Seed Seminar of Mathematics and Physics
I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.
 
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Seed Seminar of Mathematics and Physics
 
We study the volume of rigid loop-O(n) quadrangulations with a boundary of length 2p in the critical non-generic regime. We prove that, as the half-perimeter p goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen, Curien and Maillard, or alternatively (in the dilute case) as the law of the area of a suitable unit-boundary quantum disc, as determined by Ang and Gwynne. Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. This talk is based on joint work with Élie Aïdékon and XingJian Hu (Fudan University).
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Seed Seminar of Mathematics and Physics
In this talk, I will present JT gravity, a model of two-dimensional quantum gravity on constant negatively curved spacetimes,  as a model of random hyperbolic surfaces. By studying the generating function of volumes of random hyperbolic surfaces with defects, i.e. weighted geodesic boundaries, we explore critical regimes where the surfaces develop macroscopic holes. This is reminiscent of the O(N) model for random maps. We analyse the impact of this critical behavior on the density of states of the theory at the boundary, and we present a family of models that interpolate between systems with $sqrt{E}$ and E3/2, which are commonly found in models of JT Gravity coupled to dynamical end-of-the-world and FZZT branes. 
 
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Seed Seminar of Mathematics and Physics
Loop models are a class of statistical lattice models whose correlation functions can be interpreted geometrically, as sums over configurations of non-intersecting loops.
In the critical limit, the observables of these models are described by a Conformal Field Theory (CFT), which is believed to be exactly solvable.
In this talk, we will shortly review what are loop models and present recent results showing that their correlation functions are related to combinatorial objects called combinatorial maps. Then, we will relate the counting of certain maps on the torus to particular classes of maps on the sphere, and explain the CFT interpretation of this mapping.
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Seed Seminar of Mathematics and Physics
A type of BPS q-series, proposed recently as a topological invariant which provides a non-perturbative completion of complex Chern-Simons theory on closed 3-dimensional manifolds, has mathematical definitions based for example on 3D topology, quantum groups and resurgence, and key properties their integrality and behaviour under surgery formulae. These q-series invariants have displayed relations to vertex operator algebras, and structurally, they have intriguing modular properties. Their mathematical formulation is nevertheless quite constrained, relying heavily on certain negative-definite conditions, and much effort has been devoted to extending this definition. This operation is called “going to the other side”, with different interpretations from the physics, vertex algebra, and 3D topology perspectives. I will discuss different approaches to this challenge, in particular through modularity and resurgence, with implications for the related vertex operator algebras.
This talk will be streamed from Amphithéâtre Léon Motchane. The zoom link is also available by subscribing to the mailing list: sympa@listes.math.cnrs.fr
 
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Seed Seminar of Mathematics and Physics
Local measurements in many‐body quantum systems can produce surprising phenomena—from preparing long‐range entangled states to driving measurement‐induced phase transitions and modifying critical behavior. In this talk, I will introduce a new “observable‐projected” protocol: rather than performing complete projective measurements on the complement of a subsystem, we measure a single, local Hermitian operator on part of that complement. The result is a mixed‐state ensemble on the subsystem that remains analytically tractable within a conformal field theory framework. I will then show how this ensemble’s entanglement structure can be computed, and present a detailed case study of the free compact boson.
 
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Seed Seminar of Mathematics and Physics
One of the features of the critical behavior of the long-range Ising model (LRI) with interactions decaying as 1/rs+d is that it has a line of interacting fixed points, and hence corresponds to an family of interacting d-dimensional CFT, for d/2 < s < s∗. Both ends of this range of s admit a weakly-coupled description. For small s, the theory is described as a generalized free field with a φ4 interaction. For large s, the theory can be described by the d-dimensional short-range Ising model (SRI) coupled to a generalized free field. While these descriptions have been used to study the LRI in dimensions d > 1, in d = 1 it leads to a fascinating puzzle. The 1d LRI still has a line of interacting fixed points for 1/2 < s < 1. However, it is well-known that the SRI does not have a second-order phase transition in 1d and leaves the question what happens when s approaches . In this talk I will discuss our model which provides a weakly-coupled description for the 1d LRI around s = 1. At s = 1, our model becomes an exactly solvable conformal boundary condition for the 2d free scalar. We perform a number of consistency checks of our proposal and calculate CFT data around s = 1 using perturbation theory and the analytic bootstrap. This is based on https://arxiv.org/pdf/2412.12243.
 
