Title: Dimers and embeddings

One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits “nice” discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. We discuss a concept of `perfect t-embeddings’ of weighted bipartite planar graphs. We believe that these embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model.

Based on:
joint work with R. Kenyon, W. Lam, S. Ramassamy;
and joint work with D. Chelkak, B. Laslier.

Title: Outliers for zeros of random polynomials and Coulomb gases

I will describe a model of zeros of random polynomials and a specific class of Colulomb gases in dimension 2 which share a lot of properties. The empirical measures of these two models have the sale deterministic limit, and they satisfy very similar large deviations principles. The goal of the talk is to study the particles (or zeroes) which lie outside of the support of the limit of empirical measures. We will show that the point process of the outliers in both models converge towards a universal point process: the Determinanal Point Process associated to the Bergman kernel.

Title: Strong asymptotic freeness for representations of independent Haar unitary matrices

We consider the unitary representation of U(n) of dimension n^q obtained by taking q tensor products of a unitary matrix or its transpose. For a finite number of independent Haar unitary matrices in U(n) or orthogonal matrices in O(n), we prove that their representations are strongly asymptotically free in the orthogonal of a vector space of small dimension. In the simplest case q = 1, we recover a result of Collins and Male with a completely different argument. The proof is based on non-backtracking operators, Weingarten calculus and high trace method. This is a joint work with B. Collins.

Title: Gaussianity of the 4D Ising model

In this talk, we will discuss the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian and its implications from the point of view of Euclidean Field Theory. Similar statements will be discussed for the λϕ4 fields over R^4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances which are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.

Title: Two "physical" characterizations of the Sine-beta process

The Sine-beta process (defined and studied by Valko-Virag and Killip-Stoiciu) describes the limiting microscopic behavior of eigenvalues for certain random matrices. One can also look at it as an infinite-volume Gibbs measure for a statistical physics system in dimension 1. With this point of view, we give two "physical" characterizations: through DLR equations, and as the unique minimiser of a free energy functional. Joint work with Dereudre-Hardy-Maïda and Erbar-Huesmann.

Title: Kosterlitz-Thouless transition and statistical reconstruction of the Gaussian free field

The Berezinskii-Kosterlitz-Thouless transition (BKT transition) is a phase transition which occurs in dimension two for spin systems such as the plane rotator model (or XY model). This phase transition was discovered by these three physicists as the first example of a topological phase transition and was rigorously understood by Fröhlich and Spencer in the 80's. I will spend the main part of my talk explaining what are these topological phase transitions. I will then survey the contributions of Fröhlich and Spencer to this theory and I will end with new results we obtained recently with Avelio Sepúlveda in this direction. The talk will be based mostly on the preprint: https://arxiv.org/abs/2002.12284

Title: On the diameter of Gaussian free field excursion clusters away from criticality

We consider the Gaussian Free Field (GFF) on $\mathbb{Z}^d$, for $d\geq 3$, and its excursions above a given real height $h$. As $h$ varies, this defines a natural percolation model with slow decay of correlations. Sharpness of phase transition has been established very recently for this model through renormalization techniques. As a direct consequence, the probability that an excursion cluster has a diameter $N$ decays at least stretched exponentially in $N$ for any $h\neq h_*$, where $h_*$ is the percolation critical point. In this talk we shall discuss sharp bounds this tail. In dimensions $d\geq 4$, the decay is exponential, similarly to Bernoulli percolation; while in dimension $d=3$ it decays as $\exp[-\frac{\pi}{3}(h-h_*)^2\frac{N}{\log N}]$. Remarkably, such a precise bound is not known even for Bernoulli percolation. We will explain how the so called "entropic repulsion phenomenon" allows one to prove such large deviation results for the GFF. Joint work with S. Goswami and P.-F. Rodriguez.

Title: Mean-Field Quantum Spin Glasses

The theory of classical mean-field spin glasses is a well-established and celebrated field within probability theory. The addition of a constant perpendicular magnetic field introduces a non-commuting term into the energy of such spin glasses and hence causes quantum effects. The main aim of this talk is to give an overview over some of the motivations for the study of quantum spin glasses. I will also review some first mathematical results in this field. Among them is a derivation of the key features of the thermal phase diagram of the simplest of all mean-field spin glasses, the quantum random energy model and its generalisations. (Based on joint works with C. Manai).

Title: Field theory as a limit of interacting quantum Bose gases

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to a classical field theory in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the field theory is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.

Title: The eight-vertex model via dimers

The eight-vertex model is an ubiquitous description that generalizes several spin systems, the more common six-vertex model, Ashkin-Teller models, and others. In a special "free-fermion" regime, it is known since the work of Fan, Lin, Wu in the late 60s that the model can be mapped to non-bipartite dimers. However, no general theory is known for dimers in the non-bipartite case, contrary to the extensive rigorous description of Gibbs measures by Kenyon, Okounkov, Sheffield for bipartite dimers. In this talk I will show how to transform these non-bipartite dimers into bipartite ones, on generic planar graphs. I will mention a few consequences: computation of long-range correlations, criticality and critical exponents, and their "exact" application to Z-invariant regimes on isoradial graphs.

