On Some Lattice Random Walks Conditioned to Stay in Weyl Chambers} \runtit{Centered random walks in Weyl chambers

V. Despax

2018, v.24, Issue 5, 733-758

ABSTRACT

This result has been generalized in~\cite{LLP}/~\cite{LLP3} for a large class of random walks/random paths on weight lattices defined from representations of simple Lie algebras/Kac\tire Moody algebras and their conditionings to always stay in Weyl chambers. In these various works, the drift of the considered random walk is assumed to be in the interior of the cone.
In this paper, where we consider the random walks that are studied in~\cite{LLP}, we introduce the conditionings to always stay in closed Weyl chambers until an instant that we let tend to infinity. Our approach allows us to recover the main result of~\cite{LLP} and, furthermore, it yields a relevant notion of conditioning to always stay in the cone in the driftless case, where the probability of this event is zero. We also prove that the laws of the Markov chains so obtained can be recovered by letting the drift be zero in the transitions matrices obtained by~\cite{LLP}. Finally, we conjecture that our result remains true for any drift in the closure of the cone.

Keywords: lattice random walks, conditioning, Weyl chambers, harmonic functions

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