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Register a new userFluctuation and Entropy in Spectrally Constrained random fields
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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.
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We consider the Constrained-degree percolation model with random constraints on the square lattice and prove a non-trivial phase transition. In this model, each vertex has an independently distributed random constraint $jin {0,1,2,3}$ with probability $rho_j$. Each edge $e$ tries to open at a random uniform time $U_e$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when $rho_3$ is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, together with a coarse-graining argument.
Let $X$ be the constrained random walk on ${mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. $p_n doteq P_x(tau_n < tau_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_nrightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article develops a new method to approximate $p_n$ and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of $tau_n$, take limits to remove some of the constraints on the dynamics, this yields a limit unstable constrained walk $Y$ 2) Construct a basis of harmonic functions of $Y$ and use them to apply the classical superposition principle of linear analysis. The basis functions are linear combinations of $log$-linear functions and come from solutions of harmonic systems, which are graphs whose vertices represent points on the characteristic surface of $Y$, the edges between the vertices represent conjugacy relations between the points, the loops represent membership in the boundary characteristic surfaces. Using our method we derive explicit, simple and almost exact formulas for $P_x(tau_n < tau_0)$ for $d$-tandem queues, similar to the product form formulas for the stationary distribution of $X$. The same method allows us to approximate the Balayage operator mapping $f$ to $x rightarrow {mathbb E}_x left[ f(X_{tau_n}) 1_{{tau_n < tau_0}} right]$ for a range of stable constrained random walks in $2$ dimensions. We indicate how the ideas of the paper relate to more general processes and exit boundaries.
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 $n to infty$.
We present a general framework for uncertainty quantification that is a mosaic of interconnected models. We define global first and second order structural and correlative sensitivity analyses for random counting measures acting on risk functionals of input-output maps. These are the ANOVA decomposition of the intensity measure and the decomposition of the random measure variance, each into subspaces. Orthogonal random measures furnish sensitivity distributions. We show that the random counting measure may be used to construct positive random fields, which admit decompositions of covariance and sensitivity indices and may be used to represent interacting particle systems. The first and second order global sensitivity analyses conveyed through random counting measures elucidate and integrate different notions of uncertainty quantification, and the global sensitivity analysis of random fields conveys the proportionate functional contributions to covariance. This framework complements others when used in conjunction with for instance algorithmic uncertainty and model selection uncertainty frameworks.
We consider weighted random balls in $real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $real^d$. This procedure amounts to be very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and of the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields $alpha$-stable stationary isotropic self-similar generalized random fields which recovers Takenaka fields, Telecom process or fractional Brownian motion.
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