Title: Sharp asymptotics of correlation functions in the subcritical long-range random-cluster and Potts models

For a family of random-cluster models with cluster weights q ≥ 1 defined on Z^d, we will prove that the probability that 0 is connected to x is asymptotically equal to β/q χ(β)^2 J(0,x). Our proof will be based on the OSSS inequality. The new method presented in this talk has the potential to be applied to study subcritical phases of long-range spin models for which there exists a random-cluster representation which is one-monotonic.

Title: Expander graphs and the spectral gap of random regular graphs

Expander graphs are sparse graphs with good connectivity properties. I review some basics of expander graphs, and in particular explain a spectral characterization of the expansion properties of regular graphs in terms of the spectral gap. I then explain how fluctuations of the spectral gap can be analysed for random regular graphs whose degree grows with the number of vertices. This shows that approximately 27% of all regular graphs are optimal expanders.

Title: Edge Universality for non-Hermitian Random Matrices

We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. Previously non-Hermitian edge universality had only been established under the condition of four matching moments [Tao, Vu (2015)]. The proof relies on the recently obtained optimal local law in the cusp regime of Hermitian random matrices, and a supersymmetric estimate on the least singular value of shifted Ginibre ensembles. This estimate on the least singular value improves the classical smoothing bound from [Sankar, Spielman, Teng (2006)] in the transitional edge regime.

Title: Fluctuations and mixing of Internal DLA on cylinders

Internal DLA models the growth of a random cluster by subsequent aggregation of particles. At each step, a new particle starts inside the cluster, and it performs a simple random walk until reaching an unoccupied site, where it settles. When particles move on a cylinder graph GxZ this defines a positive recurrent Markov chain on cluster configurations. In this talk I will address the following questions: How does a typical configuration look like? How long does it take for the process to forget its initial profile? Partly based on joint work with Lionel Levine (Cornell).

Title: Anchored expansion in supercritical percolation on nonamenable graphs

Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G. We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n. We deduce that:

1. Every infinite cluster has anchored expansion (a relaxation of having positive Cheeger constant), and so is nonamenable in some weak sense. This answers positively a question of Benjamini, Lyons, and Schramm (1997).

2. Various observables, including the percolation probability and the truncated susceptibility (which was not even known to be finite!) are analytic functions of p throughout the entire supercritical phase.

3. A RW on an infinite cluster returns to the origin at time 2n with probability whose logarithm is proportional to minus cubic root of n.

Joint work with Tom Hutchcroft.

Title: Nonlinear Gibbs measures as the limit of equilibrium quantum Bose gases

I will discuss the rigorous derivation of nonlinear Gibbs measures starting from a linear quantum system of bosons at a positive temperature. In three dimensions, the nonlinear classical measures have to be defined via a renormalization procedure, while the quantum system is well defined without any renormalization. By simply tuning the chemical potential, we obtain a counter-term for the diverging interactions which implements the Wick renormalization of the limit classical theory. The proof relies on a novel method to control the quantum variance. This is joint work with Mathieu Lewin and Nicolas Rougerie.

Title: Spectral theory of one-dimensional quasicrystals

We give an introduction into the field of aperiodic order also known as mathematical quasicrystals. We then focus on spectral theory of Hamiltonians arising in the quantum mechanical treatment of one-dimensional models. We explain that these Hamiltonians have a tendency to singular continuous spectrum of Lebesgue measure zero. We will also indicate some recent applications obtained in joined work with Rostislav Grigorchuk, Tatiana Nagnibeda and Daniel Sell.

Title: Stability of the Laughlin phase in presence of interactions

The Laughlin wave function is at the basis of the description of the fractional quantum Hall effect (FQHE), nevertheless, many of its fundamental properties are yet to be understood. After an introduction to Laughlin theory, I will present a model for the response of FQHE charge carriers to variations of the external potential and of the inter-particle interaction.

Our main result is that the energy is asymptotically captured by the minimum of an effective func- tional with variational constraints fixed by the incompressibility of the Laughlin phase. Moreover, as was already known for the non-interacting case, we show that the one-body density converges to the characteristic function of a set.

Joint work with Nicolas Rougerie.

Title: How much can the eigenvalues of a random Hermitian matrix fluctuate?

One of the most classical results of random matrix theory is Wigner's theorem, which says that the eigenvalues of a large random Hermitian matrix, drawn from the Gaussian Unitary Ensemle (GUE), are approximately distributed according to the semicircle distribution. I will discuss some recent joint work with T. Claeys (UCLouvain) B. Fahs (Imperial College London), and G. Lambert (University of Zürich) describing how much the eigenvalues can fluctuate around the semicircle distribution.

