arXiv subjects

Jaelin Kim

Publications and source records attributed to Jaelin Kim.

4 recordsLinked to original sources

arXiv2026

Extreme value theorem for geodesic flow on the quotient of the theta group

We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $Θ\backslash\mathbb{H}^2$.

math.DS↗

arXiv2026

Quenched path limits and periodization stability for tilted Brownian motion in Poissonian potentials on $\mathbb{H}^d$

We analyze the existence of Brownian motion tilted by a potential of full support on hyperbolic spaces $\mathbb{H}^d$. On compact spaces, it is classical that these path limits, called Q-processes, exist and can be directly defined using the ground state of the corresponding Schrödinger operator. On non-compact spaces like $\mathbb{H}^d$, the existence fails in general. We show that for \emph{stationary random} potentials on $\mathbb{H}^d$ with suitable spectral and sup norm bounds, the Q-processes exist a.s. For potentials that are factors of a Poisson point process, the method works up to sup norm $(d-1)^2/8$. In this case, we also show that the path limit can be approximated by periodic potentials. As a tool, we use the foliated space defined by the point process. It turns out that the global ground state of this foliated space serves as a substitute for the non-existing $L^2$ ground states on the leaves of the foliation. Restricting the global ground state to a leaf gives a generalized eigenwave that can be plugged into the usual machinery to get the Q-process.

math.PR↗

arXiv2025

The Uniform Random Walk on graphs, loop processes and graphings

We define the Uniform Random Walk (URW) on a connected, locally finite graph as the weak limit of the uniform walk of length $n$ starting at a fixed vertex. When the limit exists, it is necessarily Markovian and is independent of the starting point. For a finite graph, URW equals the Maximal Entropy Random Walk (MERW). We investigate the existence and phase transitions of URW for loop perturbed regular graphs and their limits. It turns out that for a sequence of finite graphs, it is the global spectral theory of the limiting graphing that governs the behavior of the finite MERWs. In the delocalized phase, we use a "membrane argument", showing that the principal eigenfunction of an expander graphing is stable under a small diagonal perturbation. This gives us: 1) The existence of URW on leaves; 2) The URW is a unique entropy maximizer; 3) The MERW of a finite graph sequence Benjamini-Schramm converges to the URW of the limiting graphing. In the localized phase, the environment seen by the particle takes the role of a finite stationary measure. We show that for canopy trees, the URW exists, is transient and maximizes entropy. We also show that for large finite graphs where most vertices have a fixed degree, localization of MERW is governed by the adjacency norm.

math.PR↗

arXiv2021

Central limit theorem of Brownian motions in pinched negative curvature

We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.

math.DG↗