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Let \(\mathscr{X}\) be a thick affine building of rank \(r+1\). We consider a finite range
isotropic random walk on vertices of \(\mathscr{X}\). We show sharp lower and upper estimates
on \(p_n\), the \(n\)'th iteration of the transition operator, uniform in the region
\[
\text{dist}\big((\delta(x) + \rho)/(n+r), \partial \mathcal{M}\big) \geq K n^{-1/(2\eta)}
\]
where \(\delta\) is a generalized distance from the origin and \(\mathcal{M}\) is the convex envelop
of the set
\[
\big\{\delta(x) \in P^+ : p(O, x) > 0\big\}.
\]
In particular,
Theorem: For any \(\epsilon > 0\) small enough
\[
p_n(x) \asymp n^{-r/2-|\Phi_0^+|} \rho^n
P_{\delta(x)}(0)
\exp\big({-n\phi\big(\delta(x)/ n\big)}\big)
\]
uniformly on \(\{x \in supp\ p_n :
dist(\delta(x)/n, \partial \mathcal{M}) \geq \epsilon\}\).
The basic tool in the study of isotropic random walks is the spherical Fourier transform. In
the 1970s, Macdonald developed spherical harmonic analysis for groups of \(p\)-adic
type. An application of the spherical Fourier transform results in an oscillatory integral which
we analyse by the steepest descent method. Thanks to some geometric properties of the support of
the spherical Fourier transform of \(p\), the integral can be localized. Therefore, the proof reduces
to establishing the asymptotic behaviour, as \(n\) approaches infinity, of
\[
I_n(x) = \int\limits_{|u| \leq \epsilon} e^{n \varphi(x, u)} f(x, u) du,
\]
uniformly with respect to \(x \in \mathfrak{a}_+\) where \(\mathfrak{a}_+\) is the Weyl chamber of
the underlying root system. If \(x\) lies on the wall of \(\mathfrak{a}_+\) then the function
\(\varphi(x, \cdot)\) retain symmetries in the directions orthogonal to the wall. Close to the wall
we take advantage of this by expanding \(I_n\) into its Taylor series and using combinatorial methods
to identify remaining cancellations.
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