Guillaume Barraquand

I am a CNRS researcher at École normale supérieure in the LPENS (Laboratoire de physique).


Address: Laboratoire de physique de l'école normale supérieure, 24 rue Lhomond, 75005 Paris, France.
Office: Room LE106.
Tel: 01 44 32 25 96.
Email: guillaume.barraquand ''at''


Probability theory and Mathematical Physics:
Integrable probability, KPZ universality class, interacting particle systems, random directed polymers, random walks in random environment, random matrix theory, vertex models, symmetric functions.


Statistical physics Forum at ENS (organized with Camille Aron, Marko Medenjak and Valentina Ros).

Séminaire MEGA (organized with Laure Dumaz).

Conferences and travel

Conference on random graphs and interacting particle systems, June 2-5 2020, at ENS.

Upcoming travel
February 24-28 2020 Third Haifa probability school, Israel
March 16-20 2020 Integrable probability FRG meeting, New York City
May 2020 Random Matrix EurAsia, Singapore
23-24 October UCD, Dublin
May 26 -- June 8 2019 Integrable probability summer school, University of Virginia, Charlottesville
March 11-15 2019 Coulomb Gas, Integrability and Painlevé Equations, at CIRM, Marseille
April 8-12 2019 Integrability and Randomness in Mathematical Physics and Geometry, at CIRM, Marseille
November 4-18 2018 NYU Shanghai
October 18-19 2018 Bonn
September 17-21 2018 Warwick
June 18-23 2018 2018 summer school on random matrices, University of Michigan.
June 1-12 2018 Paris
May 14-18 2018 Integrable probability Boston 2018.
June 26-July 15 2017 PCMI summer session on random matrices, Park city.
June 5-17 2017 Stochastic Dynamics Out of Equilibrium , Institut Henri Poincaré, Paris.
April 24-28 2017 Conference Qualitative Methods in KPZ Universality , CIRM Marseille.
January 4-7 2017 Joint Mathematics meeting AMS-MAA, Atlanta.
April 8, 2016 Columbia Princeton Probability Day 2016, Columbia Mathematics Building, Room 203.
January 11 - February 19 2016 KITP program New approaches to non-equilibrium and random systems: KPZ integrability, universality, applications and experiments, Santa Barbara.
December 27-30 2015 Master Lectures on the Current Topics in Mathematical Physics and Probability: Horng-Tzer Yau, TSIMF Sanya, China.




  • Large deviations for sticky Brownian motions, with Mark Rychnovsky, preprint.

    We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

  • Half-space Macdonald processes, with Alexei Borodin and Ivan Corwin, preprint.

    Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar-Parisi-Zhang (KPZ) equation and a number of other models in its universality class. In this paper we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Non-rigorous saddle point asymptotics yield convergence of the directed polymer free energy to either the Tracy-Widom GOE, GSE or the Gaussian distribution depending on the average size of weights on the boundary.

Published papers

  • Tracy-Widom asymptotics for a river delta model, with Mark Rychnovsky, in Stochastic Dynamics Out of Equilibrium, Institut Henri Poincaré, Paris, France, 2017, Springer Proceedings in Mathematics & Statistics, volume 282, 2019.
    [ArXiv:1807.01824] [Journal version]

    We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [4] to show that, at any fixed positive time, the width of a river delta of length L approaches a constant times L^{2/3} with Tracy-Widom GUE fluctuations of order L^{4/9}. This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.

  • Facilitated exclusion processes, with Jinho Baik, Ivan Corwin and Toufic Suidan, Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham.
    [ArXiv:1707.01923][Journal version]

    We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root time scale, while the statistics in the bulk of the rarefaction fan are GUE. This uses a mapping with last-passage percolation in a half-quadrant which is exactly solvable through Pfaffian Schur processes. Our results further probe the question of how first particles fluctuate for exclusion processes with downward jump discontinuities in their limiting density profiles. Through the Facilitated TASEP and a previously studied MADM exclusion process we deduce that cube-root time fluctuations seem to be a common feature of such systems. However, the statistics which arise are shown to be model dependent (here they are GSE, whereas for the MADM exclusion process they are GUE). We also discuss a two-dimensional crossover between GUE, GOE and GSE distribution by studying the multipoint distribution of the first particles when the rate of the first one varies. In terms of half-space last passage percolation, this corresponds to last passage times close to the boundary when the size of the boundary weights is simultaneously scaled close to the critical point.

  • Stochastic six-vertex model in a half-quadrant and half-line open ASEP, with Alexei Borodin, Ivan Corwin and Michael Wheeler, Duke Mathematical Journal.
    [ArXiv:1704.04309] [Journal version]

    We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ) according to the Tracy-Widom GOE distribution on the τ^1/3 scale. This is the first example of KPZ asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models. Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall-Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to ASEP) using a Yang-Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogs via a refined Littlewood identity.

  • Pfaffian Schur processes and last passage percolation in a half-quadrant, with Jinho Baik, Ivan Corwin and Toufic Suidan, Annals of Probability, Volume 46, Number 6 (2018), 3015-3089.
    [ArXiv:1605.00525] [Journal version]

    We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy-Widom distributed, GOE Tracy-Widom distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy-Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.

