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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Seed Seminar of Mathematics and Physics
Fall’ 25: Random Forests and Fermionic Field Theories 
For many real-world networks, such as the World Wide Web, the degree distribution follows a power law. It is therefore useful to have simple random graph models whose limiting degree distribution exhibits this same feature. With this motivation, physicists Albert-László Barabási and Réka Albert introduced the preferential attachment model that now bears their name. A further advantage of this model is that it incorporates temporal dynamics: starting from an initial graph $mathcal{G}_0$, the graph at time $n+1$ is obtained from the graph at time $n$, denoted $mathcal{G}_n$, by adding a new vertex $v_{n+1}$. This vertex then attaches to one or several vertices of $mathcal{G}_n$ according to a preferential attachment rule, meaning that the probability of connecting to a given vertex of $mathcal{G}_n$ is proportional to its degree.
We present an extension of this model in which each vertex of the graph is assigned a community (or type), and in which the preferential attachment is sublinear; that is, the probability of attaching to a vertex $u$ is proportional to $deg(u)^gamma$, where $gamma$ is a parameter taking values in $(0,1)$.
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Seed Seminar of Mathematics and Physics
Fall’ 25 : Random Forests and Fermionic Field Theories  
The frozen Erdős-Rényi random graph is a variant of the standard dynamical Erdős-Rényi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter p∈[0,1] that quantifies the slowdown. In this talk, we will study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. The proof is based on the free forest property of the frozen model: the forest part of the graph is a uniform random forest. In order to prove the fluid limit results, we will explain how to study and count forests using conditioned random walks.
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Seed Seminar of Mathematics and Physics
We consider FK models with $q$ in $[0,4]$ on the square lattice and the whole plane. We assume the convergence of height functions to GFF and in particular we assume that we know the variance $sigma^2$ of the GFF. Then, we sketch an approach to get the exponent $alpha_1$ describing the probability of having a primal crossing of an annulus. The basis for this approach is the BKW coupling relating the height function to the interface loops of FK. We show that by choosing appropriate test functions (viewd as placing charges on the plane), we can get relations between $sigma^2$, $alpha_1$, and a factor accounting for local concentration of small interface loops. 
 
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Running Seminar
We will introduce our themes and show some of the objects we will work on. The seminar will focus on the arithmetic and Hodge-theoretic aspects of differential equations, including accessory parameter problems, multiplication kernels, periods, character varieties, knot invariants, and related topics.
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The nonabelian Hodge correspondence provides a parametrization of the character variety of a surface group in a semisimple Lie group G by objects called Higgs bundles (which we will introduce). Among them, cyclic Higgs bundles stand out as candidates for representations with strong geometric properties. The Slodowy slice is a natural set of deformations of a Fuchsian representation in G, whose properties are yet to be described. In this talk, I will present some cases in which we are able to prove that representations in the cyclic Slodowy slice are Anosov, hence discrete and faithful, and correspond to some geometric structure that we describe. This is joint work with Colin Davalo and Qiongling Li.