arXiv subjects

Mamoru Tanaka

Publications and source records attributed to Mamoru Tanaka.

4 recordsLinked to original sources

arXiv2016

A percolation on directed graphs

Suppose each site independently and randomly chooses some sites around it, and it is weakly (strongly) connected with them (if there choose each other). What is the probability that the weak (strong) connected cluster is infinite? We investigate a percolation model for this problem, which is a generalization of site percolation. We give a relation between the probability of the number of chosen sites around a site and the size of clusters. We also see the expected number of infinite clusters, and the exponential tail decay of the radius and the size of a cluster.

math.PR↗

arXiv2015

Property $(T_{L^Φ})$ and property $(F_{L^Φ})$ for Orlicz spaces $L^Φ$

An Orlicz space $L^Φ(Ω)$ is a Banach function space defined by using a Young function $Φ$, which generalizes the $L^p$ spaces. We show that, for a reflexive Orlicz space $L^Φ([0,1])$, a locally compact second countable group has Kazhdan's property $(T)$ if and only if it has property $(T_{L^Φ([0,1])})$, which is a generalization of Kazhdan's property $(T)$ for linear isometric representations on $L^Φ([0,1])$. We also prove that, for a Banach space $B$ whose modulus of convexity is sufficiently large, if a locally compact second countable group has Kazhdan's property $(T)$, then it has property $(F_{B})$, which is a fixed point property for affine isometric actions on $B$. Moreover, we see that, for an Orlicz sequence space $\ell^{ΦΨ}$ such that the Young function $Ψ$ sufficiently rapidly increases near $0$, hyperbolic groups (with Kazhdan's property $(T)$) don't have property $(F_{\ell^{ΦΨ}})$. These results are generalizations of the results for $L^p$-spaces.

math.GR↗

arXiv2015

Property $(T_B)$ and Property $(F_B)$ restricted to a representation without non-zero invariant vectors

In this paper, we give a necessary and sufficient condition for which a finitely generated group has a property like Kazhdan's Property $(T)$ restricted to one isometric representation on a strictly convex Banach space without non-zero invariant vectors. Similarly, we give a necessary and sufficient condition for which a finitely generated group has a property like Property $(FH)$ restricted to the set of the affine isometric actions whose linear part are one isometric representation on a strictly convex Banach space without non-zero invariant vectors. If the Banach space is the $\ell^p$ space ($1<p<\infty$) on a finitely generated group, these conditions are regarded as an estimation of the spectrum of the $p$-Laplace operator on the $\ell^p$ space and on the $p$-Dirichlet finite space respectively.

math.GR↗

arXiv2013

Multi-way expansion constants and partitions of a graph

In this paper, we consider a relation between $k$-way expansion constant of a finite graph and the expansion constants of subgraphs in a $k$-partition of the graph. Using this relation, we show that a sequence of finite graphs which have uniformly bounded $k+1$-way expansion constants and uniformly bounded degrees can be divided into $k$ or less sequences of expanders. Furthermore, we prove that such sequence of finite graphs is not coarsely embeddable into any Hilbert space.

math.CO↗