IS05: Random Planar Geometry
Organizer: Wei Qian (City University of Hong Kong)
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.
Schramm-Loewner evolution contains a topological Sierpi'nski carpet when \(\kappa\) is close to 8
Zijie Zhuang (University of Pennsylvania)
In this talk, I will present a result showing that there exists \(\delta_0>0\) such that for \(\kappa \in (8 - \delta_0,8)\), the range of an SLE\(_\kappa\) curve almost surely contains a topological Sierpiński carpet. Combined with a result of Ntalampekos (2021), this implies that SLE\(_\kappa\) is almost surely conformally non-removable in this parameter range. I will explain the main intuition coming from Mandelbrot’s fractal percolation and discuss some open questions. Based on joint work with Haoyu Liu (PKU).