Mathematical Physics Seminar
# Mathematical Physics Seminar at Geneva University 2019-2020

### Time and Place:

Monday 16:15-17:15

Room SM17, 2nd floor

2-4 rue du Lièvre, Geneva
[ Directions ]
### Next Talks:

16/12/2019. Speaker: Lucas Benigni

Title: Fermionic eigenvector moment flow
**Abstract:**
We first present known results and open problems on the study of eigenvector statistics of large random matrices such as complete delocalization, quantum unique ergodicity or asymptotic entry distribution. We willl see how we can obtain some of these properties dynamically using the Dyson Brownian motion and the Bourgade-Yau eigenvector moment flow: it consists of a parabolic equation followed by eigenvector moments. We will then present new moment observables constructed from Grassmann variables which follow a similar equation and which can be seen as a Fermionic counterpart to the (Bosonic) original ones. By combining the information obtained through the study of these two families of observables, we can compute, previously intractable, correlations between eigenvectors.

### Previous Talks:

16/09/2019. Speaker: Antti Knowles

Title: Expander graphs and the spectral gap of random regular graphs
**Abstract:**
Expander graphs are sparse graphs with good connectivity properties. I review some basics of expander graphs, and in particular explain a spectral characterization of the expansion properties of regular graphs in terms of the spectral gap. I then explain how fluctuations of the spectral gap can be analysed for random regular graphs whose degree grows with the number of vertices. This shows that approximately 27% of all regular graphs are optimal expanders.

23/09/2019. Speaker: Dominik Schröder

Title: Edge Universality for non-Hermitian Random Matrices
**Abstract:**
We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. Previously non-Hermitian edge universality had only been established under the condition of four matching moments [Tao, Vu (2015)]. The proof relies on the recently obtained optimal local law in the cusp regime of Hermitian random matrices, and a supersymmetric estimate on the least singular value of shifted Ginibre ensembles. This estimate on the least singular value improves the classical smoothing bound from [Sankar, Spielman, Teng (2006)] in the transitional edge regime.

30/09/2019. Speaker: Vittoria Silvestri

Title: Fluctuations and mixing of Internal DLA on cylinders
**Abstract:**
Internal DLA models the growth of a random cluster by subsequent aggregation of particles. At each step, a new particle starts inside the cluster, and it performs a simple random walk until reaching an unoccupied site, where it settles. When particles move on a cylinder graph GxZ this defines a positive recurrent Markov chain on cluster configurations. In this talk I will address the following questions: How does a typical configuration look like? How long does it take for the process to forget its initial profile? Partly based on joint work with Lionel Levine (Cornell).

07/10/2019. Speaker: Jonathan Hermon

Title: Anchored expansion in supercritical percolation on nonamenable graphs
**Abstract:**
Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G. We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n. We deduce that:

1. Every infinite cluster has anchored expansion (a relaxation of having positive Cheeger constant), and so is nonamenable in some weak sense. This answers positively a question of Benjamini, Lyons, and Schramm (1997).

2. Various observables, including the percolation probability and the truncated susceptibility (which was not even known to be finite!) are analytic functions of p throughout the entire supercritical phase.

3. A RW on an infinite cluster returns to the origin at time 2n with probability whose logarithm is proportional to minus cubic root of n.

Joint work with Tom Hutchcroft.

14/10/2019. Speaker: Phan Thành Nam

Title: Nonlinear Gibbs measures as the limit of equilibrium quantum Bose gases
**Abstract:**
I will discuss the rigorous derivation of nonlinear Gibbs measures starting from a linear quantum system of bosons at a positive temperature. In three dimensions, the nonlinear classical measures have to be defined via a renormalization procedure, while the quantum system is well defined without any renormalization. By simply tuning the chemical potential, we obtain a counter-term for the diverging interactions which implements the Wick renormalization of the limit classical theory. The proof relies on a novel method to control the quantum variance. This is joint work with Mathieu Lewin and Nicolas Rougerie.

21/10/2019. Speaker: Daniel Lenz

Title: Spectral theory of one-dimensional quasicrystals
**Abstract:**
We give an introduction into the field of aperiodic order also known as mathematical quasicrystals.
We then focus on spectral theory of Hamiltonians arising in the quantum mechanical treatment of one-dimensional models.
We explain that these Hamiltonians have a tendency to singular continuous spectrum of Lebesgue measure zero. We will also indicate some recent applications obtained in joined work with Rostislav Grigorchuk, Tatiana Nagnibeda and Daniel Sell.

28/10/2019. Speaker: Alessandro Olgiati

Title:
Stability of the Laughlin phase in presence of interactions
**Abstract:**
The Laughlin wave function is at the basis of the description of the fractional quantum Hall effect (FQHE), nevertheless, many of its fundamental properties are yet to be understood. After an introduction to Laughlin theory, I will present a model for the response of FQHE charge carriers to variations of the external potential and of the inter-particle interaction.

Our main result is that the energy is asymptotically captured by the minimum of an effective func- tional with variational constraints fixed by the incompressibility of the Laughlin phase. Moreover, as was already known for the non-interacting case, we show that the one-body density converges to the characteristic function of a set.

Joint work with Nicolas Rougerie.

