具有乘积型和径向空间相关性的布朗定向聚合物的无序阈值与自由能
Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations
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- Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
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中文总结 AI 辅助
该研究分析时间白噪声、空间长程相关高斯环境中布朗定向聚合物的无序阈值，确定乘积型协方差下的无序转变条件，解决径向协方差的遗留问题，给出自由能的渐近行为并说明证明方法。
中文摘要 AI 辅助
我们研究时间为白噪声、空间为具有长程相关性的有色噪声的中心高斯环境中的布朗定向聚合物。对于乘积型协方差 \\(Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-\alpha_j}\\)（其中 \\(\alpha_j\in(0,1)\\)，\\(\kappa=\sum_j\alpha_j\\)），我们确定了在边际值 \\(\kappa=2\\) 处的无序转变：当 \\(\kappa>2\\) 时，足够小的逆温度下存在弱无序；当 \\(\kappa<2\\) 时，淬火自由能 \\(p(\beta)\\) 满足 \\(-p(\beta)\asymp\beta^{4/(2-\kappa)}\\)（\\(\beta\downarrow0\\)）；当 \\(\kappa=2\\) 时，所有 \\(\beta>0\\) 下存在强无序，且足够小的 \\(\beta\\) 时 \\(p(\beta)=0\\)，故 \\(\beta_c=0<\bar\beta_c\\)。我们还考虑径向协方差情形 \\(Q(x)\asymp(1+|x|)^{-\vartheta}\\)，该情形是 Lacoin[2011] 遗留的未解决问题：当 \\(d=2\\)、\\(\vartheta>2\\) 时，\\(\ln(-p(\beta))\asymp-\beta^{-2}\\)；当 \\(d=2\\)、\\(\vartheta=2\\) 时，\\(\ln(-p(\beta))\asymp-\beta^{-1}\\)。证明方法包括副本耦合、Feynman--Kac 变分公式、重叠方法及带有有序 Wiener 混沌测度变换的连续空间分数矩。
英文摘要
We study a Brownian directed polymer in a centered Gaussian environment that is white in time and colored in space having long-range spatial correlations. For product-type covariances \(Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-α_j}\), with \(α_j\in(0,1)\) and \(κ=\sum_jα_j\), we identify the disorder transition at the marginal value \(κ=2\). For \(κ>2\), weak disorder holds at sufficiently small inverse temperature; for \(κ<2\), the quenched free energy $p(β)$ satisfies \(-p(β)\asympβ^{4/(2-κ)}\) as \(β\downarrow0\). For \(κ=2\), strong disorder holds for every \(β>0\), while \(p(β)=0\) for all sufficiently small \(β\), so \(β_c=0<\barβ_c\). We also consider the radial covariance cases, where $ Q(x)\asymp(1+|x|)^{-\vartheta}$, when \(d\ge3,\vartheta=2\) and \(d=2,\vartheta\ge2\), which was left unanswered in Lacoin~\cite{Lacoin2011}. When $d=2$, we get \(\ln(-p(β))\asymp-β^{-2}\) for \(\vartheta>2\) and \(\ln(-p(β))\asymp-β^{-1}\) for \(\vartheta=2\). The proofs consist of replica coupling, Feynman--Kac variational formula, overlap methods, and continuous-space fractional moments with ordered Wiener-chaos changes of measure.
