ICTS

In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.

The dimer model is a model of uniform perfect matching and is one of the fundamental models of statistical physics. It has many deep and intricate connections with various other models in this field, namely the Ising model and  the six-vertex model. I will focus on dimers on general (locally) planar graphs in this course.

This model has received a lot of attention in the mathematics community in the past two decades. The primary reason behind such popularity is that this model is integrable, in particular, the correlation functions can be represented exactly in a determinental form. This gives rise to a rich interplay between algebra, geometry, probability and theoretical physics. However, using the integrability seem only possible on graphs with special local structures, while the dimer model is supposed to have GFF (a canonical conformally invariant Gaussian field) type fluctuations in a much more general setting.

I will explain a new approach to understanding the dimer problem which uses the connection of this model to uniform spanning trees. In particular, it turns out, that the “winding” of uniform spanning trees (UST) is related to the height function. Using well-known SLE technology pioneered by Schramm, the scaling limit of UST is well understood. I will explain how to use this technology to also understand its limiting winding field, and then relate it to the Gaussian free field via the theory of imaginary geometry. This methods seems to be robust enough to extend to the somewhat uncharted territory of dimers on more general surfaces, e.g., the pair of pants.

I will give a general overview of the method, and ongoing and upcoming results on this approach in the first lecture.

In the second lecture, I will give the details of the proof in the simply connected case, in particular the relation to imaginary geometry, and then explain what is still left to be understood in general Riemann surfaces.

This talk is based on the following papers and some works in progress with N. Berestycki (U. Vienna) and B. Laslier (U. Paris--Diderot.).

-https://projecteuclid.org/euclid.aop/1585123322

-https://arxiv.org/abs/1610.07994

-https://arxiv.org/abs/1908.00832

Let G be a topological group and Sub(G) be the set of all closed subgroups of G. In this talk, we will define the Chabauty topology on Sub(G) and discuss its basic properties, including its relationship with geometric convergence of manifolds. We'll also spend some time discussing the topology of Sub(G) for specific G, e.g. G=R^n and G=PSL(2,R).

TBA

TBA

In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length in F). About the same time Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory.

In this series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The « dimension » of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.

Given a group G acting properly by isometries on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. We are interested in the following question: when do H and G have the same exponential growth rate ?

This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuk and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length in F). About the same time Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuk and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory.

In this series of lectures, we will revisit this problem in the general context of a group G acting on a CAT(-1) space, or more generally Gromov hyperbolic space X. We will first introduce measure-theoretic tools to study the dynamics of the geodesic flow on X. Then, given a subgroup H of G, we will explain the construction of a Hilbert-valued measure on the boundary at infinity of X. The « dimension » of this measure will allow us to characterize the equality of the exponential growth rates of H and G, in terms of the algebraic properties of the quotient G/H.

In this minicourse we will review some recent developments in the study of percolation of Gaussian free field level-sets on the hypercubic lattice in three and more dimensions. As a canonical percolation model with slow algebraic decay of correlations, the new methodologies that emerged in the course of these developments form a robust toolkit for answering some classical questions in percolation theory in non-classical contexts. One of the main focuses of the course is to work towards a proof of equality of several natural critical parameters associated with this model. This result yields, among other important properties of the model, the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  In the last of the four lectures, we will discuss a very recent result where --- in dimension 3 --- we pin down the precise large deviations behavior of the probability of connecting two points through a finite cluster (the so-called ``truncated two-point function'') at all but the critical parameter. It is possible to derive such precise rates ---- which are often impossible to obtain in the non-planar case even for the more classical percolation models --- partially thanks to the rich interplay the model enjoys with potential theory of random walks which is its another appealing feature. The proofs involve several instances of the application of rigorous renormalization methods and we will try to go through the details of at least one of them. No prior knowledge of percolation theory is necessary. The contents are mostly based on the joint works of the speaker with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.