The KTH Probability and Mathematical Physics seminar will be held Wednesdays 13.15-14.15 in the seminar room 3721.
Organisers: Lukas Schoug and Levi Haunschmid-Sibitz.
Below follows the list of speakers.
Fall 2025. (Previous semesters can be found here)
10 September 2025: Janne Junnila, KTH
Title: Real opers and curves with a geodesic property
Abstract: I will discuss complex projective structures with real holonomy on Riemann surfaces, in particular the punctured sphere. Such structures can be identified with certain differential operators known as real opers that appear in the semiclassical limit of Liouville quantum gravity and also play a prominent role in the analytic Langlands correspondence. A canonical example is given by the Fuchsian projective structure on a hyperbolic Riemann surface, and a result of Goldman says that on closed surfaces all other real opers can be obtained via a grafting procedure applied to the Fuchsian structure. With Bonk, Rohde and Wang we showed a similar grafting result in a certain non-compact setting, where the projective structure is induced by a piecewise geodesic Jordan curve on the punctured sphere. We also proved that the associated Schwarzian derivative is related to the derivative of the Loewner energy of the curve, a result analogous to a famous formula conjectued by Polyakov and proven by Takhtajan and Zograf in the Fuchsian case. In addition to explaining these results and giving some general background on complex projective structures, I will also showcase some explicit examples of real opers with reflection symmetry about the unit circle.
17 September 2025: No talk.
24 September 2025: Kurt Johansson, KTH
Title: Planar Coulomb gases and Loewner energy
Abstract: The asymptotics of the partition function of a planar Coulomb gas confined to a Jordan curve in the complex plane is related to the Loewner energy of that curve. I will discuss this connection and extensions to the case of a Jordan arc. Based on joint work with Klara Courteaut and Fredrik Viklund.
1 October 2025: Ellen Powell, Durham University
Title: Scaling limits of critical FK-decorated maps at q=4
Abstract: The critical Fortuin–Kasteleyn random planar map with parameter q>0 is a model of random (discretised) surfaces decorated by loops, related to the q-state Potts model. For q<4, Sheffield established a scaling limit result for these discretised surfaces, where the limit is described by a so-called Liouville quantum gravity surface decorated by a conformal loop ensemble. At q=4 a phase transition occurs, and the correct rescaling needed to obtain a limit has so far remained unclear. I will talk about joint work with William Da Silva, XinJiang Hu, and Mo Dick Wong, where we identify the right rescaling at this critical value and prove a number of convergence results.
8 October 2025: No talk.
15 October 2025: Léonie Papon, TU Wien. (OBS: time is 13.00-14.00!)
Title: On the scaling limit of interfaces in the critical planar Ising model perturbed by a magnetic field
Abstract: In this talk, I will present some recent results on the scaling limit of interfaces in the critical planar Ising model perturbed by a magnetic field. I will first consider the case when the field is a deterministic function. In this case, I will show that in the so-called near-critical regime and when the Ising model has Dobrushin boundary conditions, the interface separating $+1$ and $-1$ spins has a scaling limit whose law is conformally covariant and absolutely continuous with respect to SLE$_3$. Its limiting law is a massive version of SLE$_3$ in the sense of Makarov and Smirnov. I will also discuss the scaling limit of this interface when the magnetic perturbation is not near-critical.
In the second part of the talk, I will discuss some ongoing work with Fenglin Huang and Aoteng Xia in which we look at the case when the magnetic field is given by a collection of iid centered Gaussian random variables, one for each vertex. In this setting, in the near-critical regime, we show that almost surely in the disorder, the scaling limit of the quenched law of the $\pm 1$ interface is absolutely continuous with respect to SLE$_3$. We then show that this contrasts with the scaling limit of the quenched law of the collection of nested spin loops, which turns out to be almost surely singular with respect to nested CLE$_3$. This also contrasts with the deterministic case where it is known that in the near-critical regime, any subsequential limit of the collection of nested spin loops is absolutely continuous with respect to nested CLE$_3$.
22 October 2025: Alex Karrila, Åbo Akademi
Title: Smoothness of martingale observables and generalized Feynman-Kac formulas
Abstract: In this talk I will explicate connections between three closely related concepts:
1) parabolic linear PDEs of the form G(x)f(x, t) + df(x,t)/dt +h(x,t)=0, where G(x) is a positive semi-definite second-order operator in the spatial variables;
2) Feynman-Kac formulas, representing, in some cases, solutions to boundary-value problems for (1) as conditional expectations in terms of an Ito process X_t, in the simplest case with the generator G(x) + d/dt;
3) martingale observables, i.e., in the simplest case functions f(x,t) such that f(X_t,t) is a local martingale.
Our main result is that, assuming only the classic Hörmander criterion on the Ito process X (no ellipticity, no boundedness of the diffusion coefficients, no infinite life-time), all its martingale observables are smooth. As a consequence, we also obtain a comparatively general Feynman-Kac type formula, that provides smooth solutions to boundary-value problems for (1), while allowing for degenerate diffusions, unbounded coefficients, as well as a spatial boundary, under very mild assumptions on boundary regularity. Another application (and the speaker's original motivation) comes from Schramm-Loewner evolutions, for which the result makes a certain Girsanov transform martingale accessible via Ito calculus.
Joint work with Lauri Viitasaari (Aalto U., Finland).
29 October 2025: Sampad Lahiry, KU Leuven
Title: Birth of a gap: A critical phenomena in 2D Coulomb gas
Abstract: We investigate a family of radially symmetric Coulomb gas systems at inverse temperature β=2. The family is characterised by the property that the density of the equilibrium measure vanishes on a ring at radius r∗, which lies strictly inside the droplet. The large n expansion of the logarithm of the partition function is obtained up to a novel n^1/4 term. We perform a double scaling limit of the correlation kernel at the n^1/4 scale and obtain a new limiting kernel in the bulk, which differs from the well-known Ginibre kernel.
