## Gaussian curvature for LQG surfaces and random planar maps.

Abstract: Liouville quantum gravity is a canonical model for random surfaces conjectured to be the scaling limit of various planar maps. Given that curvature is a central concept in Riemannian geometry, it is natural to ask whether this can be extended to LQG surfaces. Here, we define the Gaussian curvature for LQG surfaces (despite their low regularity) and study the relations with its discrete counterparts. We conjecture that this definition of Gaussian curvature is the scaling limit of the discrete curvature, we prove that the discrete curvature on the $\epsilon$-CRT map with a Poisson...

Read More## Balanced excited random walk.

Resumen: We introduce the balanced excited random walk and review recent results. In particular we give non-trivial upper and lower bounds on the range of the balanced excited random walk in two dimensions, and verify a conjecture of Benjamini, Kozma and Schapira. These are the first non-trivial results for the 2-dimensional model. This talk is partially based on a joint work with Omer Angel (University of British Columbia) and Mark Holmes (University of Melbourne).

Read More## Limiting distributions of Spherical and Spin O(N) models: Appearance of GFF.

Resumen: Spherical model is a mathematical model of a ferromagnet introduced by Berlin and Kac in 1952 as a rough but analytically convenient modification of the Ising model. Since its inception it has enjoyed considerable popularity among the mathematicians and physicists as an exactly soluble model exhibiting a phase transition. In this talk we will explain its relation to the Gaussian free field in the infinite volume limit and to the spin O(N) model in the infinite spin-dimensionality limit of the latter.

Read More## Contour methods for -dimensional Long-Range Ising Model.

Resumen: On the -dimensional lattice with , the phase transition of the nearest-neighbor ferromagnetic Ising model can be proved by using Peierls argument, that requires a notion of contours, geometric curves on the dual of the lattice to study the spontaneous symmetry breaking. It is known that the one-dimensional nearest-neighbor ferromagnetic Ising model does not undergo a phase transition at any temperature. On the other hand, if we add a polynomially decaying long-range interaction given by for , the works by Dyson and Fr\”{o}hlich-Spencer show the phase transition at low...

Read More## Effective Mass of the Fröhlich Polaron: Recent Progress and Open Question.

Resumen: Landau and Pekar (in 1948) and Spohn (in 1987) conjectured that the effective mass $m(\alpha)$ of the Fröhlich Polaron at coupling parameter $\alpha$ grows as $\alpha^4$ as $\alpha\to\infty$ with an explicit pre-factor. In a recent joint work with C. Mukherjee, M. Sellke, and S. R. S. Varadhan, we prove the lower bound $m(\alpha) \geq C \alpha^4$, which matches (up to a constant) the corresponding sharp upper bound shown recently by combining the results from Brooks and Seiringer (2022) and Polzer (2023).

Read More## Limiting distributions of Spherical and Spin O(N) models: Appearance of GFF.

Resumen: Spherical model is a mathematical model of a ferromagnet introduced by Berlin and Kac in 1952 as a rough but analytically convenient modification of the Ising model. Since its inception it has enjoyed considerable popularity among the mathematicians and physicists as an exactly soluble model exhibiting a phase transition. In this talk we will explain its relation to the Gaussian free field in the infinite volume limit and to the spin O(N) model in the infinite spin-dimensionality limit of the latter.

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