Numerical bounds for critical exponents of crossing Brownian motion

We consider $d$-dimensional crossing Brownian motion in a truncated Poissonian potential conditioned to reach a fixed hyperplane at distance $L$from the starting point. The transverse fluctuation of the path is expected to be of order $L^\xi$. We prove that for $d\ge2$: $\xi \le 3/4$. As a second critical exponent we introduce $\chi^{(2)}$, which describes the fluctuations of naturally defined distance functions for crossing Brownian motion. The numerical bound we obtain is an improvement of Corollary 3.1 in Scaling identity for crossing Brownian motion in a Poissonian potential (Probab. Theory Related Fields 112 (1998), 299-319), resulting in $\chi^{(2)} \ge 1/5$ if $d=2$ and if the killing rate $\lambda$ is strictly positive ($\lambda>0$).