5 November 2025: Teodor Bucht, KTH
Title: Quantitative Tracy-Widom laws for sparse random matrices
Abstract: In this talk I will consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erdős-Rényi graph $G(N, p)$. I will discuss edge universality for this model and present joint work with Kevin Schnelli and Yuanyuan Xu. Our main result is that the fluctuations of the largest eigenvalue converge to the Tracy-Widom law at a rate almost $O(N^{-1/3} + p^{-2} N^{-4/3})$ in the regime $p \gg N^{-2/3}$. Our proof builds upon the Green function comparison method initiated by Erdős, Yau, and Yin (2012). To show a Green function comparison theorem for fine spectral scales, we implement algorithms for symbolic computations involving averaged products of Green function entries.
12 November 2025: Asbjørn Bækgaard Lauritsen, Université Paris-Dauphine
Title: Asymptotic behaviour of the Bardeen--Cooper--Schrieffer theory of superconducitivity
Abstract: I will introduce the Bardeen--Cooper--Schrieffer (BCS) theory of superconducitvity and explain its variational and spectral theoretic formulation. For this theory, one is in particular interested in the critical temperature, being physically the temperature below which the material is superconducting, and the energy gap. I will discuss the analysis of these quantities in various limits of the theory and explain the spectral theoretic methods (the Birman--Schwinger principle) used in the proofs.
Joint work with Joscha Henheik, Edwin Langmann, and Barbara Roos.
19 November 2025: Titus Lupu, Sorbonne Université [CANCELLED DUE TO ILLNESS]
Title: Relation between the geometry of sign clusters of the 2D GFF and its Wick powers
Abstract: In 1990 Le Gall showed an asymptotic expansion of the epsilon-neighborhood of a planar Brownian trajectory (Wiener sausage) into powers of 1/|log eps|, that involves the renormalized self-intersection local times. In my talk I will present an analogue of this in the case of the 2D GFF. In the latter case, there is an asymptotic expansion of the epsilon-neighborhood of a sign cluster of the 2D GFF into half-integer powers of 1/|log eps|, with the coefficients of the expansion being related to the renormalized (Wick) powers of the GFF.
26 November 2025: TBD
3 December 2025: Sylvain Chabredier, ENS Paris
Title: A journey through the local behaviour of various point processes
Abstract: Finding the roots of a fixed polynomial is an old and hard question, but now for a random polynomial with Gaussian i.i.d. coefficient, can we say something on the statistics on the zeroes? We will see that when the degree N is big, the local behavior of these points can be compared to a statistical physics model of positively charged particles (i.e. a Coulomb gas) and we will explore the theorems that are shown for this Coulomb gas and the associated conjectures for the zeroes of the random polynomial.
10 December 2025: Semyon Klevtsov, Strasbourg. (Postponed.)
Title: Wiegmann-Zabrodin conjecture for $\beta=1$ in the hyperbolic case.
Abstract: Wiegmann-Zabrodin conjecture concerns the $O(1)$ terms in the large-N asymptotic expansion of the Coulomb gas partition function,
or $beta$-ensemble as it is also known for random matrices. The $O(1)$ term is interesting because it reveals the gaussian free field, arising in the large N limit,
and thus a correspondence with a 2d conformal field theory.
In a particular case of $beta=1$ is determinantal. I will explain how to define the partition function in this case on Riemann surfaces,
and how to obtain it’s asymptotics in the compact case. Then in the non-compact case I will compute the order $O(1)$ term, which turns out to differ for compact case.
Work in common with P. Wiegmann.
10 December 2025: Christian Webb, University of Helsinki. NOTE: Second talk of the week. Ordinary time: 13.15-14.15.
Title: The sine-Gordon model
Abstract: I will discuss a probabilistic construction of a Euclidean quantum field theory known as the sine-Gordon model. The interest in the model stems from its connection to many interesting phenomena and other important models such as the 2d Coulomb gas and certain fermionic models such as the near critical Ising model. Time permitting, I will discuss some of these connections. The talk is based on joint work with Roland Bauerschmidt and Scott Mason as well as ongoing work with Sung Chul Park and Tuomas Virtanen.
17 December 2025: Catherine Wolfram, Yale/ETH.
Title: Epstein curves and holography of the Schwarzian action
Abstract: The circle can be seen as the boundary at infinity of the hyperbolic plane. I will explain a construction from hyperbolic geometry, due to Epstein, to construct a curve in the disk from a diffeomorphism of the circle. It turns out that the Schwarzian action (a function of the diffeomorphism of the circle, and the action of Schwarzian field theory) can be computed in various ways from geometric data about this Epstein curve. While this construction is completely deterministic, from a mathematical physics perspective this is motivated by the proposed holographic duality between Schwarzian field theory on the circle and JT gravity in the disk. In addition to explaining how to construct the Epstein curve and how it is related to the Schwarzian action, I’ll explain how the bi-local observables of Schwarzian field theory can be interpreted as a renormalized hyperbolic length using the same Epstein construction, and time permitting discuss a bit what we know so far about the relationship between the Schwarzian action and the Loewner energy. This is joint work with Franco Vargas Pallete and Yilin Wang.
Spring 2026:
21 January 2026: Joonas Vättö, Aalto University
Title:
Abstract:
3 March 2026: Jacob Stordal Christiansen, Lund University
Title:
Abstract: