Geodesics and crossing Brownian motion in a soft Poissonian potential (Q1304923)

The author studies Brownian motion in a soft Poissonian potential \(\lambda+\sum_i W(x-x_i)\), \(\lambda>0\), conditioned to reach a remote location. By a shape theorem of \textit{A.-S. Sznitman} [Commun. Pure Appl. Math. 47, No. 12, 1655-1688 (1994; Zbl 0814.60022)] the exponential decay of the normalizing constant in this model as the location moves to infinity is described by a deterministic norm \(\alpha_{\lambda,\beta}\), which depends on the strength \(\beta\) of the potential. The main result describes the asymptotics of this norm as the strength of the potential goes to infinity. Namely, \(\alpha_{\lambda,\beta}/\sqrt{\beta}\) converges to a norm \(\mu_\lambda\), which appears as a limit in a shape theorem for an associated continuum first passage percolation model.