Self-intersection local times of Volterra Gaussian bridges with large end-to-end distance

Abstract

Self-intersection local times serve as the Hamiltonian in the Gibbs measure of continuous polymer models, for which the end-to-end distance is a key quantity. We study multiple self-intersection local times of Volterra Gaussian bridges, \emph{i.e.}, Volterra Gaussian processes conditioned on their final location $a$, as $\|a\|\to\infty$. For $d$-dimensional $(2,ζ)$-locally nondeterministic processes with $ζ\in(0,2/d)$, we prove that these local times exist in every $L^p$. Under two further conditions on the kernel, the $p$-th moment of the $m$-multiple self-intersection local time of the bridge is of exact order $\|a\|^{-γ}$ for every integer $p\geqslant1$, with $γ=p(m-1)(2-ζd)/(2-ζ)$. In the plane, for kernels $k(t-s)$ with $k(0)\neq0$ and under strong local nondeterminism, we restrict the double self-intersection local time to $\{t_2-t_1\geqslant\|a\|^{-α}\}$, $α>0$, and establish a phase transition: its expectation decays like $\exp(-c\|a\|^{2-α})$, converges to an explicit positive limit, or grows logarithmically, for $α<2$, $α=2$ and $α>2$ respectively.