Search ResultsShowing 1-18 of 18
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arXiv:2505.07145 (Published 2025-05-11)
Nonamenable Poisson zoo
Comments: 40 pages, 4 figuresIn the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $\nu$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($\lambda$) copies of them at each vertex. If the expected volume of the animals w.r.t. $\nu$ is infinite, then the whole $G$ is covered for any $\lambda>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $\lambda$ the union of the animals has only finite clusters, while for $\lambda$ large enough there are also infinite clusters. Here we show that: 1. If $G$ is a nonamenable free product, then for ANY $\nu$ with infinite second but finite first moment and any $\lambda>0$, there will be infinite clusters, despite having arbitrarily low density. 2. The same result holds for ANY nonamenable $G$, when the lattice animals are worms: random walk pieces of random finite length. It remains open if the result holds for ANY nonamenable Cayley graph with ANY lattice animal measure $\nu$ with infinite second moment. 3. We also give a Poisson zoo example $\nu$ on $\mathbb{T}_d \times \mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $\lambda>0$.
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arXiv:2409.14620 (Published 2024-09-22)
On the second moment of the determinant of randomsymmetric, Wigner, and Hermitian matrices
In this paper, we analyze the second moment of the determinant of random symmetric, Wigner, and Hermitian matrices. Using analytic combinatorics techniques, we determine the second moment of the determinant of Hermitian matrices whose entries on the diagonal are i.i.d and whose entries above the diagonal are i.i.d. and have real expected values. Our results extend previous work analyzing the second moment of the determinant of symmetric and Wigner matrices, providing a unified approach for this analysis.
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The heat flow conjecture for random matrices
Comments: 47 pages; 17 figures. Improved and expanded versionRecent results by various authors have established a "model deformation phenomenon" in random matrix theory. Specifically, it is possible to construct pairs of random matrix models such that the limiting eigenvalue distributions are connected by push-forward under an explicitly constructible map of the plane to itself. In this paper, we argue that the analogous transformation at the finite-$N$ level can be accomplished by applying an appropriate heat flow to the characteristic polynomial of the first model. Let the "second moment" of a random polynomial $p$ denote the expectation value of the square of the absolute value of $p.$ We find certain pairs of random matrix models and we apply a certain heat-type operator to the characteristic polynomial $p_{1}$ of the first model, giving a new polynomial $q.$ We prove that the second moment of $q$ is equal to the second moment of the characteristic polynomial $p_{2}$ of the second model. This result leads to several conjectures of the following sort: when $N$ is large, the zeros of $q$ have the same bulk distribution as the zeros of $p_{2},$ namely the eigenvalues of the second random matrix model. At a more refined level, we conjecture that, as the characteristic polynomial of the first model evolves under the appropriate heat flow, its zeros will evolve close to the characteristic curves of a certain PDE. All conjectures are formulated in "additive" and "multiplicative" forms. As a special case, suppose we apply the standard heat operator for time $1/N$ to the characteristic polynomial $p$ of an $N\times N$ GUE matrix, giving a new polynomial $q.$ We conjecture that the zeros of $q$ will be asymptotically uniformly distributed over the unit disk. That is, the heat operator converts the distribution of zeros from semicircular to circular.
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Length of stationary Gaussian excursions
Comments: To appear in the Proceedings of the American Mathematical SocietyCategories: math.PRGiven that a stationary Gaussian process is above a high threshold, the length of time it spends before going below that threshold is studied. The asymptotic order is determined by the smoothness of the sample paths, which in turn is a function of the tails of the spectral measure. Two disjoint regimes are studied - one in which the second spectral moment is finite and the other in which the tails of the spectral measure are regularly varying and the second moment is infinite.
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arXiv:2107.04106 (Published 2021-07-08)
Barely supercritical percolation on Poissonian scale-free networks
Comments: 20 pagesCategories: math.PRWe study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on scale-free random graphs have been observed to have incredibly subtle features that are markedly different compared to those in random graphs with converging second moment. In particular, the critical window for percolation depends sensitively on whether we consider single- or multi-edge versions of the Poissonian random graph. In this paper, and together with our companion paper with Bhamidi, we build a bridge between these two cases. Our results characterize the part of the barely supercritical regime where the size of the giant components are approximately same for the single- and multi-edge settings. The methods for establishing concentration of giant for the single- and multi-edge versions are quite different. While the analysis in the multi-edge case is based on scaling limits of exploration processes, the single-edge setting requires identification of a core structure inside certain high-degree vertices that forms the giant component.
