Perfectoid Shimura varieties of abelian type

We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.Comment: 37 pages; slightly revised version, to appear in IMR

Cell decomposition of some unitary group Rapoport-Zink spaces

In this paper we study the $p$-adic analytic geometry of the basic unitary group Rapoport-Zink spaces \M_K with signature $(1,n-1)$. Using the theory of Harder-Narasimhan filtration of finite flat groups developed by Fargues in \cite{F2},\cite{F3}, and the Bruhat-Tits stratification of the reduced special fiber \M_{red} defined by Vollaard-Wedhorn in \cite{VW}, we find some relatively compact fundamental domain \D_K in \M_K for the action of G(\Q_p)\times J_b(\Q_p), the product of the associated $p$-adic reductive groups, and prove that \M_K admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda's main theorem in \cite{Mi2}.Comment: some minor errors are corrected; to appear in Math. An

pp-adic families of automorphic forms over some unitary Shimura varieties

We construct some $n$-dimensional eigenvarieties for finite slope overconvergent eigenforms over some unitary Shimura varieties with signature $(1,n-1)\times(0,n)\times\cdots\times(0,n)$ by adapting Andreatta-Iovita-Pilloni's method. We also show that there are some Galois pseudo-characters over our eigenvarieties by studying analytic continuation of finite slope eigenforms over these Shimura varieties.Comment: 24 pages; revised version; minor changes; to appear in Math. Research Letter

Weak universality of dynamical Φ34\Phi^4_3: non-Gaussian noise

We consider a class of continuous phase coexistence models in three spatial dimensions. The fluctuations are driven by symmetric stationary random fields with sufficient integrability and mixing conditions, but not necessarily Gaussian. We show that, in the weakly nonlinear regime, if the external potential is a symmetric polynomial and a certain average of it exhibits pitchfork bifurcation, then these models all rescale to $\Phi^4_3$ near their critical point.Comment: 37 pages; updated introduction and reference