  • Random-walk in Beta-distributed random environment, with Ivan Corwin, Probability Theory and Related Fields. (2017) 167:1057.
    [ArXiv:1503.04117] [Journal version]

    We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in Z which performs nearest neighbour jumps with transition probabilities drawn according to the Beta distribution. We also describe a related directed polymer model, which is a limit of the q-Hahn interacting particle system. Using a Fredholm determinant representation for the quenched probability distribution function of the walker's position, we are able to prove second order cube-root scale corrections to the large deviation principle satisfied by the walker's position, with convergence to the Tracy-Widom distribution. We also show that this limit theorem can be interpreted in terms of the maximum of strongly correlated random variables: the positions of independent walkers in the same environment. The zero-temperature counterpart of the Beta RWRE can be studied in a parallel way. We also prove a Tracy-Widom limit theorem for this model.

  • The q-Hahn asymmetric exclusion process, with Ivan Corwin, Annals of Applied Probability 26 (2016), no. 4, 2304-2356.
    [ArXiv:1501.03445] [Journal version]

    We introduce new integrable exclusion and zero-range processes on the one-dimensional lattice that generalize the q-Hahn TASEP and the q-Hahn Boson (zero-range) process introduced in [Pov13] and further studied in [Cor14], by allowing jumps in both directions. Owing to a Markov duality, we prove moment formulas for the locations of particles in the exclusion process. This leads to a Fredholm determinant formula that characterizes the distribution of the location of any particle. We show that the model-dependent constants that arise in the limit theorems predicted by the KPZ scaling theory are recovered by a steepest descent analysis of the Fredholm determinant. For some choice of the parameters, our model specializes to the multi-particle-asymmetric diffusion model introduced in [SW98]. In that case, we make a precise asymptotic analysis that confirms KPZ universality predictions. Surprisingly, we also prove that in the partially asymmetric case, the location of the first particle also enjoys cube-root fluctuations which follow Tracy-Widom GUE statistics.

  • A phase transition for q-TASEP with a few slower particles, Stochastic Processes and their Applications 125 (2015), 2674-2699.
    [ArXiv:1404.7409] [Journal version]

    We consider a q-TASEP model started from step initial condition where all but finitely many particles have speed 1 and a few particles are slower. It is shown in [9] that the rescaled particles position of q-TASEP with identical hopping rates obeys a central limit theorem à la Tracy-Widom. We adapt this work to the case of different hopping rates and show that one observes the so-called BBP transition. Our proof is a refinement of Ferrari-Vetö's and does not require any condition on the parameter q nor the macroscopic position of particles.

  • A short proof of a symmetry identity for the q-Hahn distribution, Electron. Commun. Probab. 19 (2014), no. 50, 1–3.
    [ArXiv:1404.4265] [Journal version]

    We give a short and elementary proof of a symmetry identity for the q-moments of the q-Hahn distribution arising in the study of the q-Hahn Boson process and the q- Hahn TASEP. This identity discovered by Corwin in "The q-Hahn Boson Process and q-Hahn TASEP", Int. Math. Res. Not., 2014, was a key technical step to prove an intertwining relation between the Markov transition matrices of these two classes of discrete-time Markov chains. This was used in turn to derive exact formulas for a large class of observables of both these processes.

Physics papers

  • Moderate deviations for diffusion in time dependent random media, with Pierre Le Doussal, preprint.

    The position x(t) of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray x = v0 t, where v0 is the average drift. However, it has been found that it exhibits at large time sample to sample fluctuations characteristic of the KPZ universality class when observed in an atypical direction, i.e. along the ray x = v t with v ≠ v0. Here we show, from exact solutions, that in the moderate deviation regime x − v0 t t3/4 these fluctuations are precisely described by the finite time KPZ equation, which thus describes the crossover between the Gaussian typical regime and the KPZ fixed point regime for the large deviations. This confirms heuristic arguments given in Le Doussal-Thiery. These exact results include the discrete model known as the Beta RWRE, and a continuum diffusion. They predict the behavior of the maximum of a large number of independent walkers, which should be easier to observe (e.g. in experiments) in this moderate deviations regime.

  • Stochastic growth in time dependent environments, with Pierre Le Doussal and Alberto Rosso, preprint.

    We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance c(t) depending on time. We find that for c(t) proportional to t - α there is a transition at α=1/2. When α > 1/2, the solution saturates at large times towards a non-universal limiting distribution. When α < 1/2 the fluctuation field is governed by scaling exponents depending on \alpha and the limiting statistics are similar to the case when c(t) is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential (2) An exactly solvable discretization, the log-gamma polymer model (3) Numerical simulations.



  • Diffusions in random environment slides.
  • Half-space Macdonald processes slides.
  • ASEP and KPZ equation in a half-space slides.
  • Fluctuations of the first particle in exclusion processes slides, video.
  • On some integrable models in the KPZ universality class, PhD defense slides.
  • Random walks in Beta-distributed random environment slides.
  • KPZ scaling theory for integrable exclusion processes english, français.


No teaching this year.

Past years.