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 \(Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-α_j}\), 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.