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
May 9, 2025
Franco Severo
On the supercritical phase of the φ^4 model
The φ^4 model is a real-valued spin system with quartic potential. This model has deep connections with the classical Ising model, and both are expected to belong to the same universality class. We construct a random cluster representation for φ^4, analogous to that of the Ising model. For this percolation model, we prove that local uniqueness of macroscopic cluster holds throughout the supercritical phase. The corresponding result for the Ising model was proved by Bodineau (2005) and serves as the crucial ingredient in renormalization arguments used to study fine properties of the supercritical behaviour, such as surface order large deviations, the Wulff construction and exponential decay of truncated correlations. The unboundedness of spins in the φ^4 model imposes considerable difficulties when compared with the Ising model. This is particularly the case when handling boundary conditions, which we do by relying on the recently constructed random current representation of the model.
Joint work with Trishen Gunaratnam, Christoforos Panagiotis and Romain Panis.
March 7, 2025
Gerard Ben Arous (New York University)
The Mezard-Parisi Elastic Manifold: Topological Complexity, Free Energy and phase transitions.
The Elastic Manifold is a model of an elastic interface in a disordered medium, introduced in the 80’s in order to understand the competition between the effects of disorder and those of elasticity. This model gave us a very vast literature in statistical physics, from Daniel Fisher to Marc Mezard and Giorgio Parisi, and many more works inspired by the progress of the Parisi school on Spin Glasses, up to the more mathematical recent works by Yan Fyodorov and Pierre Le Doussal.
I will cover here recent progress, first on the topological complexity of the energy landscape for the Elastic Manifold, obtained with Paul Bourgade (Courant) and Benjamin McKenna (Georgia Tech), and then on the Parisi formula for the quenched free energy, and the nature of the glass transition at low temperature, more recently proved in a series of works, with Pax Kivimae (Courant).
February 14, 2025
Benoit Laslier (Paris Cité)
Tilted Solid on solid is liquid, at least when thawed
The Solid on Solid model is a mainstay of the modelisation of 2D interfaces in the physics literature and it has also received extensive attention in mathematics. In particular a major work of Frölich and Spencer showed that it exhibits a roughening transitions where at low temperature, the interface is extremely localized with O(1) fluctuations at the microscopic scale while at high temperature it delocalize with logarithmic variance.
However, almost all the existing literature focuses on the case where the interface is parallel to the main axis of the underlying lattice, as in a crystal facet, but of course this cannot be the case globally everywhere. We will show that, at least for a model with a small added potential, whenever the interface has a tilt the phenomenology changes completely : at low enough temperature prove that the behavior of the interface is rough (like the high temperature usual case) and provide a full scaling limit for the fluctuations. The main approach is a comparison to the zero-temperature case which can be described as lozenge tilings and a renormalization procedure to understand the asymptotic of an interacting tiling model.
February 7, 2025
Mingkun Liu (Paris 13 LAGA)
Length spectra of random metric map of large genus: a Teichmüller theory approach.
After a brief historical review, I will explain how to pick a (uniform) random hyperbolic surface of genus g. After that, we will focus on the length spectrum. More specifically, we will examine short closed geodesics on a random hyperbolic surface of genus g. It turns out that, when g is big, the lengths of these geodesics are distributed just like the short cycles in a large random graph. This is a joint work with Simon Barazer and Alessandro Giacchetto.
January 31, 2025
Yvain Bruned (Université de Lorraine)
Symmetries for singular SPDEs
In this talk, we will briefly review the main ideas for solving singular SPDEs with the use of Regularity Structures. After presenting the main symmetries known, we will focus on some recent progress concerning the chain rule symmetry in the full subcritical regime. These symmetries allow us to restrict the space of solutions.
December 13, 2024
Justin Salez (Paris-Dauphine & PSL)
A new approach to the cutoff phenomenon
The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergoned by certain Markov processes in the limit where the number of states tends to infinity. Discovered forty years ago in the context of card shuffling, it has since then been established in a variety of contexts, including random walks on graphs and groups, high-temperature spin systems, or interacting particles. Nevertheless, a general theory is still missing, and identifying the general mechanisms underlying this mysterious phenomenon remains one of the most fundamental problems in the area of mixing times. In this talk, I will give a self-contained introduction to this fascinating question, and then describe a new approach based on entropy and curvature.
December 6, 2024
Ilya Losev (Cambridge)
Probabilistic Schwarzian Field Theory
Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev--Ye--Kitaev random matrix model and Jackie--Teitelboim gravity.
In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.
