The informal seminar is open to all visitors at IHES, researchers, and students in the Paris area. We aim at keeping the informality, which implies, many (friendly) interruptions, many (naive and basic, occasionally smart) questions, spontaneous (blackboard only, sometimes unfinished) talks, and a welcoming atmosphere. We expect the seminars to be in English only.
Organizers: Thierry Bodineau, Piet Lammers, Yilin Wang
If you want to be added to the mailing list, please write to Yilin ([email protected])
Past Sessions
Time and place:
IHES, Amphi Motchane, Fridays at 2 pm (unless indicated otherwise)
Talks are 90 minutes long.
Talks are 90 minutes long.
Past Seminars
Monday June 24 - 11h (Amphi Motchane)
Chenlin Gu (YMSC, Tsinghua University)
Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process
This talk presents a quantitative homogenization for non-gradient exclusion process. The main strategy roots from the quantitative homogenization theory developed by Armstrong, Kuusi, Mourrat and Smart, and was already implemented in the previous work by Giunti-Gu-Mourrat' 22 in an interacting particle system without exclusion. The new challenges here come from the hard core constraint of the particle number and the curse of dimension, and I will explain how to overcome them by a new coarse-grained strategy. As an application, our result can be integrated into the classical work Funaki-Uchiyama-Yau' 96 and yield a quantitative hydrodynamic limit. This talk is based on a joint work with Tadahisa Funaki (BIMSA) and Han Wang (Qiuzhen College, Tsinghua University).
Monday June 10 - 14h (Amphi Motchane)
I. Stuhl, Y. Soukhov (Penn State)
Packing hard spheres on lattices
In this joint talk we focus on some recent results on packings of identical hard spheres of diameter D on 2D- and 3D- lattices and graphs (a unit triangular lattice A_2, a unit honeycomb graph H_2, a unit square lattice Z^2, a unit cubic lattice Z^3). In particular, we will use connections with the algebraic number theory. Our results identify dense-packing configurations and their random "perturbations" (extreme Gibbs distributions) describing high-density "pure phases" of the hard-sphere lattice model of statistical mechanics.
Friday May 31
Charlotte Dietze (LMU Munich)
Weyl formulae for some singular metrics
I will talk about the eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. It is joint work in progress with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat.
Friday May 24
Diederik van Engelenburg (University of Lyon 1)
On duality between continuous spin models and height functions
Classical spin models taking value in the circle are naturally dual to spin models taking value in the integers (on the dual graph). Unlike in the context of Ising/Potts models, this duality is only visible at the level of order-disorder operators, there are no bijections between the models (as far as we know). I will revisit these well known relations, and will argue how the continuous symmetry group of S^1 will help to prove different results for the spin model and dual height function model. Notably, I will show a type of Gaussian domination holds for the height function on any graph, I will mention how it can be used to prove the Berezinskii--Kosterlitz--Thouless transition and time permitting, I will present some other results. All is based on joint work with Marcin Lis.
Friday May 3 (Salle de conférence)
Thierry Lévy (Sorbonne Université)
Finite determinantal point processes, random subgraphs and random linear subspaces
On a finite connected graph, the product of the non-zero eigenvalues of the Laplacian counts the rooted spanning trees, according to a theorem often attributed to Kirchhoff (1847), or sometimes to Sylvester (1857). Among many generalisations of this classical result, those of Zaslavsky (1982), Forman (1993) and Kenyon (2011) state that when we twist the Laplacian by putting a sign or a phase, complex or quaternionic, on each edge, its determinant counts covering forests of unicycles, with appropriate weights. A common feature of all these results is that the random subgraphs naturally associated to each of these situations (uniform spanning trees and random covering forests of unicycles), seen as random subsets of the (finite) set of edges of the ambient graph, are determinantal point processes.
