环境的截断比较论证以及多项式混沌的不变性原理。由于环境的非局部性，进行了远场近场分析和多尺度分析来证明截断版本在相应尺度下不改变对数配分函数。
英文摘要
We study a $(1+1)$-dimensional directed polymer in a spatially correlated random environment generated by power-law tail variables: $ω(i,x)=\sum_{y\in\mathbb Z}ψ_{y-x}ξ(i,y), ψ_y\sim λ_r |y|^{-r}, r\in(1/2,1)$, where the variables $ξ(i,y)$ are i.i.d. and have a regularly varying right tail with exponent $α>2$. The spatial covariance of the environment has long-range decay with Hurst parameter $H=\frac32-r\in(1/2,1)$. We identify the limiting fluctuations of the log-partition function in the intermediate disorder regime and show that the critical tail exponent is $α_c=\frac{3}{H}=\frac{6}{3-2r}$. When $α>α_c$, the model has the same scaling limits as the corresponding Gaussian spatially correlated polymer: if $β_NN^{H/2}\toβ\in(0,\infty)$, the centered log-partition function converges to the logarithm of the solution of the stochastic heat equation driven by fractional spatial noise; if $β_NN^{H/2}\to0$, its normalized fluctuation converges to a centered Gaussian law. In the regime $2<α\leα_c$, at the scale $β_NN^{H/2}=βN^{H/2}/l(N^{3/2})$, the log-partition function still satisfies Gaussian fluctuation. The main ingredient is a truncation comparison argument adapted to long-range moving-average environments, together with an invariance principle for polynomial chaos. Due to the non-locality of the environments, we perform a far-near field analysis, as well as multiscale analysis, to prove that the truncated version does not change the log-partition function at the corresponding scales.
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