In this talk, we discuss geometric properties of Finitary Random Interlacements (FRI) $\mathcal{FI}^{u,T}$ in $\mathbf{Z}^d$. We prove that with probability one $\mathcal{FI}^{u,T}$ has no infinite connected component for all sufficiently small fiber length $T>0$, and a unique infinite connected component for all sufficiently large $T$. At the same time, although FRI may not enjoy global stochastic monotonicity with respect to $T$, we prove the existence of a critical $T_c(u)$ for all large $u$. Moreover, we find the chemical distance on the infinite cluster is of the same order as Euclidean distance as well as a local uniqueness result for all sufficiently large $T$. Researches joint with E.B. Procaccia, J. Ye, Y. Xiong，Z. Cai, and X. Han.

Yuan Zhang received his Ph.D. at Duke University, US, in the year of 2015. His Ph.D. advisor is Dr. Rick Durrett. He is now an assistant professor at the School of Mathematical Sciences, Peking University. His research is primarily on random geometry and interacting particle systems.