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arxiv: 2605.13512 · v1 · submitted 2026-05-13 · 🧮 math.PR

Recognition: unknown

Hydrodynamic limits for TASEP with space-time discontinuities

Jacob Butt , Nicos Georgiou , Enrico Scalas
Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:20 UTC · model grok-4.3

classification 🧮 math.PR
keywords TASEPhydrodynamic limitlast-passage percolationHamilton-Jacobi equationdiscontinuous coefficientsvariational couplingconservation law
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The pith

A height-dependent TASEP with space-time discontinuous jump rates has a hydrodynamic limit given by a Lax-Oleinik variational formula for the current.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a hydrodynamic limit for a version of TASEP in which jump rates depend on both position and current height, with the rate function allowed to jump along finitely many curves. By coupling the process to an inhomogeneous last-passage percolation model and extending variational methods, they show convergence of the height function and particle density to a deterministic limit. The limiting current is characterized variationally and satisfies a Hamilton-Jacobi equation where the speed is continuous and differentiable, or an envelope viscosity solution at jumps. This extends classical hydrodynamic theory to settings with abrupt changes in the driving rates.

Core claim

Combining the law of large numbers for the associated inhomogeneous directed last-passage percolation model with an extension of the variational coupling method, the authors prove a hydrodynamic limit for the height function and associated particle density. The limiting current is characterised by a Lax-Oleinik type variational formula built from the discontinuous last-passage shape function. At points of differentiability and continuity of the coefficient, this current solves a Hamilton-Jacobi equation with discontinuous dependence on space and height. At discontinuities, it satisfies a natural envelope-based discontinuous viscosity formulation. For coefficients with only spatial discontinu

What carries the argument

The Lax-Oleinik variational formula constructed from the discontinuous last-passage shape function, which selects the limiting current and determines the PDE structure.

If this is right

  • The particle density follows the maximal-current weak solution of the conservation law with discontinuous flux when discontinuities are only spatial.
  • The limiting current is the unique nondecreasing Lipschitz solution to the envelope-based discontinuous viscosity Hamilton-Jacobi equation.
  • The microscopic dynamics naturally select the envelope solution at discontinuity curves of the speed function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method could be adapted to other particle systems or growth models with space-time varying rates.
  • Testing the rate of convergence near discontinuity curves in simulations would provide further evidence.
  • Applications may include modeling traffic or biological transport with abrupt environmental changes.

Load-bearing premise

The speed function has discontinuities along only locally finitely many curves so that the variational coupling and envelope formulation can be controlled at the jumps.

What would settle it

A simulation of the process with a speed function discontinuous across one space-time curve in which the observed height function deviates from the predicted Lax-Oleinik variational value would falsify the limit.

Figures

Figures reproduced from arXiv: 2605.13512 by Enrico Scalas, Jacob Butt, Nicos Georgiou.

Figure 1
Figure 1. Figure 1: Graphical representation of the TASEP and evolution of the height function z n (t) for a fixed n. When a particle has space to jump, the height function will be in a configuration that a top-left corner can be flipped to a low-right corner. All these locations are the growth sites, and they are marked with a shaded square below the level of z n . In the diagram, the particle at location i − 1 will jump at … view at source ↗
Figure 2
Figure 2. Figure 2: In this figure you can see a schematic of the decomposition of the discontinuity curves allowed by Assumption 2.8. For example, while h3 and h4 can be viewed as one curve h that is a graph, the monotonicity assumption would break since there is a local maximum for h. Thus, that local maximum becomes a terminal point, and h is split into two monotone pieces. Straight line discontinuities are allowed, as sho… view at source ↗
Figure 3
Figure 3. Figure 3: Left: Representation of the envelope property, showing relationship between −z n (0) and the auxiliary processes ξ n . Each ξ is placed so that it is above −z n but touching at the starting point (k, −z n k (0)). As all coupled processes evolve, all ξ n remain above, and at least one remains tangent to −z n k (t) for any k at each spatial point. Right: The same relationship remains in the limit; the scaled… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic of the points at the first part of the proof of Theorem 2.13. The boldfaced increasing path is a near optimiser for the passage time from (z0, −v0(z0)) up to the point (x0, −v(x0, t0)) and it intersects the t − h level curve at point (xh, yh). This gives us that (4.6) 1 ≥ x0 − xA hc˜ high D (h) . Similarly, by using point B we have t ≥ t − h + 1 c˜ high D (h) (y0 − yB) = t − h + 1 c˜ high D (h) (… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic proof of the argument in Lemma 4.6. The two thick level curves are those for the passage time starting from (q0, −v0(q0)) reaching level t − h and t. The thick dashed level curve is an h-level curve starting from Q = (q, gq0 (q − q0, t − h) − v0(q0)). The thick path is a concatenation of an optimal path from (q0, −v0(q0)) to Q with weight t − h and an optimal path from Q to (x, y) with weight h. … view at source ↗
Figure 6
Figure 6. Figure 6: Diagram for the proof of Lemma 4.8. The increasing path is a near optimiser for the wedge last passage time shape function (3.5) up to (x, −v(x, t)). Since that point belongs on the t-level curve, a near optimiser will have weight t − e(h) with e(h) as near zero as we wish. It will also intersect the t − h level curve at some point (xh, yh). This naturally splits w into two pieces, where the weight of each… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proves a hydrodynamic limit for the height function (and associated density) of a height-dependent TASEP whose jump rates are sampled from a macroscopic speed function that may be discontinuous along locally finitely many curves in space-time. The argument couples the microscopic process to an inhomogeneous directed last-passage percolation model, invokes an external LLN for the LPP shape function, and extends the variational coupling method to pass to the limit; the limiting current is characterized by a Lax-Oleinik variational formula that yields a discontinuous-viscosity Hamilton-Jacobi equation whose Hamiltonian depends on both the spatial variable and the solution value itself. Uniqueness of the HJ solution (and identification of its derivative as the maximal-current weak solution of the associated conservation law) is established only when the discontinuities are purely spatial.

