We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider each tiling is weighted proportionally to $q^{\mathsf{vol}}$, where $\mathsf{vol}$ is the volume under the corresponding 3D stepped surface. We prove the following Interlacing Central Limit Theorem: as $q\rightarrow1$, the domain gets large, and the fixed top row approximates a given nonrandom profile, the vertical lozenges are distributed as the eigenvalues of a GUE random matrix and of its successive principal corners. Our results extend the GUE corners asymptotics for tilings of bounded polygonal domains previously known in the uniform (i.e., $q=1$) case. Even though $q$ goes to $1$, the presence of the $q$-weighting affects non-universal constants in our Central Limit Theorem.
We introduce and study the inhomogeneous exponential jump model — an integrable stochastic interacting particle system on the continuous half line evolving in continuous time. An important feature of the system is the presence of arbitrary spatial inhomogeneity on the half line which does not break the integrability. We completely characterize the macroscopic limit shape and asymptotic fluctuations of the height function (= integrated current) in the model. In particular, we explain how the presence of inhomogeneity may lead to macroscopic phase transitions in the limit shape such as shocks or traffic jams. Away from these singularities the asymptotic fluctuations of the height function around its macroscopic limit shape are governed by the GUE Tracy–Widom distribution. A surprising result is that while the limit shape is discontinuous at a traffic jam caused by a macroscopic slowdown in the inhomogeneity, fluctuations on both sides of such a traffic jam still have the GUE Tracy–Widom distribution (but with different non-universal normalizations).
The integrability of the model comes from the fact that it is a degeneration of the inhomogeneous stochastic higher spin six vertex models studied earlier in arXiv:1703.03857 [math.PR].
We consider the $N$-particle noncolliding Bernoulli random walk — a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in{0,1}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some location $x$ (i.e., $x$ is in the ``bulk’’), and the initial configuration is balanced in a certain sense, then the space-time local statistics at $x$ are asymptotically governed by the extended discrete sine process (which can be identified with a translation invariant ergodic Gibbs measure on lozenge tilings of the plane). We also establish similar results for certain types of random initial data. Our proofs are based on a detailed analysis of the determinantal correlation kernel for the noncolliding Bernoulli random walk.
The noncolliding Bernoulli random walk is a discrete analogue of the Dyson Brownian Motion whose local statistics are universality governed by the continuous sine process. Our results parallel the ones in the continuous case. In addition, we naturally include situations with inhomogeneous local particle density on scale $T$, which nontrivially affects parameters of the limiting extended sine process, and in a particular case leads to a new behavior.
We present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the Macdonald processes hierarchy.
First, we show how Macdonald $q$-difference operators with $t=0$ (an algebraic tool crucial for studying the corresponding Macdonald processes) can be utilized to get $q$-moments of the height function $\mathfrak{h}$ in the higher spin six vertex model first computed in arXiv:1601.05770 [math.PR] using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the value of $\mathfrak{h}$ at an arbitrary point has the same distribution as the last component $\lambda_N$ of a random partition under a specific $t=0$ Macdonald measure.
On the other hand, it is known that $\mathbf{x}_N:=\lambda_N-N$ can be identified with the location of the $N$th particle in a certain discrete time $q$-TASEP started from the step initial configuration. The second construction we present is a coupling of this $q$-TASEP and the higher spin six vertex model (with the step Bernoulli boundary condition) along time-like paths providing an independent probabilistic explanation of the equality of $\mathfrak{h}(N+1,T)$ and $\mathbf{x}_N+N$ in distribution. Combined with the identification of averages of observables between the stochastic higher spin six vertex model and Schur measures (which are $t=q$ Macdonald measures) obtained recently in arXiv:1608.01553 [math-ph], this produces GUE Tracy–Widom asymptotics for a discrete time $q$-TASEP with the step initial configuration and special jump parameters.
We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes.
Our arguments are largely based on properties of a family of symmetric rational functions (introduced in arXiv:1410.0976 [math.CO]) that can be defined as partition functions of the higher spin six vertex model for suitable domains; they generalize classical Hall-Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang-Baxter equation for the higher spin six vertex model.
These are lecture notes for a course given by A.B. at the Ecole de Physique des Houches in July of 2015. All the results and proofs presented here generalize to the setting of the fully inhomogeneous higher spin six vertex model, see arXiv:1601.05770 [math.PR] for a detailed exposition of the inhomogeneous case.
We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes.
