Heat kernel on affine buildings

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Bartosz Trojan , Uniwersytet Wroclawski, Poland
Fine Hall 314

Let $\mathscr{X}$ be a thick affine building of rank $r+1$. We consider a finite range isotropic random walk on vertices of $\mathscr{X}$. Our main focus is to obtain the optimal global upper and lower bounds for the $n$'th iteration of the transition operator uniform in the region $\text{dist}\big(n^{-1} \delta(O, x), \partial \mathcal{M}\big) \geq Kn^{-1/\eta}$ where $\delta$ is the generalized distance and $\mathcal{M}$ is the convex hull of $\big\{\delta(O, x) : p(O, x) > 0\big\}$.