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

Alexandre Krajenbrink, Pierre Le Doussal

Abstract

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).

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

Abstract

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 -TASEP in either continuous or discrete time. Each particle on jumps independently by 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 , and focus on the dynamics of the rightmost particles. In the limit and at large time (and large gaps) we identify a new intermediate "mesoscopic" (i.e. finite ) 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 -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 -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).
Paper Structure (65 sections, 4 theorems, 219 equations, 6 figures)

This paper contains 65 sections, 4 theorems, 219 equations, 6 figures.

Table of Contents

  1. Introduction and summary of main results
  2. Outline
  3. Acknowledgements
  4. Continuous time $q$-TASEP
  5. Definition of the model
  6. Fredholm determinant formulae
  7. Step initial condition
  8. Random initial condition
  9. Weak noise limit of Fredholm determinant formulae
  10. Observable of interest: step initial condition
  11. Observable of interest: random initial condition
  12. Rate function from the first cumulant method
  13. Dynamical field theory
  14. Weak noise regime and saddle point equations
  15. Weak noise action
  16. ...and 50 more sections

Key Result

Theorem 2.3

Fix $0<q<1$, $N\geqslant 1$. Fix $0<\delta<1$, and $\{ a_1,\dots, a_N\}$ such that for all $\ell$, $a_\ell>0$ and $\abs{a_\ell-1}\leqslant d$, for some constant $d<\frac{1-q^\delta}{1+q^\delta}$. Then for all $t>0$ and $u\in \mathbb{C}/ \mathbb{R}_-$, we have the following where $\langle \cdot \rangle$ denotes the expectation value with respect to the stochastic dynamics of the $q$-TASEP, and whe

Figures (6)

  • Figure 1: Representation of the continuous time $q$-TASEP on a one-dimensional lattice. The particles are ordered so that $x_i< x_{i-1}$ for all times and they can only move to the right according to a Poisson clock which rate depends on the gap between consecutive particles.
  • Figure 2: Representation of the discrete time $q$-TASEP on a one-dimensional lattice. The particles are ordered so that $x_i< x_{i-1}$ for all times and they can only move to the right according to a random variable which which probability depends on the gap between consecutive particles.
  • Figure 3: Discrete time $q$-TASEP weak noise lattice recurrence dynamics (for $z_{n,t}$ to the left, and $r_{n,t}$ to the right).
  • Figure 4: The black thick lines delimit the intersection of the light cones for the weak noise theory of the discrete time $q$-TASEP. The fact that the upper limit of the light cone is located at $n=N$ indicates that if we consider a $q$-TASEP process with $M>N$ particles and we are interested in the statistics of the $N$-th one, then we can restrict the original process from $M$ to $N$ particle without changing the observable of interest since it does not enter within the light cone.
  • Figure 5: The compatibility of a discrete integrable system is seen from the equivalence of the two paths on the lattice $(n,t)\to (n+1,t) \to (n+1,t+1)$ and $(n,t)\to (n,t+1) \to (n+1,t+1)$. This is the discrete analogous of a zero curvature condition in a continuous space.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Remark 2.1
  • Remark 2.2: Convergence to the OY polymer
  • Theorem 2.3: From Ref. borodin2014duality Theorem 3.12
  • Theorem 2.4: From Ref. Imamura2019stationaryqtasep2 Theorem 1
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9: Limit to the weak noise of the OY polymer
  • Remark 2.10
  • ...and 19 more