# Seminars & Events for Mathematical Physics Seminar

##### Regular operator mappings and multivariate geometric means

We introduce the notion of regular operator mappings of several variables generalising the notion of spectral function. This setting is convenient for studying maps more general than what can be obtained from the functional calculus, and it allows for Jensen type inequalities and multivariate non-commutative perspectives. As a main application of the theory we consider geometric means of $k$ operator va riables extending the geometric mean of $k$ commuting operators and the geometric mean of two arbitrary positive definite matrices. We propose different types of updating conditions that seems natural in many applications and prove that each of these conditions, together with a few other natural axioms, uniquely defines the geometric mean for any number of operator variables.

##### Statistical Mechanics on Sparse Random Graphs: Mathematical Perspective

Theoretical physics studies of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on random finite graphs that converge locally to trees we review recent progress in validating the `cavity' prediction for the limiting free energy per vertex and the approximation of local marginals by the belief propagation algorithm. This talk is based on joint works with Anirban Basak, Andrea Montanari, Nike Sun and Alan Sly.

##### A rigorous result on many-body localization

A one-dimensional spin chain with random interactions exhibits many-body localization. I will discuss a proof under a physically reasonable assumption that limits the amount of level attraction in the system. The construction uses a sequence of local unitary transformations to diagonalize the Hamiltonian and connect the exact many-body eigenfunctions to the original basis vectors.

##### The structure of flow in Hydrodynamics, Thermodynamics and General Relativity, from Navier Stokes to Tolman

Problems with the formulation of Relativistic Astrophysics lead to the need for a variational formulation of thermodynamics. This is easy and immensely rewarding in the non relativistic context, so long as the motion is irrotational. The main topic of this talk is to overcome this limitation. The solution is amazingly simple; one has to combine two familiar forms of hydrodynamics. but it is shocking and even revolutionary nevertheless. Some basic tenets have to be given up, with interesting consequences. The application to General Relativity will be sketched at the end.

##### Superfluid behavior of a Bose-Einstein condensate in a random potential

We investigate the relation between Bose-Einstein condensation (BEC) and superfluidity in the ground state of a one-dimensional model of interacting Bosons in a strong random potential. We prove rigorously that in a certain parameter regime the superfluid fraction can be arbitrarily small while complete BEC prevails. In another regime there is both complete BEC and complete superfluidity, despite the strong disorder. This is joint work with M. Könenberg, T. Moser and R. Seiringer.

##### Stochastic higher spin vertex models on the line

We show how transfer matrices of higher spin vertex models (generalizing the six-vertex model) can be conjugated into stochastic matrices describing interacting particle systems. Bethe ansatz produces eigenfunctions and we prove their completeness on the line. This, along with a self duality of the transfer matrices, provides a means to study the long time behavior of these stochastic systems. These considerations bring under one roof, all of the recently investigated integrable probabilistic systems in the Kardar-Parisi-Zhang universality class.

##### Height Fluctuations in Interacting Dimers

Perfect matchings of Z^2(also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g. a local energy function favoring the alignment among neighboring dimers, the model is not solvable anymore and the dimer-dimer correlation functions decay polynomially at infinity with a non-universal (interaction-dependent) critical exponent. We prove that, nevertheless, the height fluctuations remain gaussian even in the presence of interactions, in the sense that all their moments converge to the gaussian ones at large distances.

##### A novel quantum-mechanical interpretation of the Dirac equation

A novel interpretation is given of Dirac's ``wave equation for the relativistic electron'' as a quantum-mechanical one-particle equation in which electron and positron are merely the two different ``topological spin'' states of a single more fundamental particle, not distinct particles in their own right. The new interpretation is backed up by the existence of such binary particle structures in general relativity, in particular the curvature singularity of the maximal analytically extended, topologically non-trivial, electromagnetic Kerr--Newman spacetime in the zero-gravity limit. The pertinent general-relativistic zero-gravity Hydrogen problem is studied in the usual Born--Oppenheimer approximation.