04/11/2019. Speaker: Christian Webb

Title: How much can the eigenvalues of a random Hermitian matrix fluctuate?
**Abstract:**
One of the most classical results of random matrix theory is Wigner's theorem, which says that the eigenvalues of a large random Hermitian matrix, drawn from the Gaussian Unitary Ensemle (GUE), are approximately distributed according to the semicircle distribution. I will discuss some recent joint work with T. Claeys (UCLouvain) B. Fahs (Imperial College London), and G. Lambert (University of Zürich) describing how much the eigenvalues can fluctuate around the semicircle distribution.

11/11/2019. Speaker: Nicolai Reshetikhin

Title: Limit shapes in statistical mechanics
**Abstract:**
This talk is an introduction into what is known in statistical mechanics as
the limit shape phenomenon. The essence of this phenomenon is that
the states of the model in the large volume limit are exponentially unlikely
outside of a small neighborhood of a single state called the limit shape.
In the vicinity of the limit shape states are distributed according to the Gaussian distribution. The talk will be focused on the six vertex model.

12/11/2019. Speaker: Tyler Helmuth

Title: Random spanning forests and hyperbolic symmetry
**Abstract:**
The arboreal gas is a probability measure on unrooted spanning forests of a graph G that gives weight beta to each edge in a forest. This specializes to the uniform measure on unrooted forests when beta is one. While it is well-known that random rooted forests are related to the Gaussian free field, I will discuss how the arboreal gas is instead related to a spin system with hyperbolic symmetry. This symmetry is responsible for some surprising behaviour.

Based on joint work in progress with Roland Bauerschmidt, Nick Crawford, and Andrew Swan.

18/11/2019. Speaker: Mikhail Khristoforov

Title: Discrete multi-point and vector-valued observables in percolation.
**Abstract:**
As is well known in rue du Lièvre, the critical percolation model on the hexagonal grid is proven to be conformally invariant in the scaling limit.

The proof is based on a certain combinatorial object (the Smirnov's observable) that controls the crossing probability of a domain with four marked boundary points.

During the talk we will discuss a natural generalization of the observable mentioned above. This observable allows / should allow to control the probabilities of certain multi-point events (like connectivity patterns and Schramm's formula for percolation) *directly* (without using SLE).

I will formulate a problem, that being solved would make our ongoing project much less ongoing :)

PS. If you are not an expert in percolation but you feel comfortable with modular forms, Riemann–Hilbert problems or Birkhoff–Grothendieck theorem, you are especially warmly welcome.

25/11/2019. Speaker: Izabella Stuhl

Title: The hard-core model in discrete 2D: ground states, dominance, Gibbs measures
**Abstract:**
The hard-core model describes a system of non-overlapping identical hard spheres
in a space or on a lattice (more generally, on a graph). An interesting open problem is:
do hard disks in a plane admit a unique Gibbs measure at high density? It seems natural
to approach this question by possible discrete approximations where disks must have
the centers at sites of a lattice or vertexes of a graph.

In this talk, I will report on a progress achieved for the models on a unit triangular lattice A_2
and a unit square lattice Z^2 for a general value of disk diameter D (in the Euclidean metric).
We analyze the structure of Gibbs measures for large fugacities (i.e., high densities) by means
of the Pirogov-Sinai theory and its modifications, including dominance among periodic ground states.

On A_2 we give a complete description of the set of extreme Gibbs
measures; the answer is provided in terms of prime decomposition of the
Löschian number D^2 in the Eisenstein integer ring. Here, the extreme
Gibbs measures are generated by D-sub-lattices, their shifts and reflections.

On Z^2, we have to exclude the values of D with sliding; for the
remaining exclusion distances the answer is given in terms of solutions to
a discrete minimization problem. The latter is connected to Norm equations in
the cyclotomic integer ring Z[zeta], where zeta is a primitive 12th
root of unity.

Parts of our argument contain computer-assisted proofs: identification of instances of sliding,
resolution of dominance issues.

This is a joint work with A. Mazel and Y. Suhov.

02/12/2019. Speaker: Vieri Mastropietro

Title: Renormalization Group approach to universality in non-integrable systems
**Abstract:**
I will present a review of results obtained by rigorous Renormalization Group methods
on the universality problem in 2D statistical mechanics systems, including Ising, Dimer or vertex models
and their non integrable generalizations. The exponents and amplitudes are written as series, convergent for values of parameters near the free fermion point and dependent on all the parameters. The validity of universal scaling relations follows by subtle cancellations due to emergent symmetries in the series expansions.

02/12/2019. Speaker: Christopher J. Bishop

Title: Weil-Petersson curves and finite total curvature.
**Abstract:**
In 2006 Takhtajan and Teo defined a Weil-Petersson metric making universal Teichmuller space (essentially the space of planar quasicircles) into a disconnected Hilbert manifold. The Weil-Petersson class is the unique connected component containing all smooth curves, and consists of certain rectifiable quasicircles. There are several known function theoretic characterizations of WP curves, including connections to Loewner's equation and SLE. In this lecture, I will describe some new geometric characterizations that say a quasicircle is WP iff some measure of local curvature is square integrable over all locations and scales. Here local curvature can be measured using quantities such as: beta-numbers, Menger curvature, integral geometry, inscribed polygons, tangent directions, and associated convex hulls and minimal surfaces in hyperbolic 3-space.

### Contact us:

johannes.alt (at) unige.ch