Laplacian growth in a channel and Hurwitz numbers

Abstract

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in the transverse direction. Similar to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution to the hierarchy is different. We characterize this solution by the string equations and construct the corresponding dispersionless tau-function. This tau-function is shown to coincide with the genus-zero part of the generating function for double Hurwitz numbers.

28 pages, 6 figures, minor corrections, references added

References in corpus (5)

• Normal random matrix ensemble as a growth problem
• Matrix Models for Random Partitions
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
• Random Matrices in 2D, Laplacian Growth and Operator Theory
• Growth processes related to the dispersionless Lax equations

Cited by in corpus (4)

• Symmetric solutions to dispersionless 2D Toda hierarchy, Hurwitz numbers and conformal dynamics
• Formal solution to the KP hierarchy
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics