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This is a survey of Harold Widom’s work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy’s groundbreaking results concerning the distribution functions of random matrix theory, touch on the remarkable universality of the Tracy–Widom distributions in mathematics and physics, and close with Tracy and Widom’s remarkable work on the asymmetric simple exclusion process.

In many-particle diffusions, particles that move the furthest and fastest can play an outsized role in physical phenomena. A theoretical understanding of the behavior of such extreme particles is nascent. A classical model, in the spirit of Einstein's treatment of single-particle diffusion, has each particle taking independent homogeneous random walks. This, however, neglects the fact that all particles diffuse in a common and often inhomogeneous environment that can affect their motion. A more sophisticated model treats this common environment as a space-time random biasing field which influences each particle's independent motion. While the bulk (or typical particle) behavior of these two models has been found to match to high degree, recent theoretical work of Barraquand, Corwin and Le Doussal on a one-dimensional exactly solvable version of this random environment model suggests that the extreme behavior is quite different between the two models. We transform these asymptotic (in system size and time) results into physically applicable predictions. Using high precision numerical simulations we reconcile different asymptotic phases in a manner that matches numerics down to realistic system sizes, amenable to experimental confirmation. We characterize the behavior of extreme diffusion in the random environment model by the presence of a new phase with anomalous fluctuations related to the Kardar-Parisi-Zhang universality class and equation.

For ASEP with step initial data and a second class particle started at the origin we prove that as time goes to infinity the second class particle almost surely achieves a velocity that is uniformly distributed on $[-1,1]$. This positively resolves Conjecture 1.9 and 1.10 of [Amir, Angel and Valko, "The TASEP speed process", Annals of Probability 39, 1205--1242, 2011] and allows us to construct the ASEP speed process., 56 pages, 11 figures

Consider a sequence of Gibbsian line ensembles, whose lowest labeled curves (i.e., the edge) have tight one-point marginals. Then, given certain technical assumptions on the nature of the Gibbs property and underlying random walk measure, we prove that the entire spatial process of the edge is tight. We then apply this black-box theory to the log-gamma polymer Gibbsian line ensemble which we construct. The edge of this line ensemble is the transversal free energy process for the polymer, and our theorem implies tightness with the ubiquitous KPZ class $2/3$ exponent, as well as Brownian absolute continuity of all the subsequential limits. A key technical innovation which fuels our general result is the construction of a continuous grand monotone coupling of Gibbsian line ensembles with respect to their boundary data (entrance and exit values, and bounding curves). {\em Continuous} means that the Gibbs measure varies continuously with respect to varying the boundary data, {\em grand} means that all uncountably many boundary data measures are coupled to the same probability space, and {\em monotone} means that raising the values of the boundary data likewise raises the associated measure. This result applies to a general class of Gibbsian line ensembles where the underlying random walk measure is discrete time, continuous valued and log-convex, and the interaction Hamiltonian is nearest neighbor and convex., 72 pages, 4 figures. We corrected some typos and a mistake in the proof of Lemma 2.13

We introduce the $q$-Hahn PushTASEP --- an integrable stochastic interacting particle system which is a 3-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (Totally Asymmetric Simple Exclusion Process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi_3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky (2013). We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. In a $q\to 1$ limit of our process we arrive at a random recursion which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e., the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner., Comment: 29 pages, 3 figures; v3: minor corrections and improvements of the presentation. To appear in IMRN

A Correction to this paper has been published: https://doi.org/10.1007/s11040-021-09380-8

We present a complete proof of the exact formula for the one-point distribution for the narrow-wedge Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) equation. This presentation is intended to be self-contained so no previous knowledge about stochastic PDEs, or exactly solvable systems is presumed., This is a chapter in a forthcoming AMS Proceedings collection of expanded notes from the AMS Short Course "Random Growth Models," which took place in Atlanta, GA, at the AMS Joint Mathematics Meetings in January 2017. Editors: Michael Damron, Firas Rassoul-Agha, Timo Seppalainen

This is an erratum to the paper [CP16]. The aim of this note is to address two separate errors in the paper.

We consider ASEP on a bounded interval and on a half line with sources and sinks. On the full line, Bertini and Giacomin proved convergence under weakly asymmetric scaling of the height function to the solution of the KPZ equation. We prove here that under similar weakly asymmetric scaling of the sources and sinks as well, the bounded interval ASEP height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides (different parameter for each side); and likewise for the half line ASEP to KPZ on a half line. This result can be interpreted as showing that the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP., Comment: 59 pages, 2 figures

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., v2: minor edits. 106 pages, 17 figures

We prove that the free energy of the log-gamma polymer between lattice points $(1,1)$ and $(M,N)$ converges to the GUE Tracy-Widom distribution in the $M^{1/3}$ scaling, provided that $N/M$ remains bounded away from zero and infinity. We prove this result for the model with inverse gamma weights of any shape parameter $\theta>0$ and furthermore establish a moderate deviation estimate for the upper tail of the free energy in this case. Finally, we consider a non i.i.d. setting where the weights on finitely many rows and columns have different parameters, and we show that when these parameters are critically scaled the limiting free energy fluctuations are governed by a generalization of the Baik-Ben Arous-P\'ech\'e distribution from spiked random matrices with two sets of parameters., Comment: 57 pages, 9 Figures

This is a correction to Theorems 7.3 and 8.12 in [1].

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). A degeneration of the stochastic Interaction Round-a-Face (IRF) model of arXiv:1701.05239, dynamic ASEP has a jump parameter $q\in (0,1)$ and a dynamical parameter $\alpha>0$. It degenerates to the standard ASEP height function when $\alpha$ goes to $0$ or $\infty$. We consider very weakly asymmetric scaling, i.e., for $\varepsilon$ tending to zero we set $q=e^{-\varepsilon}$ and look at fluctuations, space and time in the scales $\varepsilon^{-1}$, $\varepsilon^{-2}$ and $\varepsilon^{-4}$. We show that under such scaling the height function of the dynamic ASEP converges to the solution of the space-time Ornstein-Uhlenbeck process. We also introduce the dynamic ASEP on a ring with generalized rate functions. Under the very weakly asymmetric scaling, we show that the dynamic ASEP (with generalized jump rates) on a ring also converges to the solution of the space-time Ornstein-Uhlenbeck process on $[0,1]$ with periodic boundary conditions., Comment: 63 pages, 1 figure

This is an expository note answering a question posed to us by Richard Stanley, in which we prove a limit shape theorem for partitions of $n$ which maximize the number of subpartitions. The limit shape and the growth rate of the number of subpartitions are explicit. The key ideas are to use large deviations estimates for random walks, together with convex analysis and the Hardy-Ramanujan asymptotics. Our limit shape coincides with Vershik's limit shape for uniform random partitions., Comment: 15 pages

We consider the narrow wedge solution to the Kardar-Parisi-Zhang stochastic PDE under the characteristic $3:2:1$ scaling of time, space and fluctuations. We study the correlation of fluctuations at two different times. We show that when the times are close to each other, the correlation approaches one at a power-law rate with exponent $2/3$, while when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent $-1/3$. We also prove exponential-type tail bounds for differences of the solution at two space-time points. Three main tools are pivotal to proving these results: 1) a representation for the two-time distribution in terms of two independent narrow wedge solutions; 2) the Brownian Gibbs property of the KPZ line ensemble; and 3) recently proved one-point tail bounds on the narrow wedge solution., Comment: 45 pages, 3 figure (this revision has two new figures and has added details in the proof of Proposition 4.3)

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., Comment: v4: minor typos corrected. v3: 58 pages, to appear in Annals of Probability. The initial submission arXiv:1606.00525v1 has been splitted in two parts. This paper contains only the results on last passage percolation. The results on the facilitated TASEP appear in arXiv:1707.01923

We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5/2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3/2$ at all depth in the tail., 40 pages, 2 figures

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 Gaussian orthogonal ensemble distribution on the τ[superscript 1/3]-scale. This is the first example of Kardar–Parisi–Zhang 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 the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity. Keywords: Kardar–Parisi–Zhang universality class; interacting particle systems; asymmetric simple exclusion process; six-vertex model; integrable probability; Macdonald symmetric functions; Yang–Baxter equation, National Science Foundation (U.S.) (Grant DMS-160790)

Nous montrons qu’une generalisation du processus d’exclusion asymetrique appelee $\operatorname{ASEP}(q,j)$, introduite dans (Probab. Theory Related Fields 166 (2016) 887–933), converge sous faible asymetrie vers la solution de l’equation KPZ au sens de Cole–Hopf.

We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process-an infinite particle Coulomb gas that arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straightforward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution, which demonstrate a crossover between exponential decay with exponent 3 (in the shallow left tail) to exponent 5/2 (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech. (2017) 063203JSMTC61742-546810.1088/1742-5468/aa73f8] via a WKB approximation analysis of a nonlocal, nonlinear integrodifferential equation generalizing Painleve II which Amir et al. [Commun. Pure Appl. Math. 64, 466 (2011)CPMAMV0010-364010.1002/cpa.20347] related to the KPZ one-point distribution.

Universality in Random Systems Universality in complex random systems is a striking concept which has played a central role in the direction of research within probability, mathematical physics and statistical mechanics. In this article we will describe how a variety of physical systems and mathematical models, including randomly growing interfaces, certain stochastic PDEs, traffic models, paths in random environments, and random matrices all demonstrate the same universal statistical behaviors in their long-time/large-scale limit. These systems are said to lie in the Kardar-Parisi-Zhang (KPZ) universality class. Proof of universality within these classes of systems (except for random matrices) has remained mostly elusive. Extensive computer simulations, nonrigorous physical arguments/heuristics, some laboratory experiments, and limited mathematically rigorous results provide important evidence for this belief.

For each $$t\ge 1$$ we construct an $$\mathbb {N}$$ -indexed ensemble of random continuous curves with three properties: This construction uses as inputs the diffusion that O’Connell discovered (Ann Probab 40:437–458, 2012) in relation to the O’Connell–Yor semi-discrete Brownian polymer, the convergence result of Moreno Flores et al. (in preparation) of the lowest indexed curve of that diffusion to the solution of the KPZ equation with narrow wedge initial data, and the one-point distribution formula proved by Amir et al. (Commun Pure Appl Math 64:466–537, 2011) for the solution of the KPZ equation with narrow wedge initial data. We provide four main applications of this construction

We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result which implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the q-Hahn TASEP (a discrete-time generalization of TASEP with particles' jump distribution being the orthogonality weight for the classical q-Hahn orthogonal polynomials), we write down moment formulas which characterize the fixed time distribution of the q-Hahn TASEP with general initial data. The Bethe ansatz eigenfunctions of the q-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a spectral theory, too. In particular, biorthogonality of the ASEP eigenfunctions which follows from the corresponding q-Hahn statement implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the q-Hahn system to the q-Boson particle system (dual to q-TASEP) studied in detail in our previous paper (2013). Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation / stochastic heat equation, namely, q-TASEP and ASEP., 65 pages, 13 figures; v3-v4: Removed incorrect proof of spatial biorthogonality of ASEP eigenfunctions. The statement is correct due to Tracy-Widom (arXiv:0704.2633) but does not directly follow from the main results of the present work. See Remark 7.5 for details

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.

