### Schedule and Information

The seminar runs at least monthly, on Mondays at 3.15pm-4.15pm, in ROOM F11, Lindstedtsvägen 22, F-Building, Next to the Department of Mathematics, KTH. (NOTE THE NEW LOCATION, directions to the lecture room.)

Organizers: Boualem Djehiche, Henrik Hult, Kurt Johansson, Fredrik Viklund

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Spring 2018:
[NOTE the order of talks.]
Jan 22: Avelio Sepulveda (ENS Lyon)
• Title:   Excursion theory for the 2d Gaussian free field
• Abstract:  Traditional excursion theory is concerned with the decomposition of a 1-dimensional Markov processes in two parts: its zeros and its excursions, i.e., the path starting and ending on 0 that remain the same sign. We will explain how one can emulate the excursion theory for the 2d- Gaussian free field(GFF), the analogue of the Brownian motion when the time is replaced by a two-dimensional domain. The main technical difficulty is that the GFF is not a function but only a "generalised function" (Schwartz distribution), thus a priori it makes no sense to speak about its zeros, nor about its excursions. This talk is based on joint works with Juhan Aru, Titus Lupu and Wendelin Werner.
Jan 29: Per Kurlberg (KTH)
• Title:

# The distribution of class groups for binary quadratic forms

• Abstract:   Gauss made the remarkable discovery that the set of integral binary
quadratic forms of fixed discriminant carries a composition law, i.e., two forms can be "glued together" into a third form. Moreover, as two quadratic forms related to each other via an integral linear change of
variables can be viewed as equivalent, it is natural to consider equivalence classes of quadratic forms. Amazingly, Gauss' composition
law makes these equivalence classes into a finite abelian group –  in a sense it is the first abstract group "found in nature". The fine scale structure of these groups is rather mysterious - in many ways the groups "behave randomly"; even modeling their cardinality remains a challenge. Extensive calculations led Gauss and others to conjecture that the number h(d) of equivalence classes of such forms of negative discriminant d tends to infinity with |d|, and that the class number is h(d) = 1 in exactly 13 cases: d is in {-3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163}. While this was known assuming the Generalized Rieman Hypothesis, it was only in the 1960's that the problem was solved by Alan Baker and by Harold Stark. We will outline the resolution of Gauss' class number one problem and survey some known results regarding the growth of h(d).  We will also consider how often a fixed abelian group occur as a class group  --- is there a natural probability measure on the set of abelian groups?  Using a probabilistic model we will address the question: do all abelian groups occur, or are there "missing" class groups?
March 16: Yilin Wang (ETH)
• Title:

# The Loewner energy of a simply connected domain on the Riemann sphere

• Abstract:   Loewner's equation provides a way to encode a simply connected domain via a real-valued driving function of its boundary. The Loewner energy of the domain is the Dirichlet energy of the driving function. It depends a priori on the parametrization of the boundary. However it was shown previously that there is no such dependence. In this talk I will present an intrinsic interpretation of the Loewner energy and a characterization of finite energy domains.
April 16: Alexandre Stauffer (Bath)
• Title:
• Abstract
April 23: Tyler Helmuth (Bristol)
• Title:
• Abstract
May 14: Lukas Schoug (KTH)
• Title:
• Abstract
May 21: Wendelin Werner (ETH Zurich)
• Title:
• Conformal Loop Ensembles on Liouville Quantum Gravity (or how to construct a plane as a patchwork of disks).

• AbstractI will describe an ongoing project with Jason Miller and Scott Sheffield. One of its goals it to show the special natural interplay in the continuous setting between Conformal Loop Ensembles (which are the natural conformally invariant collections of loops that one can define on a Riemann surface) and the Liouville Quantum Gravity surfaces (which are random equivalence classes of domains equipped with an area measure defined via the Gaussian Free Field). One shows for instance that the conformal loop ensembles can be obtained via conformal welding of an appropriately ordered infinite family of independent quantum surfaces.
Fall 2017
December 11: Torben Krueger (Bonn)
• Title:  Spectral Universality for Random Matrices: From the Global to the Local Scale
• Abstract: The spectral statistics of large dimensional self-adjoint random matrices often exhibits universal behavior. On the global spectral scale the density of states depends only on the first two moments of the matrix entries and follows a universal shape at all its singular points, i.e. whenever it vanishes. On the local scale the joint distribution of a finite number of eigenvalues depends only on the symmetry type of the random matrix (Wigner-Dyson-Mehta spectral universality). We present recent results and methods that establish such spectral universality properties from the global down to the smallest spectral scale for a wide range of random matrix models, including matrices with general expectation and correlated entries. [Joint work with the Erdős group at IST Austria]
December 4: Huy Tran (TU Berlin)
• Title:   The continuity of Loewner map
• Abstract: The Loewner map uses a special differential equation that describes a non-crossing curve in the upper half-plane through a real-valued function, called the driving function. One of the main questions in the Loewner theory is to understand how the regularity of the curve depends on the space that the driving function lives.  The space can be the Wiener space, where the Brownian motion lives, or Holder spaces. In this talk, we will review these results. Then we will focus on another question: How stable is the Loewner map in these spaces? Interestingly, this question is more difficult. We will present a partial answer to the question in the talk.
November 20: Karen Habermann (Cambridge)
• Title:   Small-time fluctuations for sub-Riemannian diffusion loops
• Abstract:    We study the small-time fluctuations for diffusion processes which
are conditioned by their initial and final positions and whose
diffusivity has a sub-Riemannian structure. In the case where the
endpoints agree, we discuss the convergence of the suitably rescaled
fluctuations to a limiting diffusion loop, which is equal in law to
the loop we obtain by taking the limiting process of the unconditioned
starting point. The generator of the unconditioned limiting rescaled
diffusion process can be described in terms of the original
generator.

