Stochastic quantum integrable systems

We describe recent work involving interacting particle systems related to quantum
integrable systems. This theory serves as an umbrella for exactly solvable models in the Kardar
Parisi-Zhang universality class, as well as provides new examples of such systems, and new tools in
their analysis.

Stochastic quantum integrable systems in infinite volume

I will discuss the higher spin vertex model, which is a (discrete-time) stochastic quantum integrable
system on the (half-)line. In various regimes, this model degenerates to both ASEP and q-TASEP,
two well-known integrable discretizations of the Kardar-Parisi-Zhang equation. Bethe ansatz
integrability provides eigenfunctions of the model, which are nice symmetric rational functions
generalizing the Hall-Littlewood symmetric polynomials. A certain Markov (self-)duality plus
Fourier-like transforms associated with the eigenfunctions allow to write down exact formulas for
observables of the system started from an arbitrary initial data

Are limit shape equations integrable?

We will argue that limit shape equations are integrable for models in statistical mechanics where
tranfer-matrices form commuting families

Height fluctuations for the stationary KPZ equation

In this talk I will discuss the height fluctuations of models in the KPZ (Kardar-Parisi-Zhang)
universality class with stationary initial conditions. In particular, I will present the solution for the
KPZ equation, where the stationary initial height profile is a two-sided Brownian motion. This is a
joint work with Alexei Borodin, Ivan Corwin, and Bálint Vetö (arXiv: 1407.6977).

A class of (2+1)-dimensional growth process with explicit stationary measure

We introduce a class of (2+1)-dimensional random growth processes, that can be seen as
asymmetric random dynamics of discrete interfaces. Interface configurations correspond to height
functions of dimer coverings of the infinite hexagonal or square lattice. “Asymmetric” means that
the interface has an average non-zero drift. When the asymmetry parameter p − 1/2 equals zero, the
infinite-volume Gibbs measures (with given slope ) are stationary and reversible. When p eq
1/2, pi_ ho is not reversible any more but, remarkably, it is still stationary. In such stationary
states, one finds that the height function at a given point x grows linearly with time t with a nonzero
speed := <(hx(t) − hx(0))> = v t and that the typical fluctuations of Q_x(t) are smaller than any
power of t. For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics
coincides with the “anisotropic KPZ growth model” studied by A. Borodin and P. L. Ferrari. For a
suitably chosen initial condition (that is very different from the stationary state), they were able to
determine the hydrodynamic limit and the interface fluctuations, exploiting the fact that some
space-time correlations can be computed exactly, and predicted stationarity of Gibbs measures

Exact results and conjectures from the replica Bethe ansatz for KPZ growth and random directed polymers

We describe the replica Bethe Ansatz (RBA) method for the continuum KPZ equation and directed
polymer (DP) and recall some results for the height distributions of the main classes of initial
conditions and relations to random matrices statistics. We mention more recent applications of RBA
for transition classes, non-crossing paths and lattice DP models

Central Limit Theorem for discrete log-gases

A log-gas is an ensemble of N particles on the real line, for which the probability of a configuration
is the power of the Vandermonde determinant times the product of a weight w(x) over the positions
of particles. Such ensembles are widespread in the random matrix theory, while their discrete
counterparts appear in numerous statistical mechanics models such as random tilings and last
passage percolation.
I will explain a new approach which gives Central Limit Theorems for global fluctuations of
discrete log-gases for a wide class of the weights w(x). The approach is based on novel discrete
equations, which are analogues of the loop equations known in the continuous settings

Asymptotics of representations of classical Lie groups

We study the decompositions into irreducible components of tensor products and restrictions of
irreducible representations of classical Lie groups as the rank of the group goes to infinity. We
prove the Law of Large Numbers and the Central Limit Theorem for the random counting measures
describing the decomposition. It turns out that this problem is intrinsically connected with random
lozenge and domino tilings, and also with free probability.
The talk is based on a joint work with V. Gorin

Global fluctuations of non-colliding processes and non-intersecting paths

Ensembles of non-intersecting paths and non-colliding processes give rise to random surfaces in a
natural way, by viewing the paths or trajectories as the level lines. The fluctuations of such random
surfaces are expected to be universally governed by the Gaussian Free Field, which has been
verified for number of models in recent years. In this talk I will discuss a new approach to
establishing this universality. In particular, a general Central Limit Theorem for smooth multi-time
linear statistics for determinantal point process with extended kernels will be presented

Asymptotic behaviors in Schur processes

We present some asymptotic results, like the limit shape and fluctuation behavior at the edge, of
models falling in the class of Schur processes. These include pyramid partitions introduced by
Kenyon, Szendroi and Young, Aztec diamonds with non-uniform measures and steep tilings
recently introduced by Bouttier, Chapuy and Corteel. This is a joint work with D. Betea, and C.
Boutillier. We will also discuss results for models falling in the class of symmetric Schur processes.
Various limit shape results, confirming our analytic results, were obtained experimentally using a
perfect sampling algorithm for Schur processes. The algorithm which is a joint work D. Betea, C.
Boutillier, J. Bouttier, G. Chapuy and S. Corteel will also be discussed

A factorisation theorem for the number of rhombus tilings of a hexagon with trianglar holes

I shall present a curious factorisation theorem for the number of rhombus tilings of a hexagon with
vertical and horizontal symmetry axes, with triangular holes along one axis. I shall set this theorem
in relation with other factorisation theorems, and discuss some consequences and open questions.
This is joint work with Mihai Ciucu.

