I will discuss results on local eigenvalue statistics for uniform random regular graphs. Under mild growth assumptions on the degree, we prove that the local semicircle law holds at the optimal scale, and that the bulk eigenvalue statistics (gap statistics and averaged energy correlation functions) are given by those of the GOE. Joint work with…

We describe some recent advances in the study of the fundamental coupling of a canonical model of random paths, the Schramm--Loewner Evolution (SLE), to a canonical model of random surfaces, Liouville Quantum Gravity (LQG). The latter is expected to be the conformally invariant continuum limit of various models of random planar maps. Via the…

In 1960 Wigner marveled about ``the unreasonable effectiveness of mathematics in the natural sciences," referring mainly to physics. In that spirit we shall first explain how a posteriori relativity and quantum mechanics can be obtained from previously known theories using the mathematical theory of deformations. After a tachyonic overview of how…

Quantum graph models are extremely useful but they also have some drawbacks. One of them concerns the physical meaning of the vertex coupling. The self-adjointness requirement leaves a substantial freedom expressed through parameters appearing in the conditions matching the wave function at the graph vertices. It is a longstanding problem whether…

"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…

TBA

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…

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…

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…

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^…

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…

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…

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…