I will present some results of an ongoing joint work with With Adrien Kassel (CNRS, ENS Lyon) in which we investigate further extensions of these results to the covariant Laplacian associated with an arbitrary unitary connection, that is, to the Laplacian twisted by a unitary matrix on each edge.
In a first part, I will describe the classical results of Kirchhoff and Forman, then (from a perhaps slightly unorthodox point of view) determinantal point processes on finite sets, and explain what the ones have to do with the others. In a second part, I will describe the measures on Grassmannians that we introduced with Adrien Kassel, explain why they are relevant to the understanding of the twistes Laplacian, and finally describe, to the extent that we understand them, the new random objects that appear over a graph endowed with a unitary connection.
Friday April 26
Fredrik Viklund (KTH)
Free energy of a Coulomb gas on a Jordan domain
Consider a Coulomb gas restricted to a Jordan domain in the complex plane. How does the asymptotic expansion of the free energy depend on the geometry of the domain, as the number of particles tends to infinity? I will explain how this problem is related to the Grunsky operator -- a classical tool in complex analysis -- and how this in turn reveals a close connection to the Loewner energy and other interesting domain functionals. I will further discuss the effect of corners, which turns out to be universal in a certain sense. Most main players will be introduced in the talk. This is joint work with Kurt Johansson (KTH).
Friday April 19
Clément Mouhot (Cambridge & IHES)
Quantitative hydrodynamic limit for interacting particle systems: panorama and recent advances
We present an introduction to the theory of hydrodynamic limit for interacting particle systems; focused on jump and spin processes on a lattice. This theory is born in the 1970s but the first quantitative results were only obtained in the last decade. We will review recent advances on quantitative methods in the parabolic scaling, and present a recent joint work.
Friday March 15
Shuta Nakajima (Meiji University)
Upper Tail Large Deviation and Its Application to Maximal Edge-Traversal Time in First-Passage Percolation
In this talk, we will consider the maximal edge-traversal time in First-passage percolation. This is the maximum value that optimal paths can take when traversing random weighted graphs. We will discuss our use of upper tail large deviation and the resampling argument as primary analytical tools. These methods have enabled us to determine the leading-order asymptotic of the maximal edge-traversal time for several Weibull distributions. This talk is based on two works: one with Clément Cosco and the other with Ryoki Fukushima.
Friday March 1
Tomohiro Sasamoto (Tokyo institute of technology)
One-dimensional Kardar-Parisi-Zhang (KPZ) equation and its universality
In 1986, Kardar, Parisi and Zhang introduced a model equation for a growing surface, in the form of a nonlinear partial differential equation with noise[1]. In the original paper they applied a dynamical renormalization group analysis to demonstrate its universal nature, which is one of the first identified non-equilibrium universality classes (KPZ universality class). Since then their equation (KPZ equation) has been accepted as a standard model in non-equilibrium statistical mechanics.
In this talk, we focus on its one dimensional version because it has attracted particular attention in the last decade or so. Mathematically there had been an issue of well-definendness of the equation itself, which was solved by a few different ideas. There is also a high precision experiment using liquid crystal. An important step was the discovery of an exact solution in 2010[2], which confirmed that the height fluctuation is of O(t^(1/3)) and its universal distribution is given by the Tracy-Widom distribution from random matrix theory. Since then there have been a large amount of studies on its generalizations, which now forms a field of “integrable probability”. The activity still continues. Universal behaviors for general initial conditions can now be studied (“KPZ fixed point”). Very recently we have found a direct connection between KPZ systems and free fermion at finite temperature[3].
A remarkable aspect of one dimensional KPZ is its unexpectedly wide universality. For example, KPZ universality is expected to appear in long time behaviors of many one-dimensional Hamiltonian dynamical systems such as anharmonic chains [4]. This is surprising because time-evolution of such systems are deterministic and there are apparently no growing surface with noise. More recently people have observed appearance of KPZ behaviors in dynamical properties of quantum spin chains[5], first in numerical simulations but more recently in real experiments. These discoveries have been attracting considerable attention but theoretical foundations are not yet satisfactory.
