We establish the lower bound $p_{2t}(e,e)\succsim exp\left(-t^{1/3}\right)$, for the large times asymptotic behaviours of the probabilities $p_{2t}(e,e)$ of return to the origin at even times $2t$, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer $r$, such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to $r$.)

Let $G={*}_{j=1}^{q+1}{G}_{n_j+1}$ be the product of $q+1$ finite groups each having order $n_j+1$ and let $\mu$ be the probability measure which takes the value $p_j/n_j$ on each element of $G_{n_j+1}\setminus \left\{e\right\}$. In this paper we shall describe the point spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $n_j$. We also compute the continuous spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ in several cases. A family of irreducible representations of $G$, parametrized on the continuous spectrum of $\mu$,...

This paper studies the on- and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs.

The affine group of a local field acts on the tree $𝕋\left(𝔉\right)$ (the Bruhat-Tits building of $\mathrm{GL}(2,𝔉)$) with a fixed point in the space of ends $\partial 𝕋\left(F\right)$. More generally, we define the affine group $Aff\left(𝔉\right)$ of any homogeneous tree $𝕋$ as the group of all automorphisms of $𝕋$ with a common fixed point in $\partial 𝕋$, and establish main asymptotic properties of random products in $Aff\left(𝔉\right)$: (1) law of large numbers and central limit theorem; (2) convergence to $\partial 𝕋$ and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary...

We show that, given an n-dimensional normed space X, a sequence of $N={(8/\epsilon )}^{2n}$ independent random vectors ${\left(X_i\right)}_{i=1}^N$, uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $\Gamma :ℝ\to {ℝ}^N$ defined by $\Gamma x={(⟨x,X_i⟩)}_{i=1}^N$ embeds X in ${\ell }_{\infty }^N$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into ${\ell }_{\infty }^N$ with asymptotically best possible relation between N, n, and ε.

In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time...

In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time behavior...