The Z-invariant massive Laplacian on isoradial graphs

After having explained the notion of Z-invariance for models of statistical mechanics, we introduce
a one-parameter family (depending on the elliptic modulus k) of Z-invariant weights for the
discrete massive Laplacian defined on isoradial graphs. We prove an explicit formula for its inverse,
the massive Green function, which has the remarkable property of only depending on the local
geometry of the graph. We mention consequences of this result for the model of spanning forests, in
particular the proof of an order two phase transition with the critical spanning tree model on
isoradial graphs introduced by Kenyon. Finally, we consider the spectral curve of this massive
Laplacian, and prove that it is a Harnack curve of genus 1. This is joint work with Cédric Boutillier
and Kilian Raschel

Simple approaches to arctic curves for Alternating Sign Matrices

We consider Alternating Sign Matrices (ASM) with a weight w per -1 in the matrix. Many things
are known from the mapping on the 6-Vertex Model with domain-wall boundary conditions
(DWBC). Differently from models on periodic tori, the DWBC force extensive regions to be
almost-surely “frozen”, and an asymptotic boundary between the frozen and unfrozen regions,
called Arctic Curve, emerges.
At w=2, we are at the free-ermion point. We have quite detailed informations on the system. In
particular, the arctic curve is found to be a circle. At generic w, (and notably at w=1, the uniform
case), the arctic curve has been established in a series of papers of Colomo and Pronko. For a
certain F=F(z;x,y), involving the 1-boundary refined enumeration, the curve (in x-y plane) is
determined by F = d/dz F = 0 (call this the “Colomo-Pronko formula”). The derivation involves
both exact results from Integrable Systems, and a (non-rigorous) asymptotic analysis through a
mapping to a quite complicated random-matrix model.
Here we describe a procedure, that we call “the tangent method”, which provides a rederivation of
the Colomo-Pronko formula, through a completely different method avoiding the détour on random
matrices. Joint work with Filippo Colomo.

Gradient Gibbs measures with disorder

We consider - in uniformly strictly convex potentials case - two versions of random gradient
models. In model A) the interface feels a bulk term of random fields while in model B) the disorder
enters though the potential acting on the gradients itself. It is well known that without disorder there
are no Gibbs measures in infinite volume in dimension d=2, while there are gradient Gibbs
measures describing an infinite-volume distribution for the increments of the field, as was shown by
Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in model A)
prohibits the existence of such gradient Gibbs measures for general interaction potentials in d=2.
Cotar and Kuelske proved the existence of shift-covariant gradient Gibbs measures for model A)
when d>=3 and the expectation with respect to the disorder is zero, and for model B) when d>=2.
In recent work with Kuelske, we proved uniqueness of shift-covariance gradient Gibbs measures
with expected given tilt under the above assumptions. We also proved decay of covariances for both
models. We will also discuss in our talk new work on non-convex potentials with disorder

Planar maps, circle patterns, conformal point processes and 2D gravity

I present a model of random planar triangulations (planar maps) based on circle patterns and discuss
its properties. It exemplifies the relations between discrete random geometries in the plane,
conformally invariant point processes and two dimensional quantum gravity (Liouville theory and
topological gravity). This is joint work with Bertrand Eynard

Integrability, Topology and Discrete “Holomorphicity”

Integrable systems have applications ranging from experimental physics to profound mathematics.
An example of the latter is the fundamental role of the Temperley-Lieb algebra of statistical
mechanics in evaluating the Jones polynomial of knot and link invariants. A seemingly distinct
example is discrete “holomorphicity”, which gives a powerful tool both for proving conformal
invariance and finding integrable Boltzmann weights. In this talk I will explain the deep relations
between these two examples. In particular, I will show how utilizing topological invariants enables
discretely “holomorphic” quantities to be found easily. This allows both a deeper understanding of
why they occur, and a great generalization of where they occur. Applying these results to quantum
spin chains yields exact zero modes, such as a Fibonacci zero mode in the hard-square/golden
chain

Logarithmic correlations in percolation and other geometrical critical phenomena

The purpose of renormalisation group and quantum field theory approaches to critical phenomena is
to diagonalise the dilatation operator. Its eigenvalues are the critical exponents that determine the
power law decay of correlation functions. However, in many realistic situations the dilatation
operator is, in fact, not diagonalisable. Examples include geometrical critical phenomena, such as
percolation, in which the correlation functions describe fluctuating random interfaces. These
situations are described instead by logarithmic (conformal) field theories, in which the power-law
behavior of correlation functions is modified by logarithms. Such theories can be obtained as limits
of ordinary quantum field theories, and the logarithms originate from a resonance phenomenon
between two or more operators whose critical exponents collide in the limit. We illustrate this
phenomenon on the geometrical Q-state Potts model (Fortuin-Kasteleyn random cluster model),
where logarithmic correlation functions arise in any dimension. The amplitudes of the logarithmic
terms are universal and can be computed exactly in two dimensions, in fine agreement with
numerical checks. In passing we provide a combinatorial classification of bulk operators in the Potts
model in any dimension.

Random Matrices, Growth and Hydrodynamics Singularities

I review the approach to Hele-Shaw problem, stochastic aggregation and hydrodynamics
singularities through models of random matrices.