In the talk we start from recalling basics of KPZ and then explain these recent developments.
References
[1] M. Kardar, G. Parisi, and Y. C. Zhang, Dynamic scaling of growing interfaces,
Phys. Rev. Lett., 56, 889–892 (1986).
[2] T. Sasamoto and H. Spohn, One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality, Phys. Rev. Lett., 104:230602 (2010);
[3] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approach through
periodic and free boundary Schur measures, arxiv2204.08420.
[4] H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154,
1191–1227 (2014).
[5] M. Ljubotina, M. Znidaric, T. Prosen, Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet,
Phys. Rev. Lett. 122, 210602 (2019).
Friday Feburary 2, 2024
Laurent Ménard (Paris Nanterre)
Random triangulations coupled with Ising model
In this talk, we will investigate geometric properties of random planar triangulations coupled with an Ising model. This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps. I will briefly explain this phenomenon, and why it hints at a different universality class than the Brownian sphere.
In the second part of the talk, we will focus on the geometry of spin clusters in the infinite volume setting. We will exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures. A lot of precise information can be derived in all regimes. In particular, we will see that in the whole supercritical temperature regime, critical exponents for spin clusters are the same as for critical Bernoulli site percolation on uniform planar triangulations.
Based on joint works with Marie Albenque and Gilles Schaeffer
Friday January 26, 2024
Roger Van Peski (KTH & Columbia university)
New limits in discrete random matrix theory
Random matrices over the integers and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory, topology and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices. I will outline this area (no background in discrete random matrix theory will be assumed), discuss exact results and their parallels with classical random matrix theory, and give probabilistic results for products of random matrices. The latter yield interesting new local limit objects analogous to the extended sine and Airy processes in classical random matrix theory.
Friday January 19, 2024
Hao Wu (YMSC Tsinghua University)
Multiple SLEs and Dyson Brownian motion
Multiple SLEs come naturally as the scaling limit of multiple interfaces in 2-dimensional statistical physics models. Dyson Brownian motion usually describes the movement of trajectory of independent Brownian motions under mutual repulsion. In this talk, we will describe the connection between multiple SLEs and Dyson Brownian motion. The talk has two parts.
In the first part, we take critical FK-Ising model as an example and explain the emergence of multiple SLEs. We give the connection probabilities of multiple SLEs. Such probabilities are related to solutions to BPZ equations in conformal field theory.
In the second part, we explain the connection between multiple SLEs and Dyson Brownian motion. It turns out that, under proper time-parameterization, and conditioning on a rare event, the driving function of multiple SLEs becomes Dyson Brownian motion. Using such a connection, we may translate estimates on Dyson Brownian motion to estimates on multiple SLEs.
Friday December 1
Lucas D'Alimonte (Universite de Fribourg)
Ornstein—Zernike theory for the 2D near-critical random cluster model
In this talk, we will discuss the classical Ornstein--Zernike theory for the random-cluster models (also known as FK percolation). In its modern form, it is a very robust theory, which most celebrated output is the computation of the asymptotically polynomial corrections to the pure exponential decay of the two-points correlation function of the random-cluster model in the subcritical regime. We will present an ongoing project that extends this theory to the near-critical regime of the two-dimensional random-cluster model, thus providing a precise understanding of the Ornstein—Zernike asymptotics when p approaches the critical parameter p_c. The output of this work is a formula encompassing both the critical behaviour of the system when looked at a scale negligible with respect to its correlation length, and its subcritical behaviour when looked at a scale way larger than its correlation length. Based on a joint work with Ioan Manolescu.
