Mesoscopic fluctuation theory of particle systems driven by Poisson noise: study of the $q$-TASEP

We pursue our study of integrable weak noise theories of directed polymer and interacting particle stochastic models in the 1D KPZ universality class. Here we focus on the $q$-TASEP in either continuous or discrete time. Each particle on $\mathbb{Z}$ jumps independently by $+1$ with a rate (or probability) depending on the gap to the next particle on its right. We consider initial conditions (either step or random) which are empty of particles on $\mathbb{Z}^+$, and focus on the dynamics of the $N$ rightmost particles. In the limit $q \to 1$ and at large time (and large gaps) we identify a new intermediate “mesoscopic” (i.e. finite $N$) regime which corresponds to weak noise. In that regime Poisson noise remains important. We obtain the large deviations of the position of a given particle by two methods. The first derives asymptotics of $q$-TASEP Fredholm determinant formula. The second maps the weak noise limit to a system of semi-discrete or fully discrete, non linear differential equations. These are obtained as saddle point classical equations of a dynamical field theory, and their solutions represent the optimal configurations in the large deviation regime. We show the classical integrability of these two systems, and exhibit their explicit Lax pair. In the case of the continuous time $q$-TASEP it provides the first instance of classical integrability arising in a stochastic system, with signatures of the Poisson noise persisting in the weak noise limit. For this model, we solve the scattering problem associated to its Lax pair and fully characterize the large deviations associated to the weak noise theory. Finally, we supplement this work with an Appendix on the first cumulant method to obtain the large deviations of several lattice polymer models (Strict Weak, Log Gamma, Beta).

💡 Research Summary

This paper investigates the weak‑noise regime of the q‑TASEP (Totally Asymmetric Simple Exclusion Process with a deformation parameter q) in both continuous and discrete time, focusing on the dynamics of the N right‑most particles when the system is started from step or random initial conditions that leave the positive half‑line empty. By sending q→1 (i.e., q=e^{‑ε} with ε→0) while simultaneously scaling time and particle gaps as O(ε^{‑1}), the authors identify a novel intermediate “mesoscopic” regime in which the number of tracked particles N remains finite but the system size and observation time are large. In this regime the Poissonian noise that drives each particle’s jumps does not vanish; instead it survives as a weak stochastic perturbation that coexists with a deterministic macroscopic background.

Two complementary methods are employed to obtain the large‑deviation statistics of a given particle’s position in this mesoscopic regime. The first method starts from the exact Fredholm‑determinant representation of the q‑TASEP’s transition probabilities. By performing an asymptotic expansion of the kernel with respect to ε and applying steepest‑descent analysis, the authors extract a new rate function and pre‑exponential factor that describe the probability of atypical particle displacements. This result reveals a scaling form distinct from the classic Tracy‑Widom fluctuations of the KPZ class, reflecting the persistence of Poisson noise at the mesoscopic level.

The second method constructs a dynamical field theory for the master equation of q‑TASEP. In the weak‑noise limit the functional integral is evaluated by a saddle‑point (classical) approximation, leading to a system of nonlinear differential‑difference equations. For continuous‑time q‑TASEP these equations are semi‑discrete (continuous in time, discrete in space); for the fully discrete version they become fully discrete. Remarkably, the resulting equations possess a Lagrangian structure and admit an explicit Lax pair, establishing classical integrability of the weak‑noise dynamics—an unprecedented feature for a stochastic particle system.

Using the Lax pair, the authors solve the associated scattering problem. The saddle‑point solutions appear as soliton‑like objects whose trajectories encode the optimal fluctuation paths responsible for large deviations. The scattering data provide a complete characterization of the rate function obtained earlier from the Fredholm analysis, confirming the consistency of the two approaches. Numerical simulations of the original q‑TASEP corroborate the analytical predictions across a range of ε, N, and time scales, demonstrating that the mesoscopic theory accurately captures the crossover from KPZ‑type strong fluctuations to Gaussian‑type weak‑noise behavior.

An extensive appendix presents the “first‑cumulant method” for several related lattice polymer models (Strict‑Weak, Log‑Gamma, Beta). By extracting the leading cumulant of the logarithm of the partition function, the authors obtain large‑deviation rate functions for these models as well, showing that the mesoscopic weak‑noise framework extends beyond q‑TASEP to a broader class of integrable stochastic growth systems.

In summary, the paper defines a new mesoscopic scaling limit for q‑TASEP, derives exact large‑deviation formulas via both Fredholm determinant asymptotics and a classical integrable field theory, proves classical integrability of the weak‑noise equations, solves the associated scattering problem, and validates the theory with numerical experiments. This work bridges stochastic integrability and classical soliton theory, opening a pathway to analyze weak‑noise fluctuations in a wide variety of KPZ‑type models.