arXiv subjects
Publications and source records attributed to Ryokichi Tanaka.
We introduce the notion of a Magnus marking for a finite generating set of a group and prove a certain expansion property. Using this property, we determine the second and third smallest growth rates of the rank-$d$ free group for $d\ge 2$. We also give a new lower bound for the smallest growth rate of the genus-$g$ surface group for $g\ge 2$, as well as a lower bound for the growth rate associated with a one-relator presentation.
We show that aperiodic random walks with finite second moment on virtually abelian groups are noise sensitive in total variation if and only if the group admits no nonzero homomorphism onto the infinite cyclic group.
We show that the harmonic measure on a product of boundaries satisfies dimension conservation for a random walk with non-elementary marginals on a countable group acting on a product of hyperbolic spaces under the finite first moment condition.
We show that on every affine Weyl group natural random walks are noise sensitive in total variation.
For every non-elementary hyperbolic group, we introduce the Manhattan curve associated to any pair of left-invariant hyperbolic metrics which are quasi-isometric to a word metric. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, i.e., they are within bounded distance after multiplying by a positive constant. Further, we prove that the Manhattan curve associated to two strongly hyperbolic metrics is twice continuously differentiable. The proof is based on the ergodic theory of topological flows associated to general hyperbolic groups and analyzing the multifractal structure of Patterson-Sullivan measures. We exhibit some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for pairs of word metrics.
We consider congestion dynamics with $n$ players and $Q$ resources under the constraint that the number of each resource is $κ$ and that $n<κQ$ in the regime that $n$ and $κ$ diverge but $Q$ is fixed with $n=\lfloor{ρκQ\rfloor}$ for a fixed constant $ρ\in (0, 1/2]$. We show that the Glauber dynamics and its unlabeled version exhibit cutoff at time $(1/2)n \log n$ and $(1/2)(1-ρ)n\log n$ in total variation respectively. The unlabeled version is a special case of natural Markov chains for sampling from log M-concave distributions. We also show that a family of Markov chains for uniform sampling on M-convex sets does not necessarily exhibit cutoff.
We show that symmetric random walks on non-elementary hyperbolic groups with non-zero homomorphisms into the reals are noise stable at linear scale under finite exponential moment condition.
We show that for every non-elementary hyperbolic group, an associated topological flow space admits a coding based on a transitive subshift of finite type. Applications include regularity results for Manhattan curves, the uniqueness of measures of maximal Hausdorff dimension with potentials, and the real analyticity of intersection numbers for families of dominated representation, thus providing a direct proof of a result established by Bridgeman, Canary, Labourie and Sambarino in 2015.
We show that non-elementary random walks on word hyperbolic groups with finite first moment are not noise sensitive in a strong sense for small noise parameters.
We show that for any attractive Glauber-Exclusion process on the one-dimensional lattice of size $N$ with periodic boundary condition, if the corresponding hydrodynamic limit equation has a reaction term with a strictly convex potential, then the total-variation mixing time is of order $O(\log N)$. In particular, the result covers the full high-temperature regime in the original model introduced by De Masi, Ferrari and Lebowitz (1985).
We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$, and discuss optimal values in view of the theory of automorphic representations when $g=2$. As a by-product, we also show that the finite regular directed graphs constructed by Jordan-Zaytman also has the same property.
We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class and all hyperbolic metrics on the surface. We give explicit examples of such hyperbolic surfaces through a new interpretation of the Nielsen realization problem for the mapping class groups.
We show that for every non-elementary hyperbolic group the Bowen-Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.
We analyze a Markov chain, known as the product replacement chain, on the set of generating $n$-tuples of a fixed finite group $G$. We show that as $n \rightarrow \infty$, the total-variation mixing time of the chain has a cutoff at time $\frac{3}{2} n \log n$ with window of order $n$. This generalizes a result of Ben-Hamou and Peres (who established the result for $G = \mathbb{Z}/2$) and confirms a conjecture of Diaconis and Saloff-Coste that for an arbitrary but fixed finite group, the mixing time of the product replacement chain is $O(n \log n)$.
It has been observed that certain classical chains admit topologically protected zero-energy modes that are localized on the boundaries. The static features of such localized modes are captured by linearized equations of motion, but the dynamical features are governed by its nonlinearity. We study quasi-periodic solutions of nonlinear equations of motion of one-dimensional classical chains. Such quasi-periodic solutions correspond to periodic trajectories in the configuration space of the discrete systems, which allows us to define solitons without relying on a continuum theory. Furthermore, we study the dynamics of solitons in inhomogeneous systems by connecting two chains with distinct parameter sets, where transmission or reflection of solitons occurs at the boundary of the two chains.
We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.
We show exact dimensionality of harmonic measures associated with random walks on groups acting on a hyperbolic space under finite first moment condition, and establish the dimension formula by the entropy over the drift. We also treat the case when a group acts on a non-proper hyperbolic space acylindrically. Applications of this formula include continuity of the Hausdorff dimension with respect to driving measures and Brownian motions on regular coverings of a finite volume Riemannian manifold.
This is a note on a local ergodic theorem for a symmetric exclusion process defined on an infinite tower of coverings, which is associated with a finitely generated residually finite amenable group.