Friday November 17
Paul Melotti (Paris - Saclay)
Dynamics of discrete holomorphic functions via combinatorics
There exists several ways to discretize holomorphic functions. One of them is based on Schramm's orthogonal circle patterns, and their generalization to so-called "cross-ratio maps" and "P-nets". These systems are naturally associated with a discrete time dynamics. I will mention results and open problems about this dynamics, in particular the "Devron" property, that states that singularities cannot be escaped by reversing time. I will show that these questions can be tackled by identifying those (birational) dynamics with the dSKP equation, which itself can be identified with partition functions of (oriented) dimers, a famously integrable model of statistical mechanics. Based on joint works with Niklas Affolter, Béatrice de Tilière, Jean-Baptiste Stiegler.
Friday November 10
Masha Gordina (University of Connecticut)
Ergodicity for Langevin dynamics with singular potentials
We discuss Langevin dynamics of N particles on R^d interacting through a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus (Bakry-Emery). In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on joint work with F.Baudoin and D.Herzog.
Friday November 3
Elton Hsu (Northwestern University)
The Parisi Formula via Stochastic Analysis
The Parisi formula is a fundamental result in spin glass theory. It gives a variational characterization of the asymptotic limit of the expected free energy. The upper bound is a consequence of an interpolation identity due to F. Guerra and the lower bound is a celebrated result of M. Talagrand.
In this talk I will present a new approach to (an enhanced version of) Guerra's identity using stochastic analysis, more specifically Brownian motion and Ito’s calculus. This approach is suggested by the form of the Parisi formula in which the solution of a Hamilton-Jacobi equation is involved. It helps in many ways to illuminate the original method of Guerra and suggests some possible approaches to the significantly deeper lower bound, which has been intensively studied since Talagrand’s work. Among the techniques from stochastic analysis we will use include path space integration by parts for the Wiener measure, Girsanov’s transform (i.e., exponential martingales), and probabilistic representation of solutions to (linear) partial differential equations. The key observation is that the nonlinear Hamilton-Jacobi partial differentiation equation figuring in Parisi’s variation formula becomes linear after differentiating with respect to Guerra’s interpolation parameter, thus bringing the full strength of stochastic analysis based on Ito’s calculus into play. It is hoped that this approach will shed some lights on the much more difficult lower bound in the Parisi formula.
Friday October 13
Max Fathi ( LJLL and LPSM)
Globally lipschitz transport maps
One way of proving probabilistic functional inequalities (concentration inequalities, logarithmic Sobolev inequalities...) is to use a change of variables, to transfer them from a simple reference measure (typically, Gaussian) to more general settings. One example of this is the Caffarelli contraction theorem, which states that uniformly log-concave measures can be realized as images of standard Gaussian measures by globally lipschitz maps, using the L2 optimal transport map. One open problem in this direction is to find an analogue of Caffarelli's theorem in the Riemannian setting.
In this talk, I will present a stochastic construction of non-optimal maps, due to Kim and Milman, and Lipschitz estimates in various settings, including certain measures on Riemannian manifolds. Joint work with Dan Mikulincer and Yair Shenfeld.
Friday September 22
Nicolas Curien (Paris Saclay)
Ideal Poisson-Voronoi tiling
We study the limit in low intensity of Poisson--Voronoi tessellations in hyperbolic spaces.
In contrast to the Euclidean setting, a limiting non-trivial ideal tessellation appears as the intensity tends to 0. The tessellation obtained is a natural Möbius-invariant decomposition of the hyperbolic space into countably many infinite convex polytopes, each with a unique end. We study its basic properties, in particular the geometric features of its cells.
Based on joint works with Matteo d'Achille, Nathanel Enriquez, Russell Lyons and Meltem Unel.
Friday September 15
Marco Carfagnini (UC San Diego)
Spectral gaps via small deviations
In this talk we will discuss spectral gaps of second order differential operators and their connection to limit laws such as small deviations and Chung’s laws of the iterated logarithm. The main focus is on hypoelliptic diffusions such as the Kolmogorov diffusion and horizontal Brownian motions on Carnot groups. If time permits, we will discuss spectral properties and existence of spectral gaps on general Dirichlet metric measure spaces.This talk is based on joint works with Maria (Masha) Gordina and Alexander (Sasha) Teplyaev.
