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

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 $Q(x)\asymp {\prod }_{j=1}^d(1+\mathrm{\mid }{x}_j\mathrm{\mid }{)}^{-{\alpha }_j}$, 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 $-p(\beta )\asymp {\beta }^{4\mathrm{/}(2-\kappa )}$ as $\beta \downarrow 0$. 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 $Q(x)\asymp (1+\mathrm{\mid }x\mathrm{\mid }{)}^{-\vartheta }$, when $d\ge 3,\vartheta =2$ and $d=2,\vartheta \ge 2$, 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.