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Abstract We study Busemann functions, semi-infinite geodesics, and competition interfaces in the exactly solvable last-passage percolation with inhomogeneous exponential weights. New phenomena concerning geodesics arise due to inhomogeneity. These include novel Busemann functions associated with flat regions of the limit shape and thin rectangles, semi-infinite geodesics with intervals of asymptotic directions, non-trivial axis-directed geodesics, intervals with no geodesic directions, and isolated geodesic directions. We further observe a new dichotomy for competition interfaces and second-class customers in a series of memoryless continuous-time queues with inhomogeneous service rates: a second-class customer either becomes trapped or proceeds through the service stations at strictly positive speed.
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This content will become publicly available on September 18, 2026
Large-parameter asymptotics of generalized Hastings–McLeod functions
Abstract The generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths, and non-intersecting Brownian bridges on the circle. We establish the leading-order asymptotic behavior of the generalized Hastings–McLeod functions as the inhomogeneous parameter approaches infinity using the Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems. This analysis is done in both the pole-free region and pole region. The asymptotic formulae show excellent agreement with numerically computed solutions in both regions.
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- Award ID(s):
- 2108019
- PAR ID:
- 10671868
- Publisher / Repository:
- IOP
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 38
- Issue:
- 9
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 095027
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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