Chenlin Gu (YMSC, Tsinghua University)
Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process
This talk presents a quantitative homogenization for non-gradient exclusion process. The main strategy roots from the quantitative homogenization theory developed by Armstrong, Kuusi, Mourrat and Smart, and was already implemented in the previous work by Giunti-Gu-Mourrat' 22 in an interacting particle system without exclusion. The new challenges here come from the hard core constraint of the particle number and the curse of dimension, and I will explain how to overcome them by a new coarse-grained strategy. As an application, our result can be integrated into the classical work Funaki-Uchiyama-Yau' 96 and yield a quantitative hydrodynamic limit. This talk is based on a joint work with Tadahisa Funaki (BIMSA) and Han Wang (Qiuzhen College, Tsinghua University).
Monday June 10 - 14h (Amphi Motchane)
I. Stuhl, Y. Soukhov (Penn State)
Packing hard spheres on lattices
In this joint talk we focus on some recent results on packings of identical hard spheres of diameter D on 2D- and 3D- lattices and graphs (a unit triangular lattice A_2, a unit honeycomb graph H_2, a unit square lattice Z^2, a unit cubic lattice Z^3). In particular, we will use connections with the algebraic number theory. Our results identify dense-packing configurations and their random "perturbations" (extreme Gibbs distributions) describing high-density "pure phases" of the hard-sphere lattice model of statistical mechanics.
Friday May 31
Charlotte Dietze (LMU Munich)
Weyl formulae for some singular metrics
I will talk about the eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. It is joint work in progress with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat.
Friday May 24
Diederik van Engelenburg (University of Lyon 1)
On duality between continuous spin models and height functions
Classical spin models taking value in the circle are naturally dual to spin models taking value in the integers (on the dual graph). Unlike in the context of Ising/Potts models, this duality is only visible at the level of order-disorder operators, there are no bijections between the models (as far as we know). I will revisit these well known relations, and will argue how the continuous symmetry group of S^1 will help to prove different results for the spin model and dual height function model. Notably, I will show a type of Gaussian domination holds for the height function on any graph, I will mention how it can be used to prove the Berezinskii--Kosterlitz--Thouless transition and time permitting, I will present some other results. All is based on joint work with Marcin Lis.
Friday May 3 (Salle de conférence)
Thierry Lévy (Sorbonne Université)
Finite determinantal point processes, random subgraphs and random linear subspaces
On a finite connected graph, the product of the non-zero eigenvalues of the Laplacian counts the rooted spanning trees, according to a theorem often attributed to Kirchhoff (1847), or sometimes to Sylvester (1857). Among many generalisations of this classical result, those of Zaslavsky (1982), Forman (1993) and Kenyon (2011) state that when we twist the Laplacian by putting a sign or a phase, complex or quaternionic, on each edge, its determinant counts covering forests of unicycles, with appropriate weights. A common feature of all these results is that the random subgraphs naturally associated to each of these situations (uniform spanning trees and random covering forests of unicycles), seen as random subsets of the (finite) set of edges of the ambient graph, are determinantal point processes.
I will present some results of an ongoing joint work with With Adrien Kassel (CNRS, ENS Lyon) in which we investigate further extensions of these results to the covariant Laplacian associated with an arbitrary unitary connection, that is, to the Laplacian twisted by a unitary matrix on each edge.
In a first part, I will describe the classical results of Kirchhoff and Forman, then (from a perhaps slightly unorthodox point of view) determinantal point processes on finite sets, and explain what the ones have to do with the others. In a second part, I will describe the measures on Grassmannians that we introduced with Adrien Kassel, explain why they are relevant to the understanding of the twistes Laplacian, and finally describe, to the extent that we understand them, the new random objects that appear over a graph endowed with a unitary connection.