October 9 : Atul Shekhar (KTH)
• Title:  Regularization of Planar Boundaries under Stochastic Evolution
• Abstract:  Brownian motion $B$ exhibits a curious property called regularization by noise which can be attributed to its quadratic variation process. It was shown by A.M. Davie that differential equations of form $dX_t = f(X_t)dt + dB_t$ admits a unique solution for almost surely all Brownian sample paths even if $f$ is only a bounded measurable function. We will consider the case when $f$ is a holomorphic map in an open set which is irregular as the boundary is approached. We will show that there is a unique flow $\varphi(z)$ associated to the above equation and the complex derivative $\varphi^{'}(z)$ admits a continuous extension to the boundary. The result is compared to classical results from complex analysis on boundary behaviour of derivative of conformal maps.
September 11: Daniel Ahlberg (SU)
• Title: Random coalescing geodesics in first-passage percolation
• AbstractSince the work of Kardar-Parisi-Zhang in the 1980s, it has been widely believed that a large class of two-dimensional growth models should obey the same asymptotic behaviour. To rigorously understand the predictions of KPZ-theory has since been one of the most central themes in mathematical physics. One prominent model believed to belong to this class is known as first-passage percolation. It can be interpreted as the random metric on Z^2 obtained by assigning non-negative i.i.d. weights to the edges of the nearest neighbour lattice. We shall discuss properties of geodesics in this metric and their connection to KPZ-theory. We develop an ergodic theory for infinite geodesics via the study of what we shall call random coalescing geodesics’. As an application of this theory we answer a question posed by Benjamini, Kalai and Schramm in 2003, that has come to be known as the midpoint problem’. This is joint work with Chris Hoffman.
September 4: Greg Lawler (Chicago)
• Title: Two-sided loop-erased random walk
• AbstractThe loop-erased random walk is measure on non self-intersection paths obtained by erasing loops from a simple random walk.  I will review what is known about this, including the dependence on the spatial dimension, and then discuss a new result showing the existence of the two-sided walk in all dimensions.  The two-sided walk can be considered as the distribution of the middle’’ of a loop-erased walk.
August 28: [joint analysis seminar] Ilia Binder (Toronto)
• Title: Integrability of the KdV equation with almost periodic initial data.
• AbstractIn 2008, P. Deift conjectured that the solution of KdV equation with almost periodic initial data is almost periodic in time.  I will discuss the proof of this conjecture (as well as the uniqueness) in the case of the so-called Sodin-Yuditskii type initial data, i.e. the initial data for which the associated Schroedinger operator has purely absolutely continuous spectrum which satisfies certain thickness conditions. In particular, this proves the existence, uniqueness,  and almost periodicity of the solutions with small analytic quasiperiodic initial data with Diophantine frequency vector.This is a joint work with D. Damanik (Rice), M. Goldstein (Toronto) and M. Lukic (Rice).

Spring 2017
February 6: Petter Bränden (KTH)
• Title: Strong Rayleigh measures
• AbstractWe give a brief introduction to Strong Rayleigh measures. This is a class of discrete probability measures exhibiting strong negative dependence properties. The measures have recently been used in different settings such as interacting particle systems, the van der Waerden conjecture on permanents, the Kadison-Singer problem and the Traveling Salesman Problem.
March 6: Anders Szepessy (KTH)
• Title: Quantum particle dynamics approximated by stochastic molecular dynamics
• AbstractI will first show how Irvine and Zwanzig derived the fundamental conservation laws of continuum fluid dynamics
from quantum mechanics and then modify it to include approximation by stochastic molecular dynamics.
A motivation for the molecular dynamics derivation is e.g. to have a computational method to determine the stress tensor and the heat flux, for any temperature, and estimate the accuracy.
April 3: Tatyana Turova (Lund)
• Title: Phase Transitions in the One-dimensional Coulomb Gas Ensembles
• AbstractWe consider the system of particles on a finite interval with pair-wise
nearest neighbours interaction and external force. This model was
introduced by Malyshev (2015) to study the flow of charged particles on
a rigorous mathematical level. It is a simplified version of a
3-dimensional classical Coulomb gas model. We study Gibbs distribution
at finite positive temperatureextending recent results on the zero
temperature case (ground states). We derive the asymptotics for the mean and for the variances of the  distances between the neighbouring charges. We prove that depending on
the strength of the external force there are several phase transitions  in the local structure of the configuration of the particles in the  limit when the number of particles goes to infinity. We identify 5 different phases for any positive temperature.