##### Rigidity phenomena in random point sets

In several naturally occurring (infinite) point processes, the number the points inside a finite domain can be determined, almost surely, by the point configuration outside the domain. There are also other processes where such ''rigidity'' extends also to a number of moments of the mass distribution. The talk will focus on point processes with such curious "rigidity" phenomena, and their implications. We will also talk about applications to stochastic geometry and some questions in harmonic analysis.

##### Airy diffusion and N^{1/3} fluctuations in the 2D Ising model

For the two-dimensional Ising model at low temperatures consider a floating droplet of the (+) phase floating in the sea of (-) phase, pressed against a horizontal wall within a box of linear size N. I will explain that the fluctuations of the boundary of the droplet near the contact with the wall are of the order of N^{1/3}. When scaled by N^{1/3} vertically and by N^{2/3} horizontally, the limiting behavior of the boundary as N goes to infinity is given by the Airy diffusion process. This diffusion process has appeared earlier in a paper by Ferrari and Spohn, where the brownian motion above the parabolic barrier is considered. Work in progress with D. Ioffe and Y. Velenik.

##### On the decay rates of determinantal correlation functionals

Determinantal correlation functionals arise as the correlation functions of non-interacting fermions, and also in other determinantal point processes, such as the eigenvalues of random matrices, and the zeros of random polynomials. It is then of interest to know what implication does fast decay of the two-point function have on the decay of the n-point functional. In this talk I will explain how in one-dimensional lattice models exponential decay of the two-point function implies exponential decay of the determinant. The decay is in the symmetrized maximal distance of the particle configurations, and the bounds presented will be uniform in the number of particles. (Joint work with R. Sims).

##### When the Dust Settles

Small aerosols drift down a temperature or turbulence gradient since faster particles fly longer distances before equilibration. That fundamental phenomenon is known since Maxwell and it was universally believed that particles moving down the kinetic energy gradient must concentrate in minima (say, on walls in turbulence). Here, I show that this is incorrect: escaping minima is possible for inertial particles whose time of equilibration is longer than the time to reach the minimum. "The best way out is always through": particles escape by flying through minima or reflecting from walls. I present the analytical solution of this problem, which has surprising analogies with multiple phenomena, from Anderson localization to non-equilibrium steady states and modified fluctuation-dissipation theorem.

##### A simple renormalization flow setup for FK-percolation models

We will present a simple setup in which one can make sense of a renormalization flow for FK-percolation models in terms of a simple Markov process on a state-sace of discrete weighted graphs. We will describe how to formulate the universality conjectures in this framework (in terms of stationary measures for this Markov process), and how to prove this statement in the very special case of the two-dimensional uniform spanning tree (building on existing results on this model). This is partly based on joint work with Stéphane Benoist and Laure Dumaz.

##### Singular Eigenvalue Perturbation Theory

Eigenvalue Perturbation Theory is central to the theory of nonrelativistic quantum mechanics going back to Schrodinger's first papers. This lecture will review what is known about the eigenvalues in physical situations where one doesn't have simple convergence to a new isolated eigenvalue. Included are the anharmonic oscillator and Zeeman effect (divergent series and summability), autoionizing states in atoms (complex scaling and resonances), Stark effect (exponentially small resonances) and double wells (instantons).

##### A class of gapped Hamiltonians on quantum spin chains and its classification

The MPS (matrix product state) formalism gives a recipe to construct Hamiltonians in quantum spin chains from $n$-tuples of $k\times k$- matrices. This $n$-tuple defines a completely positive map and the existence of the uniform spectral gap of the Hamiltonian is related to the spectral property of the associated CP map. I would like to talk about a classification problem of this class of Hamiltonians. Through the relation between Hamiltonians and CP maps, the problem is reduced to the question of path connectedness of a class of CP maps.