Friday April 26
Fredrik Viklund (KTH)
Free energy of a Coulomb gas on a Jordan domain
Consider a Coulomb gas restricted to a Jordan domain in the complex plane. How does the asymptotic expansion of the free energy depend on the geometry of the domain, as the number of particles tends to infinity? I will explain how this problem is related to the Grunsky operator -- a classical tool in complex analysis -- and how this in turn reveals a close connection to the Loewner energy and other interesting domain functionals. I will further discuss the effect of corners, which turns out to be universal in a certain sense. Most main players will be introduced in the talk. This is joint work with Kurt Johansson (KTH).
Friday April 19
Clément Mouhot (Cambridge & IHES)
Quantitative hydrodynamic limit for interacting particle systems: panorama and recent advances
We present an introduction to the theory of hydrodynamic limit for interacting particle systems; focused on jump and spin processes on a lattice. This theory is born in the 1970s but the first quantitative results were only obtained in the last decade. We will review recent advances on quantitative methods in the parabolic scaling, and present a recent joint work.
Friday March 15
Shuta Nakajima (Meiji University)
Upper Tail Large Deviation and Its Application to Maximal Edge-Traversal Time in First-Passage Percolation
In this talk, we will consider the maximal edge-traversal time in First-passage percolation. This is the maximum value that optimal paths can take when traversing random weighted graphs. We will discuss our use of upper tail large deviation and the resampling argument as primary analytical tools. These methods have enabled us to determine the leading-order asymptotic of the maximal edge-traversal time for several Weibull distributions. This talk is based on two works: one with Clément Cosco and the other with Ryoki Fukushima.
Friday March 1
Tomohiro Sasamoto (Tokyo institute of technology)
One-dimensional Kardar-Parisi-Zhang (KPZ) equation and its universality
In 1986, Kardar, Parisi and Zhang introduced a model equation for a growing surface, in the form of a nonlinear partial differential equation with noise[1]. In the original paper they applied a dynamical renormalization group analysis to demonstrate its universal nature, which is one of the first identified non-equilibrium universality classes (KPZ universality class). Since then their equation (KPZ equation) has been accepted as a standard model in non-equilibrium statistical mechanics.
In this talk, we focus on its one dimensional version because it has attracted particular attention in the last decade or so. Mathematically there had been an issue of well-definendness of the equation itself, which was solved by a few different ideas. There is also a high precision experiment using liquid crystal. An important step was the discovery of an exact solution in 2010[2], which confirmed that the height fluctuation is of O(t^(1/3)) and its universal distribution is given by the Tracy-Widom distribution from random matrix theory. Since then there have been a large amount of studies on its generalizations, which now forms a field of “integrable probability”. The activity still continues. Universal behaviors for general initial conditions can now be studied (“KPZ fixed point”). Very recently we have found a direct connection between KPZ systems and free fermion at finite temperature[3].
A remarkable aspect of one dimensional KPZ is its unexpectedly wide universality. For example, KPZ universality is expected to appear in long time behaviors of many one-dimensional Hamiltonian dynamical systems such as anharmonic chains [4]. This is surprising because time-evolution of such systems are deterministic and there are apparently no growing surface with noise. More recently people have observed appearance of KPZ behaviors in dynamical properties of quantum spin chains[5], first in numerical simulations but more recently in real experiments. These discoveries have been attracting considerable attention but theoretical foundations are not yet satisfactory.
In the talk we start from recalling basics of KPZ and then explain these recent developments.
References
[1] M. Kardar, G. Parisi, and Y. C. Zhang, Dynamic scaling of growing interfaces,
Phys. Rev. Lett., 56, 889–892 (1986).
[2] T. Sasamoto and H. Spohn, One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality, Phys. Rev. Lett., 104:230602 (2010);
[3] T. Imamura, M. Mucciconi, T. Sasamoto, Solvable models in the KPZ class: approach through
periodic and free boundary Schur measures, arxiv2204.08420.