The proofs rely on a conditional central limit theorem for non-identical
random variables, which may have an interest on its own.
April 10: Johan Tykesson (Chalmers)
• Title: Generalized Divide and Color Models
• Abstract:
• In this talk, we consider the following model: one starts with a finite or countable set

$V$, a random partition of $V$ and a parameter $p\in [0,1]$. The corresponding

Generalized Divide and Color Model is the $\{0,1\}$-valued process indexed by $V$ obtained by

independently, for each partition element in the random partition

chosen, with probability $p$, assigning all the elements of the partition element the value 1,

and with probability $1-p$, assigning all the elements of the partition element the value 0.

A very special interesting case of this is the

Divide and Color Model'' (which motivates the name we use)

introduced and studied by Olle Häggström.

Some of the questions which we study here are the following. Under what situations can different

random partitions give rise to the same color process?

What can one say concerning exchangeable random partitions?

What is the set of product measures that a color process stochastically dominates?

For random partitions which are translation invariant, what ergodic properties

do the resulting color processes have?

The motivation for studying these processes is twofold; on the one hand, we believe that this is

a very natural and interesting class of processes that deserves investigation and

on the other hand, a number of quite varied well-studied processes

actually fall into this class such as the Ising model, the stationary distributions for the Voter Model, random walk in random scenery and of course the original Divide and Color Model.

May 15: Antti Knowles (Geneva)
• Title: Extreme eigenvalues of sparse random graphs
• Abstract: I review some recent work on the extreme eigenvalues of sparse random graphs, such as inhomogeneous Erdos-Renyi graphs. Let n denote the number of vertices and d the maximal mean degree. We establish a crossover in the behaviour of the extreme eigenvalues at the scale d = log n. For d >> log n, we prove that the extreme eigenvalues converge to the edges of the support of the asymptotic eigenvalue distribution. For d << log n, we prove that these extreme eigenvalues are governed by the largest degrees, and that they exhibit a novel behaviour, which in particular rules out their convergence to a nondegenerate point process.
​ Joint work with Florent Benaych-Georges and Charles Bordenave.​

May 22: Juhan Aru (ETH Zurich)
• Title: How to describe the 2D Gaussian free field
• Abstract:
2D continuum Gaussian free field (GFF) is a canonical model for random surfaces. It has gained a central place due to its varied connections to SLE processes of Schramm, to Brownian loop soups and to what is sometimes called the Liouville measure. In these connections, the 2D GFF is often used as a tool - for example, it is used to explain the reversibility of SLE curves, and the Liouville measure is constructed as an exponential of the GFF. In this talk we will switch the focus and concentrate on the geometric and probabilistic properties of the GFF itself. We will highlight the usefulness of thinking of the 2D GFF as a generalization of Brownian motion, and see that even though the 2D GFF is not defined pointwise, one can talk of its level sets or study points that are above or below a certain height. The talk is based on joint works with T. Lupu, E. Powell, A. Sepúlveda and W. Werner.
June 12: Pierre Nolin (ETH Zurich)
• Title: Near-critical percolation and self-organized criticality in two dimensions
• AbstractWe discuss several two-dimensional lattice models displaying a form of self-organized criticality. The behavior of these models is related to the phase transition of independent site percolation, which is now very well understood in two dimensions. In particular, we study the frozen percolation model (where connected components of vertices stop growing when they get too large), and we present a connection with forest-fire processes (where lightning hits independently each vertex with a small rate, and burns its entire connected component immediately). This talk is based on joint works with Rob van den Berg (CWI and VU, Amsterdam) and Demeter Kiss.
Fall 2016
September 5: Nathanael Berestycki (Cambridge)
• Title: The dimer model: universality and conformal invariance.
• Abstract:  The dimer model on a finite bipartite planar graph is a uniformly chosen set of edges which cover every vertex exactly once. It is a classical model of statistical mechanics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s who computed its partition function.
I will discuss some recent joint work with Benoit Laslier and Gourab Ray, where we prove a general result which shows that when the mesh size tends to 0, the fluctuations are described by a universal and conformally invariant limit known as the Gaussian free field.
A key novelty in our approach is that the exact solvability of the model plays only a minor role. Instead, we rely on a connection to imaginary geometry, where Schramm--Loewner Evolution curves are viewed as flow lines of an underlying Gaussian free field. Hence the technique is quite robust and applies in a variety of situations.
September 26: Alan Sola (Stockholm)
• Title: Disks and slits: scaling limits in conformal aggregation models
• AbstractThis talk is devoted to a natural variation of the Hastings-Levitov model in planar aggregation, where clusters are grown by composing simple random conformal maps. I will explain how the choice of model parameter and imposition of regularization lead to different small-particle scaling limits: strong regularization leads to growing disks while weak regularization and strong feedbacks results in randomly oriented slits.

