# Seminars & Events for Mathematical Physics Seminar

##### Towards crystallization in Coulomb systems

We are interested in the statistical mechanics of (classical) two-dimensional Coulomb gases and one-dimensional log gases in a confining potential. We connect the Hamiltonian to the "renormalized energy", a way to compute the total Coulomb interaction of an infinite jellium, and whose minimum is expected to be achieved by the triangular lattice in 2D, and is achieved by the lattice Z in 1D. We apply this to the study of the finite temperature situation. Results include computations of the next order term in the partition function, equidistribution of charges, and concentration to the minimizers of the renormalized energy as the temperature tends to zero. This is based on joint works, mostly with Etienne Sandier.

##### Atoms, Nuclei, and 3d Triangulations

Based on the work of Durhuus-Jonsson and Benedetti-Ziegler, we revisit the question of the number of triangulations of the 3-ball. We introduce a notion of nucleus (a triangulation of the 3-ball without internal nodes, and with each internal face having at most 1 external edge). We show that every triangulation can be built from trees of nuclei. This leads to a new reformulation of Gromov's question: We show that if the number of rooted nuclei with N tetrahedra is exponentially bounded in N, then the number of rooted triangulations with N tetrahedra is also exponentially bounded. This is joint work with Pierre Collet and Maher Younan.

##### Quantum Information Functionals

Quantum information functionals and their algebraic properties play an important role in quantum information theory. Recent developments in the theory of operator monotone and operator convex functions and related topics have simplified earlier results and also led to new insights. One example is the convexity of chi-square divergences. We discuss an intriguing strong order relation for measures of quantum information exemplified by the Wigner-Yanase-Dyson skew information measures and other WYD-like skew information measures. We finally introduce and discuss the notion of classical mixing relative to a preferred quantum statistics.

##### Quantum diffusion and delocalization for random band matrices

I give a summary of recent progress in establishing the diffusion approximation for random band matrices. We obtain a rigorous derivation of the diffusion profile in the regime W > N^{4/5}, where W is the band width and N the dimension of the matrix. As a corollary, we prove complete delocalization of the eigenvectors. Our proof is based on a new self-consistent equation for the Green function. Joint work with L. Erdos, H.T. Yau, and J. Yin.

##### Surface depinning under the two-dimensional hammock potential

We consider a random two-dimensional surface satisfying a Lipschitz constraint. The surface is uniformly chosen from the set of all real-valued Lipschitz functions on a two-dimensional discrete torus. Our main result is that the surface delocalizes, having fluctuations whose variance is at least logarithmic in the size of the torus. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Universal finite size corrections and the central charge in non solvable Ising models

We investigate a non solvable two-dimensional ferromagnetic Ising model with nearest neighbor plus weak finite range interactions. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Parafermionic observables in planar Potts models and Self-Avoiding Walks

In this talk, we will discuss the role of parafermionic observables in the study of several planar statistical physics models. These objects have been introduced recently by Smirnov and Cardy and have been instrumental in Smirnov's proof of conformal invariance of the Ising model. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Striped states at the ferromagnetic transition

PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. We consider a two-dimensional Ising model with nearest neighbor ferromagnetic and long-range, power-law decaying, antiferromagnetic interactions. We assume that the decay exponent of the antiferromagnetic part is larger than 4.