We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $\Delta\to 1^+$ so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the KPZ equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic Burgers equation. Our proofs rely upon the Markov (self) duality of our model. The starting point is an exact microscopic Hopf-Cole transform for the stochastic six vertex model which follows from the model's known one-particle Markov self-duality. Given this transform, the crucial step is to establish self-averaging for specific quadratic function of the transformed height function. We use the model's two-particle self-duality to produce explicit expressions (as Bethe ansatz contour integrals) for conditional expectations from which we extract time-decorrelation and hence self-averaging in time. The crux of our Markov duality method is that the entire convergence result reduces to precise estimates on the one-particle and two-particle transition probabilities. Previous to our work, Markov dualities had only been used to prove convergence of particle systems to linear Gaussian SPDEs (e.g. the stochastic heat equation with additive noise)., Comment: 74 pages, 21 figures

We study ASEP in a spatially inhomogeneous environment on a torus $ \mathbb{T}^{(N)} = \mathbb{Z}/N\mathbb{Z} $ of $ N $ sites. A given inhomogeneity $ \widetilde{a}(x) \in (0,\infty) $, $x \in \mathbb{T} $, perturbs the overall asymmetric jumping rates $ r < \ell \in (0,1) $ at bonds, so that particles jump from site $ x $ to $ x+1 $ with rate $ r\widetilde{a}(x) $ and from $x+1$ to $x$ with rate $ \ell\widetilde{a}(x) $ (subject to the exclusion rule in both cases). Under the limit $ N\to\infty $, we suitably tune the asymmetry $ (\ell-r) $ to zero like $N^{-\frac{1}{2}}$ and the inhomogeneity $ \widetilde{a}(x) $ to unity, so that the two compete on equal footing. At the level of the G\"{a}rtner (or microscopic Hopf--Cole) transform, we show convergence to a new SPDE -- the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise. Equivalently, at the level of the height function we show convergence to the Kardar--Parisi--Zhang equation with a mix of spatial and spacetime additive noise. Our method applies to a general class of $ \widetilde{a}(x) $, which, in particular, includes i.i.d., long-range correlated, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the \emph{kernel} of the semigroup $ \mathcal{Q}(t):=e^{t\mathcal{H}} $ for a Hill-type operator $ \mathcal{H}:= \frac12\partial_{xx} + \mathcal{A}'(x) $, and its discrete analog, where $ \mathcal{A} $ (and its discrete analog) is a generic H\"{o}lder continuous function., Comment: 45 pages, 2 figures

We introduce a new interacting (stochastic) particle system $$q$$ -PushASEP which interpolates between the $$q$$ -TASEP of Borodin and Corwin (Probab Theory Relat Fields 158(1–2):225–400, 2014; see also Borodin et al., Ann Probab 42(6):2314–2382, 2014; Borodin and Corwin, Int Math Res Not 2:499–537, 2015; O’Connell and Pei, Electron J Probab 18(95):1–25, 2013; Borodin et al., Comput Math, 2013) and the $$q$$ -PushTASEP introduced recently (Borodin and Petrov, Adv Math, 2013). In the $$q$$ -PushASEP, particles can jump to the left or to the right, and there is a certain partially asymmetric pushing mechanism present. This particle system has a nice interpretation as a model of traffic on a one-lane highway. Using the quantum many body system approach, we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals. We also discuss relevant Fredholm determinantal formulas for the distribution of the location of each particle, and connections of the model with a certain two-sided version of Macdonald processes and with the semi-discrete stochastic heat equation.

We introduce an exactly-solvable model of random walk in random environment that we call the Beta RWRE. This is a random walk in $\mathbb{Z}$ 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., Comment: 55 pages. v5: updated version correcting minor mistakes in the published version

We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q-TASEP. We call these geometric and Bernoulli discrete time q-TASEP. We obtain concise formulas for expectations of a large enough class of observables of the systems to completely characterize their fixed time distributions when started from step initial condition. We then extract Fredholm determinant formulas for the marginal distribution of the location of any given particle. Underlying this work is the fact that these expectations solve closed systems of difference equations which can be rewritten as free evolution equations with k−1 two-body boundary conditions—discrete q-deformed versions of the quantum delta Bose gas. These can be solved via a nested contour integral ansatz. The same solutions also arise in the study of Macdonald processes, and we show how the systems of equations our expectations solve are equivalent to certain commutation relations involving the Macdonald first difference operator., National Science Foundation (U.S.) (Grant DMS-1056390), National Science Foundation (U.S.) (Grant DMS-1208998), Microsoft Research (Schramm Memorial Fellowship), Clay Mathematics Institute (Research Fellowship)

