arXiv subjects

Zhizhou Liu

Publications and source records attributed to Zhizhou Liu.

3 recordsLinked to original sources

arXiv2026

Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters

We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both the level sets of the Gaussian free field and the vacant set of random interlacements undergo non-trivial percolation phase transitions at deterministic critical levels. A key aspect of the proof is the development of certain quenched controls over tree embeddings, permitting the application of a static renormalization scheme in the presence of spatial irregularities, which may be of independent interest.

math.PR↗

arXiv2026

Solidification estimates for random walks on supercritical percolation clusters

We consider the simple random walk on the infinite cluster of a general class of percolation models on $\mathbb{Z}^d$, $d\geq 3$, including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost every realization of the percolation configuration, we obtain uniform controls on the absorption probability of a random walk by certain "porous interfaces" surrounding the discrete blow-up of a compact set $A$. These controls substantially generalize previous results obtained in arXiv:1706.07229 for Brownian motion in $\mathbb{R}^d$ and in arXiv:2012.05230 for random walks on $\mathbb{Z}^d$ equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.

math.PR↗

arXiv2025

Numerical Unique Ergodicity of Monotone SDEs driven by Nondegenerate Multiplicative Noise

We first establish the unique ergodicity of the stochastic theta method (STM) with $θ\in [1/2, 1]$ for monotone SODEs, without growth restriction on the coefficients, driven by nondegenerate multiplicative noise. The main ingredient of the arguments lies in constructing new Lyapunov functions involving the coefficients, the stepsize, and $θ$ and deriving a minorization condition for the STM. We then generalize the arguments to the Galerkin-based full discretizations for a class of monotone SPDEs driven by infinite-dimensional nondegenerate multiplicative trace-class noise. Applying these results to the stochastic Allen--Cahn equation indicates that its Galerkin-based full discretizations are uniquely ergodic for any interface thickness. Numerical experiments verify our theoretical results.

math.NA↗