Significance. If the technical controls hold, the result supplies the first hydrodynamic theory for TASEP-type models whose coefficients are allowed to jump in both space and time. The combination of an inhomogeneous LPP law of large numbers with a variational-coupling argument, together with the envelope formulation that selects the correct weak solution at discontinuities, provides a reusable template for other particle systems with discontinuous rates. The explicit identification of the selected PDE structure is a clear strength.

major comments (2)
  1. [§5] §5 (variational coupling extension): local finiteness of the discontinuity curves is invoked to control the coupling and envelope formulation at jumps, yet the argument supplies no uniform bound on the number of crossings per path under hydrodynamic scaling; repeated crossings could permit o(1) errors to accumulate before the limit is taken, and this bound is load-bearing for the space-time case.
  2. [§6] §6 (uniqueness): uniqueness of the Hamilton-Jacobi solution in the class of nondecreasing Lipschitz currents is proved only for purely spatial discontinuities; the space-time case therefore rests entirely on the variational characterization without an independent uniqueness statement, leaving open the possibility that other solutions to the envelope formulation exist.
minor comments (2)
  1. [§2] The notation distinguishing the macroscopic speed function from the microscopic rate function is introduced only in the abstract and could be restated explicitly at the beginning of §2.
  2. Figure 1 (schematic of discontinuity curves) would benefit from a caption that explicitly labels the local-finiteness assumption.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and for identifying these two technical points that require clarification. We address each major comment below.

read point-by-point responses

  1. Referee: [§5] §5 (variational coupling extension): local finiteness of the discontinuity curves is invoked to control the coupling and envelope formulation at jumps, yet the argument supplies no uniform bound on the number of crossings per path under hydrodynamic scaling; repeated crossings could permit o(1) errors to accumulate before the limit is taken, and this bound is load-bearing for the space-time case.

    Authors: We agree that an explicit uniform bound on the number of crossings is needed for rigor. Because the discontinuity curves are locally finite and the variational paths are 1-Lipschitz, the number of crossings per path in any compact space-time region is bounded by a constant depending only on the local geometry and the Lipschitz constant. We will insert a short lemma establishing this bound under hydrodynamic scaling and verify that the resulting error remains o(1) uniformly; the revised manuscript will contain this estimate. revision: yes

  2. Referee: [§6] §6 (uniqueness): uniqueness of the Hamilton-Jacobi solution in the class of nondecreasing Lipschitz currents is proved only for purely spatial discontinuities; the space-time case therefore rests entirely on the variational characterization without an independent uniqueness statement, leaving open the possibility that other solutions to the envelope formulation exist.

    Authors: The referee correctly notes that uniqueness of the discontinuous-viscosity solution is proved only when discontinuities are purely spatial. In the space-time setting the limiting current is identified directly as the hydrodynamic limit of the microscopic height function, which is given by the Lax-Oleinik variational formula constructed from the LPP shape function. This variational object is the unique candidate selected by the particle system; however, we do not supply an independent uniqueness theorem for the envelope formulation when the Hamiltonian depends on both space and time. Such a result would require additional analytic tools and lies outside the scope of the present work. revision: no

standing simulated objections not resolved
  • Uniqueness of the envelope formulation for the Hamilton-Jacobi equation with space-time discontinuities

Circularity Check

0 steps flagged

No significant circularity; derivation combines external LLN with variational coupling extension

full rationale

The paper's central result is obtained by combining a law of large numbers for the inhomogeneous last-passage percolation model with an extension of the variational coupling method, then verifying that the limit satisfies a Lax-Oleinik variational formula and a discontinuous viscosity formulation of the Hamilton-Jacobi equation. No step reduces a prediction to a fitted input by construction, nor does any load-bearing claim rest solely on a self-citation whose content is unverified within the paper. The local-finiteness assumption on discontinuity curves is an explicit hypothesis used to control crossings in the coupling, not a self-definitional loop. Uniqueness of the HJ solution is proven only in the spatial-discontinuity case; the space-time case is characterized variationally without claiming uniqueness, which is consistent with the stated scope. The derivation therefore remains self-contained against the external LLN benchmark and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on the law of large numbers for inhomogeneous directed last-passage percolation (standard in the field) and an extension of variational coupling; no free parameters or new invented entities are introduced.

axioms (1)
  • domain assumption Law of large numbers for inhomogeneous directed last-passage percolation with discontinuous rates
    Invoked to obtain the limiting shape function from which the variational formula is built.

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