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1. Rewriting History in Integrable Stochastic Particle Systems

   https://doi.org/10.1007/s00220-024-05189-y  

   Petrov, Leonid; Saenz, Axel (November 2024, Communications in Mathematical Physics)

   Abstract Many integrable stochastic particle systems in one space dimension (such as TASEP—Totally Asymmetric Simple Exclusion Process—and itsq-deformation, theq-TASEP) remain integrable if we equip each particle with its own speed parameter. In this work, we present intertwining relations between Markov transition operators of particle systems which differ by a permutation of the speed parameters. These relations generalize our previous works (Petrov and Saenz in Probab Theory Relat Fields 182:481–530, 2022), (Petrov in SIGMA 17(021):34, 2021), but here we employ a novel approach based on the Yang-Baxter equation for the higher spin stochastic six vertex model. Our intertwiners are Markov transition operators, which leads to interesting probabilistic consequences. First, we obtain a new Lax-type differential equation for the Markov transition semigroups of homogeneous, continuous-time versions of our particle systems. Our Lax equation encodes the time evolution of multipoint observables of theq-TASEP and TASEP in a unified way, which may be of interest for the asymptotic analysis of multipoint observables of these systems. Second, we show that our intertwining relations lead to couplings between probability measures on trajectories of particle systems which differ by a permutation of the speed parameters. The conditional distribution for such a coupling is realized as a “rewriting history” random walk which randomly resamples the trajectory of a particle in a chamber determined by the trajectories of the neighboring particles. As a byproduct, we construct a new coupling for standard Poisson processes on the positive real half-line with different rates. 

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2. Infinite order phase transition in the slow bond TASEP

   https://doi.org/10.1002/cpa.22185  

   Sarkar, Sourav; Sly, Allan; Zhang, Lingfu (June 2024, Communications on Pure and Applied Mathematics)

   Abstract In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to for some small . Janowsky and Lebowitz  posed the well‐known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of is 0. This was ultimately resolved rigorously in Basu‐Sidoravicius‐Sly which established that . Here we study the effect of the current as tends to 0 and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular we show that the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal. 

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3. Mapping TASEP back in time

   https://doi.org/10.1007/s00440-021-01074-0  

   Petrov, Leonid; Saenz, Axel (July 2021, Probability Theory and Related Fields)

   Abstract We obtain a new relation between the distributions$$\upmu _t$$ ${\mu }_t$at different times$$t\ge 0$$ $t\ge 0$of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions$$\upmu _t$$ ${\mu }_t$backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving$$\upmu _t$$ ${\mu }_t$which in turn brings new identities for expectations with respect to$$\upmu _t$$ ${\mu }_t$. The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions. 

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4. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (June 2025, FPSAC Proceedings)

   We study random permutations corresponding to pipe dreams. Our main model is motivated by the Grothendieck polynomials with parameter ß = 1 arising in the K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of its Grothendieck polyno- mial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process, we describe the limiting permuton and fluctuations around it as the order n of the permutation grows to infinity. The fluctuations are of order n$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for B = 1 Grothendieck polynomials, and provide bounds for general B. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond. 

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5. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (July 2024, arXiv)

   We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $$\beta=1$$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $$n$$ of the permutation grows to infinity. The fluctuations are of order $$n^{\frac13}$$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $$\beta=1$$ Grothendieck polynomials, and provide bounds for general $$\beta$$. 

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