arXiv subjects

D. Yogeshwaran

Publications and source records attributed to D. Yogeshwaran.

At least 19 recordsLinked to original sources

arXiv2026

Limit theory for Lipschitz-localized statistics in random geometric models

We study sums of locally dependent scores associated with general marked (i.e., labeled) Euclidean point processes. We introduce geometric mixing conditions on the underlying point process and a Lipschitz-"localization" condition on the scores, which jointly ensure a central limit theorem for the sums of the scores as well as expectation and variance asymptotics. Our localization condition is formulated using the bounded Lipschitz metric, providing a distributional criterion. This stands in contrast to the classical stabilization conditions in stochastic geometry, which are typically based on stopping-set constructions. To demonstrate the applicability of our general framework, we consider several stochastic processes indexed by spatial random graphs. These include spin systems, interacting diffusions, and interacting particle systems. In particular, spin systems highlight the importance of our localization condition. Additional applications include empirical random fields and geostatistical Boolean models.

math.PR↗

arXiv2025

Invariant transports of stationary random measures: asymptotic variance, hyperuniformity, and examples

We consider invariant transports of stationary random measures on $\mathbb{R}^d$ and establish natural mixing criteria that guarantee persistence of asymptotic variances. To check our mixing assumptions, which are based on two-point Palm probabilities, we combine factorial moment expansion with stopping set techniques, among others. We complement our results by providing formulas for the Bartlett spectral measure of the destinations. We pay special attention to the case of a vanishing asymptotic variance, known as hyperuniformity. By constructing suitable transports from a hyperuniform source we are able to rigorously establish hyperuniformity for many point processes and random measures. On the other hand, our method can also refute hyperuniformity. For instance, we show that finitely many steps of Lloyd's algorithm or of a random organization model preserve the asymptotic variance if we start from a Poisson process or a point process with exponentially fast decaying correlation. Finally, we define a hyperuniformerer that turns any ergodic point process with finite intensity into a hyperuniform process by randomizing each point within its cell of a fair partition.

math.PR↗

arXiv2024

Stationary random measures : Covariance asymptotics, variance bounds and central limit theorems

We consider covariance asymptotics for linear statistics of a general stationary random measure in terms of its truncated pair correlation measure. These asymptotics are particularly of interest in the case of hyperuniform random measures. We give exact infinite series-expansion formulas for the covariance of smooth statistics of random measures involving higher-order integrals of the truncated correlation measures and higher-order derivatives of the test functions and also, equivalently in terms of their Fourier transforms. