Universality of random planar maps and trees

Información del proyecto

UniversalMap

Identificador del acuerdo de subvención: 101001124

DOI

10.3030/101001124

Fecha de la firma de la CE 18 Diciembre 2020

Fecha de inicio 1 Enero 2022

Fecha de finalización 31 Diciembre 2027

Financiado con arreglo a

• EXCELLENT SCIENCE - European Research Council (ERC)

Coste total

€ 1 826 250,00

Aportación de la UE

€ 1 826 250,00

Coordinado por

TEL AVIV UNIVERSITY
Israel

Periodic Reporting for period 2 - UniversalMap (Universality of random planar maps and trees)

Período documentado: 2023-07-01 hasta 2024-12-31

Resumen del contexto y de los objetivos generales del proyecto

We study various problems in statistical physics of critical systems in 2 dimensions and high dimensions. We have set forth various problems addressing random planar maps and trees. The central goal is to make a physics prediction known as "the KPZ equation" into rigorous mathematics and use it to solve long-standing open problems on stochastic processes in two dimensions. The main challenge in this endeavor is understanding statistical properties of random planar maps embedded in the plane by embedding algorithms of a “conformal type”, such as harmonic embeddings, circle packing or square tilings. Additionally, we study random trees, percolation, and hyperbolic surfaces.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

1. We have solved Problem 14 in the grant proposal, showing that the GHP limit of uniform spanning trees of the the hypercube lattice in dimension at least 5 is the Brownian continuum random tree.
2. Showed that the union of two independent USFs in Z^d are transient (and developed some general conditions and theory).
3. Showed that critical percolation on the Hamming hypercube converges the same scaling limit as in the mean-field Erdos-Renyi model, answering various questions and conjectures posed in the past.
4. Provided a sharp lower bound on the Laplacian eigenvalues of compact hyperbolic surfaces of high genus.
5. Showed that the spectral dimension of the minimal spanning forest on the Poisson-weighted infinite tree equals 3/2.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

We expect to gain some major progress on the majority of problems proposed in this project.

random hyperbolic circle packing