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arXiv:2107.03259 (Published 2021-07-07)
Percolation of worms
Comments: 50 pagesCategories: math.PRWe introduce a new correlated percolation model on the $d$-dimensional lattice $\mathbb{Z}^d$ called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and $v \in (0,\infty)$ (the intensity parameter). From each site of $\mathbb{Z}^d$ we start $\mathrm{POI}(v)$ independent simple random walks with this length distribution. We investigate the connectivity properties of the set $\mathcal{S}^v$ of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then $\mathcal{S}^v$ undergoes a percolation phase transition as $v$ varies. Our main contribution is a sufficient condition on the length distribution which guarantees that $\mathcal{S}^v$ percolates for all $v>0$ if $d \geq 5$. E.g., if the probability mass function of the length distribution is $ m(\ell)= c \cdot \ln(\ln(\ell))^{\varepsilon}/ (\ell^3 \ln(\ell)) 1[\ell \geq \ell_0] $ for some $\ell_0>e^e$ and $\varepsilon>0$ then $\mathcal{S}^v$ percolates for all $v>0$. Note that the second moment of this length distribution is only "barely" infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being "extremal" in this family of models.
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arXiv:2106.00298 (Published 2021-06-01)
Gaps between prime divisors
Erd\H{o}s considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments.
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arXiv:2007.09269 (Published 2020-07-17)
The number of saddles of the spherical $p$-spin model
Comments: 52 pages, 1 figureCategories: math.PRWe show that the quenched complexity of saddles of the spherical pure $p$-spin model agrees with the annealed complexity when both are positive. Precisely, we show that the second moment of the number of critical values of a given finite index in a given interval has twice the growth rate of the first moment.
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arXiv:2003.04965 (Published 2020-03-10)
The diameter of the directed configuration model
Comments: 33 pages, 2 figuresWe show that the diameter of the directed configuration model with $n$ vertices rescaled by $\log n$ converges in probability to a constant. Our assumptions are the convergence of the in- and out-degree of a uniform random vertex in distribution, first and second moment. Our result extends previous results on the diameter of the model and applies to many other random directed graphs.
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arXiv:1905.12342 (Published 2019-05-29)
Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets
Categories: math.PRFor a smooth vectorial stationary Gaussian random field $X : \Omega \times \mathbb{R}^d \to \mathbb{R}^d$, we give necessary and sufficient conditions to have a finite second moment for the number of roots of $X(t) - u$. The results are obtained by using a method of proof inspired on the one obtained by D. Geman for stationary Gaussian processes long time ago. Afterwards the same method is applied to the number of critical points of a scalar random field and also to the level set of a vectorial process $X : \Omega \times \mathbb{R}^D \to \mathbb{R}^d$ with $D > d$.
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arXiv:1807.04910 (Published 2018-07-13)
Pairwise Independent Random Walks can be Slightly Unbounded
Comments: 19 pagesSubjects: 60G50A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a $4$-wise independent random walk on a line over $n$ steps is $O(\sqrt{n})$. For small values of $k$, there exist $k$-wise independent random walks that can be stored in much less space than storing $n$ random bits, so these properties are often useful for lowering space bounds. In this paper, we show that for all of these examples, $4$-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in $\pm 1$ and expected maximum distance $O(\sqrt{n} \lg n)$ from the origin. We also show that this bound is tight for the first and second moment, i.e. the expected maximum square distance of a $2$-wise independent random walk is always $O(n \lg^2 n).$ Also, for any even $k \ge 4,$ we show that the $k$th moment of the maximum distance of any $k$-wise independent random walk is $O(n^{k/2}).$ The previous two results generalize to random walks tracking insertion-only streams, and provide higher moment bounds than currently known. We also prove a generalization of Kolmogorov's maximal inequality by showing an equivalent statement that requires only $4$-wise independent random variables with bounded second moments, which also generalizes a result of [5].