Our arguments are largely based on properties of a family of symmetric rational functions which can be defined as partition functions of the inhomogeneous higher spin six vertex model for suitable domains. In the homogeneous case, such functions were previously studied in arXiv:1410.0976 [math.CO]; they also generalize classical Hall-Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang-Baxter equation for the higher spin six vertex model.
We introduce and study $q$-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical ($q=0$) and geometric ($q\to 1$) RSK correspondences (the latter ones are sometimes also called tropical).
For $0< q <1$ our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on $q$-Whittaker processes (which are $t=0$ versions of Macdonald processes). We present four Markov dynamics which for $q=0$ reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries.
Our new two-dimensional discrete time dynamics generalize and extend several known constructions: (1) The discrete time $q$-TASEPs arise as one-dimensional marginals of our “column” dynamics. In a similar way, our “row” dynamics lead to discrete time $q$-PushTASEPs - new integrable particle systems in the Kardar-Parisi-Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time $q$-PushASEP conjectured by Corwin-Petrov (2013). (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the $q$-randomized column and row Robinson-Schensted correspondences introduced by O'Connell-Pei (2012) and Borodin-Petrov (2013), respectively. (3) In a scaling limit as $q\to1$, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma or strict-weak directed random polymers.
We introduce a four-parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain Markov dualities. Using this, for the systems started in step initial data we write down nested contour integral formulas for moments and Fredholm determinant formulas for Laplace-type transforms. Taking various choices or limits of parameters, this family degenerates to many of the known exactly solvable models in the Kardar-Parisi-Zhang universality class, as well as leads to many new examples of such models. In particular, ASEP, the stochastic six-vertex model, $q$-TASEP and various directed polymer models all arise in this manner. Our systems are constructed from stochastic versions of the R-matrix related to the six-vertex model. One of the key tools used here is the fusion of R-matrices and we provide a probabilistic proof of this procedure.
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.
Asymptotic representation theory of general linear groups $GL(n,q)$ over a finite field leads to studying probability measures $\rho$ on the group $U$ of all infinite uni-uppertriangular matrices over $F_q$, with the condition that $\rho$ is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures $\rho$) was conjectured by Kerov in connection with nonnegative specializations of Hall-Littlewood symmetric functions.
Vershik and Kerov also conjectured the following Law of Large Numbers. Consider an $n\times n$ diagonal submatrix of the infinite random matrix drawn from an ergodic measure coming from the Kerov’s conjectural classification. The sizes of Jordan blocks of the submatrix can be interpreted as a (random) partition of $n$, or, equivalently, as a (random) Young diagram $\lambda(n)$ with $n$ boxes. Then, as $n$ goes to infinity, the rows and columns of $\lambda(n)$ have almost sure limiting frequencies corresponding to parameters of this ergodic measure.
Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson-Schensted-Knuth (RSK) insertion algorithm which samples random Young diagrams $\lambda(n)$ coming from ergodic measures. The probability weights of these Young diagrams are expressed in terms of Hall-Littlewood symmetric functions. Our insertion algorithm is a modified and extended version of a recent construction by Borodin and the second author (arXiv:1305.5501). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).
These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.
We develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the $q$-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with $q$-TASEP, this leads to moment formulas which characterize the fixed time distribution of $q$-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions.
We consider limits of our $q$-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O’Connell-Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation / Kardar-Parisi-Zhang equation.
We introduce a new interacting (stochastic) particle system $q$-PushASEP which interpolates between the q-TASEP introduced by Borodin and Corwin (see arXiv:1111.4408, and also arXiv:1207.5035; arXiv:1305.2972; arXiv:1212.6716) and the $q$-PushTASEP introduced recently by Borodin and Petrov (arXiv:1305.5501). 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 in which cars are able to accelerate or slow down.
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.
Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters $(q,t)$, where $0\le q, t < 1$. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes.
The classification unites known examples of such dynamics and also yields many new ones. When $t = 0$, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a $q$-deformation of the PushTASEP (= long-range TASEP). When $q = t$, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson–Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.
In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonnegative sequences.
A principal ingredient of Borodin-Olshanski’s proof is a new explicit determinantal formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). We present a simpler and more direct derivation of that formula using the Cauchy-Binet summation involving the inverse Vandermonde matrix. We also obtain a q-generalization of that formula, namely, a new explicit determinantal formula for arbitrary q-specializations of skew Schur polynomials. Its particular case is related to the q-Gelfand-Tsetlin graph and q-Toeplitz matrices introduced and studied by Gorin [arXiv:1011.1769].