We consider the ASEP and the stochastic six vertex model started with step initial data. After a long time, $T$, it is known that the one-point height function fluctuations for these systems are of order $T^{1/3}$. We prove the KPZ prediction of $T^{2/3}$ scaling in space. Namely, we prove tightness (and Brownian absolute continuity of all subsequential limits) as $T$ goes to infinity of the height function with spatial coordinate scaled by $T^{2/3}$ and fluctuations scaled by $T^{1/3}$. The starting point for proving these results is a connection discovered recently by Borodin-Bufetov-Wheeler between the stochastic six vertex height function and the Hall-Littlewood process (a certain measure on plane partitions). Interpreting this process as a line ensemble with a Gibbsian resampling invariance, we show that the one-point tightness of the top curve can be propagated to the tightness of the entire curve., Comment: 60 pages, 11 figures

We consider directed polymer models involving multiple non-intersecting random walks moving through a space-time disordered environment in one spatial dimension. For a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling (in which time and space are scaled diffusively, and the strength of the environment is scaled to zero in a critical manner) the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this paper we prove the analogous result for multiple non-intersecting random walks started and ended grouped together. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren., 50 pages, 2 figures

We consider a discrete model for anisotropic (2 + 1)-dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space-time limit of covariances to those of the (2 + 1)-dimensional additive stochastic heat equation (or Edwards-Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE. Keywords: 2+1 growth models, KPZ universality class, q-Whittaker processes, Gaussian Free Field, Space-time process, Galileo Galilei Institute for Theoretical Physics (Arcetri, Italy), Kavli Institute for Theoretical Physics, National Science Foundation (U.S.) (Grant PHY-1125915), National Science Foundation (U.S.) (Grant DMS-1056390), National Science Foundation (U.S.) (Grant DMS-1607901), Simons Foundation. Postdoctoral Fellowship, Radcliffe Institute for Advanced Study (Fellowship)

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., 40 pages,11 figures. v3: Presentation improved in Introduction and Section 4. to appear in Ann. Appl. Probab

Consider N Brownian bridges B_i:[-N,N] -> R, B_i(-N) = B_i(N) = 0, 1 infinity of these curves scaled around (0,2^{1/2} N) horizontally by a factor of N^{2/3} and vertically by N^{1/3}. If a parabola is added to each limit curve, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which curves are a.s. everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with "wanderers" and "outliers". We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property -- called the Brownian Gibbs property -- of being invariant under the following action. Select an index 1, 49 pages, 7 figures

We determine a $q\to 1$ limit of the two-dimensional $q$-Whittaker driven particle system on the torus studied previously in [Corwin-Toninelli, arXiv:1509.01605]. This has an interpretation as a $(2+1)$-dimensional stochastic interface growth model, that is believed to belong to the so-called anisotropic Kardar-Parisi-Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the $(2+1)$-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in $(2+1)$-dimension is irrelevant., 24 pages, 1 figure

The celebrated Gauss–Bonnet formula has a nice generalization to surfaces with densities, in which both arclength and area are weighted by positive functions. Surfaces with densities, especially when arclength and area are weighted by the same factor, appear throughout mathematics, including probability theory and Perelman’s recent proof of the Poincaré conjecture.

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices., Comment: 46 pages, 3 figures, LaTeX; Extended explanations in the first two sections

We consider the transition probabilities for random walks in $1+1$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE., Comment: 15 pages, revised version

We study the partition function of two versions of the continuum directed polymer in 1 + 1 dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in R, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in R_. The partition functions solve the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar-Parisi-Zhang equation.We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent 1/3 and are given by the GUE and GSE Tracy-Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified.The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb-Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions. Keywords: Kardar–Parisi–Zhang; Directed polymers; Bethe ansatz; Lieb–Liniger model; Delta Bose gas, National Science Foundation (U.S.) (Grant DMS-1056390)