IS05: Random Planar Geometry

date: 7/15/2025, time: 14:00-15:30, room: ICS 25

Organizer: Wei Qian (City University of Hong Kong)

Chair: Pierre Nolin (City University of Hong Kong)

The three-point connectivity constant for critical planar percolation

Morris Ang (UC San Diego)

For critical percolation on the triangular lattice, consider the probability that three points lie in the same cluster. The Delfino-Viti conjecture predicts that in the fine mesh limit, under suitable normalization, this probability converges to (a multiple of) the imaginary DOZZ formula from conformal field theory. We prove the Delfino-Viti conjecture, and more generally, obtain the cluster connectivity three-point function of the conformal loop ensemble. Based on joint work with Gefei Cai, Xin Sun, and Baojun Wu.

Exceptional times when bi-infinite geodesics exist in dynamical last passage percolation

Manan Bhatia (MIT)

Exponential last passage percolation (LPP) is a canonical planar directed model of random geometry in the KPZ universality class where the Euclidean metric is distorted by i.i.d. noise. One can also consider a dynamical version of LPP, where the noise is resampled at a constant rate, thereby gradually altering the underlying geometry. In fact, LPP is known to be noise sensitive in the sense that running the dynamics for a microscopic amount of time leads to a macroscopic change in the geometry. In this talk, we shall discuss the question of the existence of exceptional times in dynamical LPP at which bi-infinite geodesics exist. For static LPP, bi-infinite geodesics almost surely do not exist as was shown in Basu-Hoffman-Sly ’18 and Balasz-Busani-Seppalainen ’19.

For dynamical LPP, we show that such exceptional times are at least very close to existing; namely, we give a subpolynomial lower bound \((\Omega(1/\log n))\) on the probability that there is an exceptional time \(t\in [0,1]\) at which the origin lies on a geodesic of length \(n\). In the other direction, for a dynamics on the related Brownian LPP model, we analyse ‘geodesic switches’ to establish that the corresponding set of exceptional times almost surely has Hausdorff dimension at most \(1/2\)– we expect the correct dimension to be \(0\) as can be gathered by an intuitive non-rigorous argument.

Percolation of Gaussian free field and loop soup in dimension two

Yifan Gao (City University of Hong Kong)

We present some recent progress in the study of two-sided level set percolation of GFF. Moreover, we show that the level-set percolation for the occupation field of random walk loop soup at subcritical intensity exhibits a non-trivial phase transition. Along the way, some useful properties and estimates for the four-arm events in the loop soups are derived.