Martin boundary of a killed random walk in the Weyl chamber of the dual of SU(3)

Kilian Raschel (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider a classical random walk killed at the boundary of the quarter plane $Z_{+}^{2}$ and with drift zero, that appears quite naturally in the study of quantum random walks on the dual of SU(3). In [Quantum random walk on the dual of SU(n), Probab. Theory Related Fields, 1991], P. Biane has computed the asymptotic of the Green functions of this process along the paths of states $(i,j)$ in $Z_{+}^{2}$ such that $i+j\to \infty$ and $j/i\to\tan(\gamma)$, for any $\gamma\in]0,\pi/2[$. In this note we wish to extend his results up to $\gamma\in[0,\pi/2]$. In particular, this will allow us to prove that the Martin boundary of the process is reduced to one point.

Mots clés: absorption probabilities; Random walk; Green functions; Martin boundary; absorption probabilities.

Domaines: Probabilités [math.PR]

Kilian Raschel. Martin boundary of a killed random walk in the Weyl chamber of the dual of SU(3). 2009. ⟨hal-00425651v1⟩