Hydrodynamic limits for TASEP with space-time discontinuities
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Butt, Jacob Georgiou, Nicos Scalas, Enrico |
| author_facet | Butt, Jacob Georgiou, Nicos Scalas, Enrico |
| contents | We develop a hydrodynamic theory for a height-dependent version of the totally asymmetric simple exclusion process in which the jump rate at a growth site is sampled from a macroscopic two-dimensional speed function evaluated at the spatial coordinate and the current height level. The speed function is allowed to have discontinuities along locally finitely many curves. Through the TASEP height-function representation, the process is coupled to an inhomogeneous directed last-passage percolation model whose exponential rates vary discontinuously in the two macroscopic LPP coordinates. Combining the law of large numbers for this last-passage model with an extension of the variational coupling method, we prove a hydrodynamic limit for the height function and for the associated particle density.
The limiting current is characterised by a Lax-Oleinik type variational formula built from the discontinuous last-passage shape function. We then identify the first-order PDE structure selected by the microscopic dynamics. At points of differentiability of the limiting current and continuity of the sampled coefficient, the current solves a Hamilton-Jacobi equation whose Hamiltonian depends discontinuously on the spatial variable and on the value of the solution itself. At discontinuities, the variational formula leads to a natural envelope-based discontinuous viscosity formulation, and we prove that the limiting current satisfies this formulation. Finally, when the coefficient has only spatial discontinuities, we prove uniqueness of the Hamilton-Jacobi solution in the natural class of nondecreasing Lipschitz currents, and identify its spatial derivative as the maximal-current weak solution of the associated scalar conservation law with discontinuous flux. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13512 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hydrodynamic limits for TASEP with space-time discontinuities Butt, Jacob Georgiou, Nicos Scalas, Enrico Probability 60K35, 35L65 (Primary) 35F21, 35D40, 60K37, 82C22 (Secondary) We develop a hydrodynamic theory for a height-dependent version of the totally asymmetric simple exclusion process in which the jump rate at a growth site is sampled from a macroscopic two-dimensional speed function evaluated at the spatial coordinate and the current height level. The speed function is allowed to have discontinuities along locally finitely many curves. Through the TASEP height-function representation, the process is coupled to an inhomogeneous directed last-passage percolation model whose exponential rates vary discontinuously in the two macroscopic LPP coordinates. Combining the law of large numbers for this last-passage model with an extension of the variational coupling method, we prove a hydrodynamic limit for the height function and for the associated particle density. The limiting current is characterised by a Lax-Oleinik type variational formula built from the discontinuous last-passage shape function. We then identify the first-order PDE structure selected by the microscopic dynamics. At points of differentiability of the limiting current and continuity of the sampled coefficient, the current solves a Hamilton-Jacobi equation whose Hamiltonian depends discontinuously on the spatial variable and on the value of the solution itself. At discontinuities, the variational formula leads to a natural envelope-based discontinuous viscosity formulation, and we prove that the limiting current satisfies this formulation. Finally, when the coefficient has only spatial discontinuities, we prove uniqueness of the Hamilton-Jacobi solution in the natural class of nondecreasing Lipschitz currents, and identify its spatial derivative as the maximal-current weak solution of the associated scalar conservation law with discontinuous flux. |
| title | Hydrodynamic limits for TASEP with space-time discontinuities |
| topic | Probability 60K35, 35L65 (Primary) 35F21, 35D40, 60K37, 82C22 (Secondary) |
| url | https://arxiv.org/abs/2605.13512 |