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

Source 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 $Q(x)\asymp\prod_{j=1}^d(1+|x_j|)^{-α_j}$, 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 $κ 0$, while $p(β)=0$ for all sufficiently small $β$, so $β_c=0 2$ 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.