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An infinite sequence of real random variables $(xi_1, xi_2, dots)$ is said to be rotatable if every finite subsequence $(xi_1, dots, xi_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(xi_1, xi_2, dots)$ is rotatable if and only if $xi_j = tau eta_j$ for all $j$, where $(eta_1, eta_2, dots)$ is a sequence of independent standard Gaussian random variables and $tau$ is an independent nonnegative random variable. Freedmans theorem is equivalent to a classical result of Schoenberg which says that a continuous function $phi : mathbb{R}_+ to mathbb{C}$ with $phi(0) = 1$ is completely monotone if and only if $phi_n: mathbb{R}^n to mathbb{R}$ given by $phi_n(x_1, ldots, x_n) = phi(x_1^2 + cdots + x_n^2)$ is nonnegative definite for all $n in mathbb{N}$. We establish the analogue of Freedmans theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenbergs result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincare, and Borel which says that if $(zeta_1, ldots, zeta_n)$ is uniformly distributed on the sphere of radius $sqrt{n}$ in $mathbb{R}^n$, then, for fixed $k in mathbb{N}$, the distribution of $(zeta_1, ldots, zeta_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n to infty$.
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We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $nge 1$, $F_{n+2} = F_{n+1} pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<ple 1$). Our main result is that the exponential growth of $F_n$ for $0<ple 1$ (linear case) or for $1/3le ple 1$ (non-linear case) is almost surely given by $$int_0^infty log x d u_alpha (x), $$ where $alpha$ is an explicit function of $p$ depending on the case we consider, and $ u_alpha$ is an explicit probability distribution on $RR_+$ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<ple1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.
We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $nge 1$, $F_{n+2} = lambda F_{n+1} pm F_{n}$ (linear case) and $widetilde F_{n+2} = |lambda widetilde F_{n+1} pm widetilde F_{n}|$ (non-linear case), where each $pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<ple 1$). Our main result is that, when $lambda$ is of the form $lambda_k = 2cos (pi/k)$ for some integer $kge 3$, the exponential growth of $F_n$ for $0<ple 1$, and of $widetilde F_{n}$ for $1/k < ple 1$, is almost surely positive and given by $$ int_0^infty log x d u_{k, rho} (x), $$ where $rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ u_{k, rho}$ is an explicit probability distribution on $RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<ple 1$ in the easier case $lambdage 2$. Finally, we study the variations of the exponent as a function of $p$.
We investigate the statistical properties of translation invariant random fields (including point processes) on Euclidean spaces (or lattices) under constraints on their spectrum or structure function. An important class of models that motivate our study are hyperuniform and stealthy hyperuniform systems, which are characterised by the vanishing of the structure function at the origin (resp., vanishing in a neighbourhood of the origin). We show that many key features of two classical statistical mechanical measures of randomness - namely, fluctuations and entropy, are governed only by some particular local aspects of their structure function. We obtain exponents for the fluctuations of the local mass in domains of growing size, and show that spatial geometric considerations play an important role - both the shape of the domain and the mode of spectral decay. In doing so, we unveil intriguing oscillatory behaviour of spatial correlations of local masses in adjacent box domains. We describe very general conditions under which we show that the field of local masses exhibit Gaussian asymptotics, with an explicitly described limit. We further demonstrate that stealthy hyperuniform systems with joint densities exhibit degeneracy in their asymptotic entropy per site. In fact, our analysis shows that entropic degeneracy sets in under much milder conditions than stealthiness, as soon as the structure function fails to be logarithmically integrable.
Let $xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $xi^*(n) = max_x xi(n,x)$. It is known that $limsup xi^*(n)/n$ is a positive constant a.s. We prove that $liminf_n (logloglog n)xi^*(n)/n$ is a positive constant a.s.; this answers a question of P. Revesz (1990). The proof is based on an analysis of the {em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.
Local convergence techniques have become a key methodology to study random graphs in sparse settings where the average degree remains bounded. However, many random graph properties do not directly converge when the random graph converges locally. A notable, and important, random graph property that does not follow from local convergence is the size and uniqueness of the giant component. We provide a simple criterion that guarantees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant are also described by the local limit. We apply this novel method to the configuration model as a proof of concept, reproving a result that is well-established. As a side result this proof, we show that the proof also implies the small-world nature of the configuration model.
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