We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows arbitrarily large number of sides), we show that these fluctuations are asymptotically governed by a Gaussian free (massless) field. This complements the similar result obtained in Kenyon [Comm. Math. Phys. 281 (2008) 675-709] about tilings of regions without frozen facets of the limit shape. In our asymptotic analysis we use the explicit double contour integral formula for the determinantal correlation kernel of the model obtained previously in arXiv:1202.3901 [math.PR].
A Gelfand-Tsetlin scheme of depth $N$ is a triangular array with m integers at level $m$, $m=1,\ldots,N$, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed $N$-th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation).
This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov in arXiv:math-ph/0507007.
We also prove that at the edge of the limit
shape, the asymptotic behavior of random tilings is given by the Airy process.
In particular, our results cover the most investigated case of random boxed
plane partitions (when the polygon is a hexagon).
We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes the Kerov’s operators first appeared in [Okounkov, arXiv:math/0002135], and also stems from the study of duality of graded graphs in [Fomin, 1994].
Our main object is a countable branching graph carrying an $\mathfrak{sl}(2,\mathbb{C})$-module of a special kind. Using this structure, we introduce distinguished probability measures on the floors of the graph, and define two related types of Markov dynamics associated with these measures. We study spectral properties of the dynamics, and our main result is the explicit description of eigenfunctions of the Markov generator of one of the processes.
For the Young graph our approach reconstructs the z-measures on partitions and the associated dynamics studied by Borodin and Olshanski [arXiv:math-ph/0409075, arXiv:0706.1034]. The generator of the dynamics of [arXiv:math-ph/0409075] is diagonal in the basis of the Meixner symmetric functions introduced recently by Olshanski [arXiv:1009.2037, arXiv:1103.5848]. We give new proofs to some of the results of these two papers. Other graphs to which our technique is applicable include the Pascal triangle, the Kingman graph (with the two-parameter Poisson-Dirichlet measures), the Schur graph and the general Young graph with Jack edge multiplicities.
The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.
We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin’s measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able to express our correlation kernels (both static and dynamical) through those of the z-measures on partitions obtained previously by Borodin and Olshanski in a series of papers. The results about the fixed time case were announced in the author’s note arXiv:1002.2714. A part of the present paper contains proofs of those results.
In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice ${1,2,\ldots}$ or on the open half-line $(0,+\infty)$. The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group.
We consider a certain sequence of random walks. The state space of the n-th random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to a continuous-time Markov process. The state space of this process is the infinite-dimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The main result about the limit process is the expression of its the pre-generator as a formal second order differential operator in a polynomial algebra. Of separate interest is the generalization of Kerov interlacing coordinates to the case of shifted Young diagrams.
The aim of the paper is to introduce a two-parameter family of infinite-dimensional diffusion processes $X(\alpha,\theta)$ related to Pitman’s two-parameter Poisson-Dirichlet distributions $PD(\alpha,\theta)$. The diffusions $X(\alpha,\theta)$ are obtained in a scaling limit transition from certain finite Markov chains on partitions of natural numbers. The state space of $X(\alpha,\theta)$ is an infinite-dimensional simplex called the Kingman simplex. In the special case when parameter $\alpha$ vanishes, our finite Markov chains are similar to Moran-type model in population genetics, and our diffusion processes reduce to the infinitely-many-neutral-alleles diffusion model studied by Ethier and Kurtz (1981).
Our main results extend those of Ethier and Kurtz to the two-parameter case and are as follows: The Poisson-Dirichlet distribution $PD(\alpha,\theta)$ is a unique stationary distribution for the corresponding process $X(\alpha,\theta)$; the process is ergodic and reversible; the spectrum of its generator is explicitly described. The general two-parameter case seems to fall outside the setting of models of population genetics, and our approach differs in some aspects from that of Ethier and Kurtz. We also consider the case of degenerate series of parameters $\alpha$ and $\theta$ and conclude that the diffusions in finite-dimensional simplexes studied by Ethier and Kurtz (1981) arise as a special case of our two-parameter family of diffusions.
One of the versions of my “job talk” in Fall 2013 describing many facets of integrable probability. It is supposed to be accessible. The talk is based on [15] and on the previous two talks. More details can also be found in arXiv:1106.1596 [math.PR] by Corwin and arXiv:1212.3351 [math.PR] by Borodin and Gorin.
The talk is based on [8] and describes $\mathfrak{sl}(2,\mathbb{C})$ structures behind Markov jump processes on the Young and related branching graphs
The talk describes population genetics perspective behind infinite-dimensional diffusions preserving the two-parameter Poisson–Dirichlet distributions and related models. It is based on [2].