We provide elementary proofs of several results concerning the possible outcomes arising from a fixed profile within the class of positional voting systems. Our arguments enable a simple and explicit construction of paradoxical profiles, and we also
demonstrate how to choose weights that realize desirable results from a given profile. The analysis ultimately boils down to thinking about positional voting systems in terms of doubly stochastic matrices.
Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C$ defined on the set $mathcal{B}$ of all balls $Bsubset X$ we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts in $L^{2}(X,m
)$, is essentially self-adjoint, and has a purely point spectrum. Choosing a family ${varepsilon(B)}_{Bin mathcal{B}}$ of i.i.d. random variables, we define the perturbed function $mathcal{C}(B)=C(B)(1+varepsilon(B))$ and the perturbed hierarchical Laplacian $mathcal{L}=L_{mathcal{C}}$. All outcomes of the perturbed operator $mathcal{L}$ are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process $M$ defined in terms of $mathcal{L}$-eigenvalues. Under some natural assumptions $M$ can be approximated by a Poisson point process. Using a result of Arratia, Goldstein, and Gordon based on the Chen-Stein method, we provide total variation convergence rates for the Poisson approximation. We apply our theory to random perturbations of the operator $mathfrak{D}^{alpha }$, the $p$-adic fractional derivative of order $alpha >0$. This operator, related to the concept of $p$-adic Quantum Mechanics, is a hierarchical Laplacian which acts in $L^{2}(X,m)$ where $X=mathbb{Q}_{p}$ is the field of $p$-adic numbers and $m$ is Haar measure. It is translation invariant and the set $mathsf{Spec}(mathfrak{D}^{alpha })$ consists of eigenvalues $p^{alpha k}$, $kin mathbb{Z}$, each of which has infinite multiplicity.
In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincare and Cheeger bounds, and they raised the question as to whether the Poincare bound
is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincare, establish some sufficient conditions on the canonical paths for the Poincare bound to triumph, and show that there is always a choice of paths for which this happens.
Using Steins method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums of mean zero random variables. As a corollary, we deduce
a Berry-Esseen type theorem for the convergence of certain geometric sums. Our results make use of a second order characterizing equation and a distributional transformation which is related to zero-biasing.
