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Weyl chambers, symmetric spaces and number variance saturation. (English) Zbl 1112.58038

It is known that the Brownian motion in a Weyl chamber starting at the origin may be interpreted as the radial part of the standard Brownian motion on a flat symmetric space of Euclidean type [see D. J. Grabiner, Ann. Inst. Henri Poincaré, Probab. Stat. 35, No. 2, 177–204 (1999; Zbl 0937.60075) and Ph. Bougerol and Th. Jeulin, Probab. Theory Relat. Fields 124, No. 4, 517–543 (2002; Zbl 1020.15024)]. The authors study a similar relationship between the Brownian motion with drift in a Weyl chamber and the radial part of the Brownian motion on a non-compact symmetric space with negative curvature. In the type A case, this approach is used to give a new random matrix interpretation for a combinatorial model proposed by K. Johansson [Commun. Math. Phys. 252, No. 1–3, 111–148 (2004; Zbl 1112.82036)] as a means of constructing a point process emulating the number variance saturation behavior of the Riemann zeta zeroes.

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60B99 Probability theory on algebraic and topological structures
60J65 Brownian motion
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
15B52 Random matrices (algebraic aspects)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
53C35 Differential geometry of symmetric spaces
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