Title: Limit shapes in statistical mechanics

This talk is an introduction into what is known in statistical mechanics as the limit shape phenomenon. The essence of this phenomenon is that the states of the model in the large volume limit are exponentially unlikely outside of a small neighborhood of a single state called the limit shape. In the vicinity of the limit shape states are distributed according to the Gaussian distribution. The talk will be focused on the six vertex model.

Title: Random spanning forests and hyperbolic symmetry

The arboreal gas is a probability measure on unrooted spanning forests of a graph G that gives weight beta to each edge in a forest. This specializes to the uniform measure on unrooted forests when beta is one. While it is well-known that random rooted forests are related to the Gaussian free field, I will discuss how the arboreal gas is instead related to a spin system with hyperbolic symmetry. This symmetry is responsible for some surprising behaviour.

Based on joint work in progress with Roland Bauerschmidt, Nick Crawford, and Andrew Swan.

Title: Discrete multi-point and vector-valued observables in percolation.

As is well known in rue du Lièvre, the critical percolation model on the hexagonal grid is proven to be conformally invariant in the scaling limit.

The proof is based on a certain combinatorial object (the Smirnov's observable) that controls the crossing probability of a domain with four marked boundary points.

During the talk we will discuss a natural generalization of the observable mentioned above. This observable allows / should allow to control the probabilities of certain multi-point events (like connectivity patterns and Schramm's formula for percolation) *directly* (without using SLE).

I will formulate a problem, that being solved would make our ongoing project much less ongoing :)

PS. If you are not an expert in percolation but you feel comfortable with modular forms, Riemann–Hilbert problems or Birkhoff–Grothendieck theorem, you are especially warmly welcome.

Title: The hard-core model in discrete 2D: ground states, dominance, Gibbs measures

The hard-core model describes a system of non-overlapping identical hard spheres
in a space or on a lattice (more generally, on a graph). An interesting open problem is:
do hard disks in a plane admit a unique Gibbs measure at high density? It seems natural
to approach this question by possible discrete approximations where disks must have
the centers at sites of a lattice or vertexes of a graph.

In this talk, I will report on a progress achieved for the models on a unit triangular lattice A_2
and a unit square lattice Z^2 for a general value of disk diameter D (in the Euclidean metric).
We analyze the structure of Gibbs measures for large fugacities (i.e., high densities) by means
of the Pirogov-Sinai theory and its modifications, including dominance among periodic ground states.

On A_2 we give a complete description of the set of extreme Gibbs
measures; the answer is provided in terms of prime decomposition of the
Löschian number D^2 in the Eisenstein integer ring. Here, the extreme
Gibbs measures are generated by D-sub-lattices, their shifts and reflections.

On Z^2, we have to exclude the values of D with sliding; for the
remaining exclusion distances the answer is given in terms of solutions to
a discrete minimization problem. The latter is connected to Norm equations in
the cyclotomic integer ring Z[zeta], where zeta is a primitive 12th
root of unity.

Parts of our argument contain computer-assisted proofs: identification of instances of sliding,
resolution of dominance issues.

This is a joint work with A. Mazel and Y. Suhov.

Title: Renormalization Group approach to universality in non-integrable systems

I will present a review of results obtained by rigorous Renormalization Group methods on the universality problem in 2D statistical mechanics systems, including Ising, Dimer or vertex models and their non integrable generalizations. The exponents and amplitudes are written as series, convergent for values of parameters near the free fermion point and dependent on all the parameters. The validity of universal scaling relations follows by subtle cancellations due to emergent symmetries in the series expansions.

Title: Weil-Petersson curves and finite total curvature.

In 2006 Takhtajan and Teo defined a Weil-Petersson metric making universal Teichmuller space (essentially the space of planar quasicircles) into a disconnected Hilbert manifold. The Weil-Petersson class is the unique connected component containing all smooth curves, and consists of certain rectifiable quasicircles. There are several known function theoretic characterizations of WP curves, including connections to Loewner's equation and SLE. In this lecture, I will describe some new geometric characterizations that say a quasicircle is WP iff some measure of local curvature is square integrable over all locations and scales. Here local curvature can be measured using quantities such as: beta-numbers, Menger curvature, integral geometry, inscribed polygons, tangent directions, and associated convex hulls and minimal surfaces in hyperbolic 3-space.

Title: Fermionic eigenvector moment flow

We first present known results and open problems on the study of eigenvector statistics of large random matrices such as complete delocalization, quantum unique ergodicity or asymptotic entry distribution. We willl see how we can obtain some of these properties dynamically using the Dyson Brownian motion and the Bourgade-Yau eigenvector moment flow: it consists of a parabolic equation followed by eigenvector moments. We will then present new moment observables constructed from Grassmann variables which follow a similar equation and which can be seen as a Fermionic counterpart to the (Bosonic) original ones. By combining the information obtained through the study of these two families of observables, we can compute, previously intractable, correlations between eigenvectors.