Closed-form computation of critical exponents for general walk models

Determine for which multidimensional orthant walk models the principal Dirichlet eigenvalue governing the excursion critical exponent can be computed in closed form, and thereby identify the models whose critical exponents admit closed-form expressions.

Background

For a walk model in the multidimensional orthant, the excursion asymptotics are governed by a critical exponent expressed through the principal Dirichlet eigenvalue of a spherical polyhedral domain obtained from the model’s covariance matrix. In dimension two this eigenvalue has an explicit formula, whereas in dimensions three and higher it is generally not available in closed form.

The paper develops a classification of polyhedral domains that are nodal domains of spherical eigenfunctions and computes their principal eigenvalues. This resolves the question for an important class of models, but does not characterize all walk models for which the eigenvalue, and hence the critical exponent, can be computed explicitly.

References

While the formula eq:DW_exponent characterises the critical exponent in eq:one-term-asymp, it is not clear a priori for which walk models $\lambda_1$ can be computed in closed form.

— Enumeration of walks in multidimensional orthants and reflection groups  (2501.05654 - Gohier et al., 10 Jan 2025) in Section “Focus on two important tools,” Introduction and main results