Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three, there exists a cone such that the heat kernel (the continuous analogue of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles.

[1]  Ahmad El Soufi,et al.  Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold , 2007, 0705.1263.

[2]  P. Bérard.  Remarques sur la conjecture de Weyl , 1983 .

[3]  R. Bañuelos,et al.  Brownian motion in cones , 1997 .

[4]  Alin Bostan,et al.  Non-D-finite excursions in the quarter plane , 2012, J. Comb. Theory A.

[5]  R. B. Kellogg,et al.  Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain , 1977 .

[6]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[8]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[9]  R. DeBlassie,et al.  Exit times from cones in ℝn of Brownian motion , 1987 .

[10]  Tanguy Rivoal,et al.  On the values of G-functions , 2011, 1103.6022.

[11]  Kilian Raschel,et al.  On the functions counting walks with small steps in the quarter plane , 2012 .

[12]  Solution of an eigenvalue problem for the Laplace operator on a spherical surface. M.S. Thesis - Maryland Univ. , 1974 .

[13]  Kilian Raschel,et al.  Polyharmonic Functions And Random Processes in Cones , 2020, AofA.

[14]  Xuwen Zhu,et al.  Fundamental gaps of spherical triangles , 2021, Annals of Global Analysis and Geometry.

[15]  Stephen Melczer,et al.  Counting walks with large steps in an orthant , 2018, Journal of the European Mathematical Society.

[16]  Nicholas M. Katz,et al.  Nilpotent connections and the monodromy theorem: Applications of a result of turrittin , 1970 .

[17]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[18]  Marni Mishna,et al.  Walks with small steps in the quarter plane , 2008, 0810.4387.

[19]  Philippe Flajolet,et al.  Basic analytic combinatorics of directed lattice paths , 2002, Theor. Comput. Sci..

[20]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[21]  Andrejs Treibergs,et al.  A capture problem in Brownian motion and eigenvalues of spherical domains , 2005 .

[22]  B Bogosel,et al.  3D positive lattice walks and spherical triangles , 2020, J. Comb. Theory, Ser. A.

[23]  Yves André,et al.  Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité , 2000 .

[24]  Denis Denisov,et al.  Random walks in cones , 2015 .