November 29, 2024
Hong-Bin Chen (IHES)
On free energy in non-convex mean-field spin glass models
We start by reviewing the classical Sherrington-Kirkpatrick (SK) model. In this model, +1/-1-valued spins interact with each other subject to random coupling constants. The covariance of the random interaction is quadratic in terms of spin overlaps. Parisi proposed the celebrated variational formula for the limit of free energy of the SK model in the 80s, which was later rigorously verified in the works by Guerra and Talagrand. This formula has been generalized in various settings, for instance, to vector-valued spins, by Panchenko. However, in these cases, the convexity of the interaction is crucial. In general, the limit of free energy in non-convex models is not known and we do not have variational formulas as valid candidates. Here, we report recent progress through the lens of the Hamilton-Jacobi equation. Under the assumption that the limit of free energy exists, we show that the value of the limit is prescribed by a characteristic line; and the limit (as a function) satisfies an infinite-dimensional Hamilton-Jacobi equation "almost everywhere". This talk is based on a joint work with Jean-Christophe Mourrat.
September 13, 2024
Slava Rychkov (IHES)
Real-space renormalization of 2D lattice models with tensor networks
Tensor networks are a recent cool addition to physicist’s toolkit used to study renormalization of lattice models. However the mathematical theory of tensor network renormalization group (TNRG) is still in its infancy. I will aim to transmit my excitement about the tensor networks. Rough plan:
1. Wilson’s conjecture about renormalization group fixed points describing criticality - can we prove it?
2. Why are tensor networks better than other approaches to renormalization (e.g. spin blocking).
3. Numerical algorithms for TNRG - what do people see numerically?
4. Discrete scaling operator
5. A few mathematically rigorous results about TNRG
6. Open problems
October 11, 2024
Renan Gross (Cambridge)
A sharp lower bound on the small eigenvalues of surfaces
The Laplacian is a central operator in the analysis of surfaces (and life in general). In this talk, we investigate how small its small eigenvalues can be, giving a sharp, quadratic bound on the k-th eigenvalue of a surface in terms of k, the surface's genus g, and its global geometry via the injectivity radius. The techniques involve extremal length, spectral embedding, and volume arguments.
Joint work with Guy Lachman and Asaf Nachmias, based on the paper: https://arxiv.org/abs/2407.21780
For an exposition and overview of the paper, see here: https://sarcasticresonance.wordpress.com/2024/08/02/new-paper-on-arxiv-a-sharp-lower-bound-on-the-eigenvalues-of-surfaces/
Franco Severo
On the supercritical phase of the φ^4 model
The φ^4 model is a real-valued spin system with quartic potential. This model has deep connections with the classical Ising model, and both are expected to belong to the same universality class. We construct a random cluster representation for φ^4, analogous to that of the Ising model. For this percolation model, we prove that local uniqueness of macroscopic cluster holds throughout the supercritical phase. The corresponding result for the Ising model was proved by Bodineau (2005) and serves as the crucial ingredient in renormalization arguments used to study fine properties of the supercritical behaviour, such as surface order large deviations, the Wulff construction and exponential decay of truncated correlations. The unboundedness of spins in the φ^4 model imposes considerable difficulties when compared with the Ising model. This is particularly the case when handling boundary conditions, which we do by relying on the recently constructed random current representation of the model.
Joint work with Trishen Gunaratnam, Christoforos Panagiotis and Romain Panis.
March 7, 2025
Gerard Ben Arous (New York University)
The Mezard-Parisi Elastic Manifold: Topological Complexity, Free Energy and phase transitions.
The Elastic Manifold is a model of an elastic interface in a disordered medium, introduced in the 80’s in order to understand the competition between the effects of disorder and those of elasticity. This model gave us a very vast literature in statistical physics, from Daniel Fisher to Marc Mezard and Giorgio Parisi, and many more works inspired by the progress of the Parisi school on Spin Glasses, up to the more mathematical recent works by Yan Fyodorov and Pierre Le Doussal.
I will cover here recent progress, first on the topological complexity of the energy landscape for the Elastic Manifold, obtained with Paul Bourgade (Courant) and Benjamin McKenna (Georgia Tech), and then on the Parisi formula for the quenched free energy, and the nature of the glass transition at low temperature, more recently proved in a series of works, with Pax Kivimae (Courant).
February 14, 2025
Benoit Laslier (Paris Cité)
Tilted Solid on solid is liquid, at least when thawed
The Solid on Solid model is a mainstay of the modelisation of 2D interfaces in the physics literature and it has also received extensive attention in mathematics. In particular a major work of Frölich and Spencer showed that it exhibits a roughening transitions where at low temperature, the interface is extremely localized with O(1) fluctuations at the microscopic scale while at high temperature it delocalize with logarithmic variance.
