3-DIMENSIONAL LATTICE WALKS CONFINED TO THE POSITIVE OCTANT PROPOSED BY KILIAN RASCHEL

Many recent papers deal with the enumeration of 2-dimensional walks steps confined to the positive quarter plane N2 , see the seminal paper [4] The aim of the present research topic is to explore the analogous dimensional walks confined to the positive octant N3 . To summarize, the interface of three different areas of mathematics:

with prescribed or [3]. problem for 3this topic is at

• combinatorics; • probability theory; • geometry. On the combinatorial side, the aim is to properly construct and classify these models. For that purpose, the reference paper will be [2]. On the probabilistic side, there are many interesting questions regarding the first exit time from the octant: in particular, what is the asymptotics of the persistence probability in the cone? We will use results in [5]. Surprisingly, it is shown [5] that fine geometric properties of the cone are involved in these random walk problems. In dimension 3, different aspects of spherical geometry (see for instance the chapter 18 in [1]) will be used. References [1] Berger, Marcel. Geometry. II. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987 [2] Bostan, Alin; Bousquet-M´elou, Mireille; Kauers, Manuel; Melczer, Stephen. On 3-dimensional lattice walks confined to the positive octant. Ann. Comb. 20 (2016), no. 4, 661–704 [3] Bostan, Alin; Raschel, Kilian; Salvy, Bruno Non-D-finite excursions in the quarter plane. J. Combin. Theory Ser. A 121 (2014), 45–63 [4] Bousquet-M´elou, Mireille; Mishna, Marni. Walks with small steps in the quarter plane. Algorithmic probability and combinatorics, 1–39, Contemp. Math., 520, Amer. Math. Soc., Providence, RI, 2010 [5] Denisov, Denis; Wachtel, Vitali. Random walks in cones. Ann. Probab. 43 (2015), no. 3, 992–1044 ´de ´ration de recherche Denis Poisson & Laboratoire de Mathe ´matiques et CNRS & Fe ´orique, Universite ´ de Tours, Parc de Grandmont, 37200 Tours, France Physique The

Date: November 24, 2017. 1

3-DIMENSIONAL LATTICE WALKS CONFINED TO ...

Springer-Verlag, Berlin, 1987. [2] Bostan, Alin; Bousquet-Mélou, Mireille; Kauers, Manuel; Melczer, Stephen. On 3-dimensional lattice walks confined to the positive octant. Ann. Comb. 20 (2016), no. 4, 661–704. [3] Bostan, Alin; Raschel, Kilian; Salvy, Bruno Non-D-finite excursions in the quarter plane. J. Combin. Theory Ser ...
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