This reports on joint work (in progress) with A. Turner (Lancaster) and F. Viklund (KTH).

October 3: Sergei Zuyev (Chalmers)
• Title: TBA
• Abstract:  TBA
October 10: Kevin Schnelli (KTH)
• Title: Local law of addition of random matrices on optimal scale
• Abstract: Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by
Voiculescu on the macroscopic scale. In this talk, I show that it holds on the
microscopic scale all the way down to the eigenvalue spacing with an optimal error bound.
Joint work with Zhigang Bao and Laszlo Erdos.
November 21: Ellen Powell (Cambridge)
• Title: Level lines of the Gaussian free field with general boundary data.
• Abstract:  I will discuss the level lines of a GFF in a planar domain, with general
boundary data F. In particular I will give a criteria for these level
lines to exist, and will discuss some properties of the resulting
curves. One interesting consequence of the definition of these level
lines is the ability to define and study a new family of SLE curves,
which are the natural generalisation of SLE_4(\rho) processes with a
continuum of force points. This is joint work with Hao Wu from the
University of Geneva.

Spring 2016
March 21: Erik Broman (Uppsala)
• Title:  Continuum percolation models with infinite range
• Abstract In the classical Boolean percolation model, one starts with a homogeneous Poisson process in R^d with intensity u>0, and around each point one places a ball of radius 1. In the talk I will discuss two variants of this model, both which are of infinite range. Firstly, we will consider the so-called Poisson cylinder model, in which the balls are replaced by bi-infinite cylinders of radius 1. We then investigate whether the resulting collection of cylinders is connected, and if so, what the diameter of this set is. In particular I will compare results between Euclidean and hyperbolic geometry.

In the second case, we replace the balls with attenuation functions. That is, we let l : (0,infty) -> (0,infty) be some non-increasing function, and then define the random field Psi by letting Psi(y)=sum l(|x-y|), where we sum over all x in the Poisson process. We study the level sets Psi_\geq h which is simply the set of points where the random field Psi is larger than or equal to h: We determine for which functions I this model has a non-trivial phase transition in h: In addition, we will discuss some classical results and whether these can be transferred to this setting.

Please note: There will be some overlap with my talk during the Nordic Congress of Mathematicians, but here I will go into more detail and present additional results.

April 25: Takis Konstantopoulos (Uppsala)
• Title: Finite and infinite exchangeability
• AbstractExchangeability is ubiquitous in probability theory, with applications ranging from statistical mechanics to stochastic networks and bayesian inference. The classical result in this area is de Finetti's theorem that completely characterizes exchangeable probability measures on infinite products of a "nice" space (e.g., a Polish space). But what happensto exchangeable measures on finite products? It turns out that an analogous result hold, but the mixing measure may not be positive. We shall present a proof of this and show that no topological assumptions are needed whatsoever. We also ask the question of whether an exchangeable measure in n dimensions can be extended to n+1 or higher dimensions. (This is not always the case, and this is a problem that appears, e.g., in extensions of statistical physics models to higher dimensions). We give a necessary and sufficient condition for this, but we do require that the space be a locally compact Hausdorff space.

This is joint work with Svante Janson and Linglong Yuan.
May 16: Alexander Drewitz (Köln)
• Title: An introduction to random interlacements with a view to chemical distance
• Abstract: We start with giving an introduction to the model of 'Random Interlacements' introduced by Sznitman in 2007. Inspired by this model, we then proceed to exhibit recent developments of the notion of chemical distance in the model of random interlacements (in the unique infinite connected component) as well as for more general one parameter families of random infinite subsets of $\Z^d$. Here, we primarily focus on models with long-range correlations.
May 30: Dmitry Beliaev (Oxford)
• Title: Random plane waves and the critical percolation
• Abstract:Random plane waves are the universal model for high energy eigenfunctions of the Laplacian in domains with chaotic dynamics. We are interested in the geometry of the nodal lines (zero set) and nodal domains of the random plane wave. In this talk I will explain why this is an interesting model, present some classical and recent results and finally will discuss the conjectured connection between the random wave model and critical percolation. The talk should be accessible to the general audience, in particular no prior knowledge about the random plane wave model or percolation is required.