However, almost all the existing literature focuses on the case where the interface is parallel to the main axis of the underlying lattice, as in a crystal facet, but of course this cannot be the case globally everywhere. We will show that, at least for a model with a small added potential, whenever the interface has a tilt the phenomenology changes completely : at low enough temperature prove that the behavior of the interface is rough (like the high temperature usual case) and provide a full scaling limit for the fluctuations. The main approach is a comparison to the zero-temperature case which can be described as lozenge tilings and a renormalization procedure to understand the asymptotic of an interacting tiling model.
February 7, 2025
Mingkun Liu (Paris 13 LAGA)
Length spectra of random metric map of large genus: a Teichmüller theory approach.
After a brief historical review, I will explain how to pick a (uniform) random hyperbolic surface of genus g. After that, we will focus on the length spectrum. More specifically, we will examine short closed geodesics on a random hyperbolic surface of genus g. It turns out that, when g is big, the lengths of these geodesics are distributed just like the short cycles in a large random graph. This is a joint work with Simon Barazer and Alessandro Giacchetto.
January 31, 2025
Yvain Bruned (Université de Lorraine)
Symmetries for singular SPDEs
In this talk, we will briefly review the main ideas for solving singular SPDEs with the use of Regularity Structures. After presenting the main symmetries known, we will focus on some recent progress concerning the chain rule symmetry in the full subcritical regime. These symmetries allow us to restrict the space of solutions.
December 13, 2024
Justin Salez (Paris-Dauphine & PSL)
A new approach to the cutoff phenomenon
The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergoned by certain Markov processes in the limit where the number of states tends to infinity. Discovered forty years ago in the context of card shuffling, it has since then been established in a variety of contexts, including random walks on graphs and groups, high-temperature spin systems, or interacting particles. Nevertheless, a general theory is still missing, and identifying the general mechanisms underlying this mysterious phenomenon remains one of the most fundamental problems in the area of mixing times. In this talk, I will give a self-contained introduction to this fascinating question, and then describe a new approach based on entropy and curvature.
December 6, 2024
Ilya Losev (Cambridge)
Probabilistic Schwarzian Field Theory
Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev--Ye--Kitaev random matrix model and Jackie--Teitelboim gravity.
In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.
November 29, 2024
Hong-Bin Chen (IHES)
On free energy in non-convex mean-field spin glass models
We start by reviewing the classical Sherrington-Kirkpatrick (SK) model. In this model, +1/-1-valued spins interact with each other subject to random coupling constants. The covariance of the random interaction is quadratic in terms of spin overlaps. Parisi proposed the celebrated variational formula for the limit of free energy of the SK model in the 80s, which was later rigorously verified in the works by Guerra and Talagrand. This formula has been generalized in various settings, for instance, to vector-valued spins, by Panchenko. However, in these cases, the convexity of the interaction is crucial. In general, the limit of free energy in non-convex models is not known and we do not have variational formulas as valid candidates. Here, we report recent progress through the lens of the Hamilton-Jacobi equation. Under the assumption that the limit of free energy exists, we show that the value of the limit is prescribed by a characteristic line; and the limit (as a function) satisfies an infinite-dimensional Hamilton-Jacobi equation "almost everywhere". This talk is based on a joint work with Jean-Christophe Mourrat.
September 13, 2024
Slava Rychkov (IHES)
Real-space renormalization of 2D lattice models with tensor networks
Tensor networks are a recent cool addition to physicist’s toolkit used to study renormalization of lattice models. However the mathematical theory of tensor network renormalization group (TNRG) is still in its infancy. I will aim to transmit my excitement about the tensor networks. Rough plan:
1. Wilson’s conjecture about renormalization group fixed points describing criticality - can we prove it?
2. Why are tensor networks better than other approaches to renormalization (e.g. spin blocking).
3. Numerical algorithms for TNRG - what do people see numerically?
4. Discrete scaling operator
5. A few mathematically rigorous results about TNRG
6. Open problems
October 11, 2024
Renan Gross (Cambridge)
A sharp lower bound on the small eigenvalues of surfaces
The Laplacian is a central operator in the analysis of surfaces (and life in general). In this talk, we investigate how small its small eigenvalues can be, giving a sharp, quadratic bound on the k-th eigenvalue of a surface in terms of k, the surface's genus g, and its global geometry via the injectivity radius. The techniques involve extremal length, spectral embedding, and volume arguments.
Joint work with Guy Lachman and Asaf Nachmias, based on the paper: https://arxiv.org/abs/2407.21780
For an exposition and overview of the paper, see here: https://sarcasticresonance.wordpress.com/2024/08/02/new-paper-on-arxiv-a-sharp-lower-bound-on-the-eigenvalues-of-surfaces/
@ IHES, Bures-sur-Yvette, France