[4] H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154,
1191–1227 (2014).
[5] M. Ljubotina, M. Znidaric, T. Prosen, Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet,
Phys. Rev. Lett. 122, 210602 (2019).
Friday Feburary 2, 2024
Laurent Ménard (Paris Nanterre)
Random triangulations coupled with Ising model
In this talk, we will investigate geometric properties of random planar triangulations coupled with an Ising model. This model is known to undergo a combinatorial phase transition at an explicit critical temperature, for which its partition function has a different asymptotic behavior than uniform maps. I will briefly explain this phenomenon, and why it hints at a different universality class than the Brownian sphere.
In the second part of the talk, we will focus on the geometry of spin clusters in the infinite volume setting. We will exhibit a phase transition for the existence of an infinite spin cluster: for critical and supercritical temperatures, the root spin cluster is finite almost surely, while it is infinite with positive probability for subcritical temperatures. A lot of precise information can be derived in all regimes. In particular, we will see that in the whole supercritical temperature regime, critical exponents for spin clusters are the same as for critical Bernoulli site percolation on uniform planar triangulations.
Based on joint works with Marie Albenque and Gilles Schaeffer
Friday January 26, 2024
Roger Van Peski (KTH & Columbia university)
New limits in discrete random matrix theory
Random matrices over the integers and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory, topology and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices. I will outline this area (no background in discrete random matrix theory will be assumed), discuss exact results and their parallels with classical random matrix theory, and give probabilistic results for products of random matrices. The latter yield interesting new local limit objects analogous to the extended sine and Airy processes in classical random matrix theory.
Friday January 19, 2024
Hao Wu (YMSC Tsinghua University)
Multiple SLEs and Dyson Brownian motion
Multiple SLEs come naturally as the scaling limit of multiple interfaces in 2-dimensional statistical physics models. Dyson Brownian motion usually describes the movement of trajectory of independent Brownian motions under mutual repulsion. In this talk, we will describe the connection between multiple SLEs and Dyson Brownian motion. The talk has two parts.
In the first part, we take critical FK-Ising model as an example and explain the emergence of multiple SLEs. We give the connection probabilities of multiple SLEs. Such probabilities are related to solutions to BPZ equations in conformal field theory.
In the second part, we explain the connection between multiple SLEs and Dyson Brownian motion. It turns out that, under proper time-parameterization, and conditioning on a rare event, the driving function of multiple SLEs becomes Dyson Brownian motion. Using such a connection, we may translate estimates on Dyson Brownian motion to estimates on multiple SLEs.
Friday December 1
Lucas D'Alimonte (Universite de Fribourg)
Ornstein—Zernike theory for the 2D near-critical random cluster model
In this talk, we will discuss the classical Ornstein--Zernike theory for the random-cluster models (also known as FK percolation). In its modern form, it is a very robust theory, which most celebrated output is the computation of the asymptotically polynomial corrections to the pure exponential decay of the two-points correlation function of the random-cluster model in the subcritical regime. We will present an ongoing project that extends this theory to the near-critical regime of the two-dimensional random-cluster model, thus providing a precise understanding of the Ornstein—Zernike asymptotics when p approaches the critical parameter p_c. The output of this work is a formula encompassing both the critical behaviour of the system when looked at a scale negligible with respect to its correlation length, and its subcritical behaviour when looked at a scale way larger than its correlation length. Based on a joint work with Ioan Manolescu.
Friday November 17
Paul Melotti (Paris - Saclay)
Dynamics of discrete holomorphic functions via combinatorics
There exists several ways to discretize holomorphic functions. One of them is based on Schramm's orthogonal circle patterns, and their generalization to so-called "cross-ratio maps" and "P-nets". These systems are naturally associated with a discrete time dynamics. I will mention results and open problems about this dynamics, in particular the "Devron" property, that states that singularities cannot be escaped by reversing time. I will show that these questions can be tackled by identifying those (birational) dynamics with the dSKP equation, which itself can be identified with partition functions of (oriented) dimers, a famously integrable model of statistical mechanics. Based on joint works with Niklas Affolter, Béatrice de Tilière, Jean-Baptiste Stiegler.
Friday November 10
Masha Gordina (University of Connecticut)
Ergodicity for Langevin dynamics with singular potentials
We discuss Langevin dynamics of N particles on R^d interacting through a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus (Bakry-Emery). In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on joint work with F.Baudoin and D.Herzog.
Friday November 3
Elton Hsu (Northwestern University)
The Parisi Formula via Stochastic Analysis
The Parisi formula is a fundamental result in spin glass theory. It gives a variational characterization of the asymptotic limit of the expected free energy. The upper bound is a consequence of an interpolation identity due to F. Guerra and the lower bound is a celebrated result of M. Talagrand.
In this talk I will present a new approach to (an enhanced version of) Guerra's identity using stochastic analysis, more specifically Brownian motion and Ito’s calculus. This approach is suggested by the form of the Parisi formula in which the solution of a Hamilton-Jacobi equation is involved. It helps in many ways to illuminate the original method of Guerra and suggests some possible approaches to the significantly deeper lower bound, which has been intensively studied since Talagrand’s work. Among the techniques from stochastic analysis we will use include path space integration by parts for the Wiener measure, Girsanov’s transform (i.e., exponential martingales), and probabilistic representation of solutions to (linear) partial differential equations. The key observation is that the nonlinear Hamilton-Jacobi partial differentiation equation figuring in Parisi’s variation formula becomes linear after differentiating with respect to Guerra’s interpolation parameter, thus bringing the full strength of stochastic analysis based on Ito’s calculus into play. It is hoped that this approach will shed some lights on the much more difficult lower bound in the Parisi formula.
Friday October 13
Max Fathi ( LJLL and LPSM)
Globally lipschitz transport maps
One way of proving probabilistic functional inequalities (concentration inequalities, logarithmic Sobolev inequalities...) is to use a change of variables, to transfer them from a simple reference measure (typically, Gaussian) to more general settings. One example of this is the Caffarelli contraction theorem, which states that uniformly log-concave measures can be realized as images of standard Gaussian measures by globally lipschitz maps, using the L2 optimal transport map. One open problem in this direction is to find an analogue of Caffarelli's theorem in the Riemannian setting.
In this talk, I will present a stochastic construction of non-optimal maps, due to Kim and Milman, and Lipschitz estimates in various settings, including certain measures on Riemannian manifolds. Joint work with Dan Mikulincer and Yair Shenfeld.
Friday September 22
Nicolas Curien (Paris Saclay)
Ideal Poisson-Voronoi tiling
We study the limit in low intensity of Poisson--Voronoi tessellations in hyperbolic spaces.
In contrast to the Euclidean setting, a limiting non-trivial ideal tessellation appears as the intensity tends to 0. The tessellation obtained is a natural Möbius-invariant decomposition of the hyperbolic space into countably many infinite convex polytopes, each with a unique end. We study its basic properties, in particular the geometric features of its cells.
Based on joint works with Matteo d'Achille, Nathanel Enriquez, Russell Lyons and Meltem Unel.
Friday September 15
Marco Carfagnini (UC San Diego)
Spectral gaps via small deviations
In this talk we will discuss spectral gaps of second order differential operators and their connection to limit laws such as small deviations and Chung’s laws of the iterated logarithm. The main focus is on hypoelliptic diffusions such as the Kolmogorov diffusion and horizontal Brownian motions on Carnot groups. If time permits, we will discuss spectral properties and existence of spectral gaps on general Dirichlet metric measure spaces.This talk is based on joint works with Maria (Masha) Gordina and Alexander (Sasha) Teplyaev.
@ IHES, Bures-sur-Yvette, France