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Seed Seminar of Mathematics and Physics
 
Lipschitz transport maps between two measures are useful tools for transferring analytical properties, such as functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that the transfer of analytical properties does not depend on the optimality of the transport map. This is why several works have established Lipschitz bounds for other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we use the control interpretation of the transport vector field inducing the transport map and a coupling strategy to obtain Lipschitz bounds for this map between asymptotically log-concave measures and their Lipschitz perturbations. This talk is based on a joint work with Giovanni Conforti.
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Seed Seminar of Mathematics and Physics
Fall’ 25 : Random Forests and Fermionic Field Theories
We show that a large class of fermionic theories are dual to a q → 0 limit of the Potts model in the presence of a magnetic field. These can be described using a statistical model of random forests on a graph, generalizing the (unrooted) random forest description of the Potts model with only nearest neighbor interactions. We then apply this to find a statistical description of a recently introduced family of  OSp(1|2M) invariant field theories that provide a UV completion to sigma models with the same symmetry.
The zoom link is available by subscribing to the mailing list: sympa@listes.math.cnrs.fr
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Seed Seminar of Mathematics and Physics
Fall’ 25: Random Forests and Fermionic Field Theories 
We consider the Renormalization Group (RG) fixed-point theory associated with a fermionic $psi^4_d$ model in d=1,2,3 with fractional kinetic term, whose scaling dimension is fixed so that the quartic interaction is weakly relevant in the RG sense. The model is defined in terms of a Grassmann functional integral with interaction $V^*$, solving a fixed-point RG equation in the presence of external fields, and a fixed ultraviolet cutoff. We define and construct the field and density scale-invariant response functions, and prove that the critical exponent of the former is the naive one, while that of the latter is anomalous and analytic. We construct the corresponding (almost-)scaling operators, whose two point correlations are scale-invariant up to a remainder term, which decays like a stretched exponential at distances larger than the inverse of the ultraviolet cutoff. Our proof is based on constructive RG methods and, specifically, on a convergent tree expansion for the generating function of correlations, which generalizes the approach developed by three of the authors in a previous publication (Giuliani et al. in JHEP 01:026, 2021. doi.org/10.1007/JHEP01(2021)026). CMP 406.10 (2025): 257, joint work with A. Giuliani, V. Mastropietro and S. Rychkov.
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Seed Seminar of Mathematics and Physics
Fall’ 25: Random Forests and Fermionic Field Theories 
The study of critical models is of the more active areas of statistical mechanics. Regarding the dimer model, the convergence of the critical model towards the Gaussian free field was obtained around 25 years ago by Kenyon. More recently, perturbations of the critical model known as near-critical models have been considered, and some convergence results have been obtained, in particular for the Ising model. Convergence results have also been obtained for the near-critical dimer model, which did not allow to identify the limiting field, even though it was conjectured to be the sine-Gordon field. I will present a derivation of the limit using discrete massive holomorphy techniques, which expresses the limiting field as the solution of a certain Dirichlet problem associated with a massive Dirac operator. I will finally explain how to relate this field to the sine-Gordon field. This is based on an ongoing work with Nathanaël Berestycki and Scott Mason.
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Seed Seminar of Mathematics and Physics
Fall’ 25: Random Forests and Fermionic Field Theories 
In two-dimensional models of critical non-intersecting loops, we conjecture an exact formula for three-point functions of fields that insert legs (open loop segments) and can have nonzero conformal spin. The conjecture extends a previous result for diagonal fields, recently proved by Ang-Cai-Sun-Wu, who also proved our conjecture for three spinless two-leg operators. We discuss in details the supporting evidence for our general conjecture coming from transfer-matrix computations using the unoriented Jones-Temperley-Lieb algebra.
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