Disorder Thresholds and Free Energy of Brownian Directed Polymers with Product and Radial Spatial Correlations

Abstract

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 \(Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-\alpha_j}\), with \(\alpha_j\in(0,1)\) and \(\kappa=\sum_j\alpha_j\), we identify the disorder transition at the marginal value \(\kappa=2\). For \(\kappa>2\), weak disorder holds at sufficiently small inverse temperature; for \(\kappa<2\), the quenched free energy $p(\beta)$ satisfies \(-p(\beta)\asymp\beta^{4/(2-\kappa)}\) as \(\beta\downarrow0\). For \(\kappa=2\), strong disorder holds for every \(\beta>0\), while \(p(\beta)=0\) for all sufficiently small \(\beta\), so \(\beta_c=0<\bar\beta_c\). 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 \(\ln(-p(\beta))\asymp-\beta^{-2}\) for \(\vartheta>2\) and \(\ln(-p(\beta))\asymp-\beta^{-1}\) 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.