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arXiv:1409.5687 (Published 2014-09-19)
Moment bounds for a class of fractional stochastic heat equations
Categories: math.PRWe consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.
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Intermittency for the wave and heat equations with fractional noise in time
Comments: Published at http://dx.doi.org/10.1214/15-AOP1005 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Probability 2016, Vol. 44, No. 2, 1488-1534DOI: 10.1214/15-AOP1005Categories: math.PRKeywords: fractional noise, intermittency, second moment, malliavin calculus techniques, fractional brownian motionTags: journal articleIn this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index $H>1/2$. The solutions of these equations are interpreted in the Skorohod sense. Using Malliavin calculus techniques, we obtain an upper bound for the moments of order $p\geq2$ of the solution. In the case of the wave equation, we derive a Feynman-Kac-type formula for the second moment of the solution, based on the points of a planar Poisson process. This is an extension of the formula given by Dalang, Mueller and Tribe [Trans. Amer. Math. Soc. 360 (2008) 4681-4703], in the case $H=1/2$, and allows us to obtain a lower bound for the second moment of the solution. These results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time.
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Non-asymptotic performance analysis of importance sampling schemes for small noise diffusions
In this note we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a PDE, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for more accurate comparison among competing importance sampling schemes.
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Complexity of Multilevel Monte Carlo Tau-Leaping
Comments: 24 pages and 2 figures. Minor edits since last versionTau-leaping is a popular discretization method for generating approximate paths of continuous time, discrete space, Markov chains, notably for biochemical reaction systems. To compute expected values in this context, an appropriate multilevel Monte Carlo form of tau-leaping has been shown to improve efficiency dramatically. In this work we derive new analytic results concerning the computational complexity of multilevel Monte Carlo tau-leaping that are significantly sharper than previous ones. We avoid taking asymptotic limits, and focus on a practical setting where the system size is large enough for many events to take place along a path, so that exact simulation of paths is expensive, making tau-leaping an attractive option. We use a general scaling of the system components that allows for the reaction rate constants and the abundances of species to vary over several orders of magnitude, and we exploit the random time change representation developed by Kurtz. The key feature of the analysis that allows for the sharper bounds is that when comparing relevant pairs of processes we analyze the variance of their difference directly rather than bounding via the second moment. Use of the second moment is natural in the setting of a diffusion equation, where multilevel was first developed and where strong convergence results for numerical methods are readily available, but is not optimal for the Poisson-driven jump systems that we consider here. We also present computational results that illustrate the new analysis.
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arXiv:1106.5703 (Published 2011-06-28)
Approach to Express The Second Moment with A Class of Stochastic Completion Time
Comments: 13 pagesCategories: math.PRThis work presents an approach to express the second moment of the completion time with a preempt-repeat model job processed on a machine subject to stochastic breakdowns by some distribution characters of the uptime, downtime and processing time.
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A CLT for the third integrated moment of Brownian local time increments
Categories: math.PRLet $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of Brownian motion. Our main result is to show that for each fixed $t$ $${\int (L^{x+h}_t- L^x_t)^3 dx-12h\int (L^{x+h}_t - L^x_t)L^x_t dx-24h^{2}t\over h^2} \stackrel{\mathcal{L}}{\Longrightarrow}\sqrt{192}(\int (L^x_t)^3dx)^{1/2}\eta$$ as $h\to 0$, where $\eta$ is a normal random variable with mean zero and variance one that is independent of $L^{x}_{t}$. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments
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arXiv:math/0609682 (Published 2006-09-25)
On the second moment of the number of crossings by a stationary Gaussian process
Comments: Published at http://dx.doi.org/10.1214/009117906000000142 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Journal: Annals of Probability 2006, Vol. 34, No. 4, 1601-1607Categories: math.PRKeywords: second moment, geman condition, finite variance, centered stationary gaussian process, twice differentiable covariance functionTags: journal articleCram\'{e}r and Leadbetter introduced in 1967 the sufficient condition \[\frac{r''(s)-r''(0)}{s}\in L^1([0,\delta],dx),\qquad \delta>0,\] to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function $r$. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.