The totally asymmetric simple exclusion process (TASEP) is one of the most studied models in nonequilibrium statistical mechanics: particles hop rightward on a one-dimensional lattice, subject to the constraint that no site can hold more than one particle. The q-deformation (q-TASEP) replaces the constant hopping rate with one that depends on the gap to the next particle — an interaction that introduces correlations without sacrificing exact solvability. Both belong to the KPZ universality class, which governs a vast family of random growth processes.
Krajenbrink and Le Doussal (arXiv 2602.23209, February 2026) find that the q-TASEP's weak noise limit reveals classical integrability — not the quantum integrability of the original stochastic model, but an emergent classical structure with explicit Lax pairs and solvable scattering problems. This is the first instance of classical integrability appearing in a stochastic particle system where signatures of the Poisson noise persist in the limit.
The distinction matters. Weak noise limits of stochastic systems typically yield deterministic hydrodynamic equations — mean-field descriptions where the noise has been averaged out. The q-TASEP is different. Because the hopping is Poisson rather than Gaussian, the noise has a discrete, arrival-time character that doesn't wash out in the weak noise limit. An intermediate mesoscopic regime survives where the system is nearly deterministic but the Poisson structure still shapes the large deviation probabilities — the rare fluctuations far from typical behavior.
The weak noise equations that govern these large deviations form a system of nonlinear differential equations. The surprise is that these equations are classically integrable: they admit a Lax pair representation, which means they can be solved by inverse scattering. The Lax pair converts the nonlinear problem into a linear one — a spectral problem whose eigenvalues are constants of motion. The scattering data evolves trivially in time, and the solution is reconstructed by inverting the transform.
The large deviation probabilities for particle positions are computed exactly from the scattering solution. The results match the asymptotic expansion of the known Fredholm determinant formula for the q-TASEP, validating the approach from both the integrable-systems side and the exact-solution side independently.
Stochastic noise, in the right limit, organizes itself into an integrable structure. The randomness doesn't destroy the order — it generates it.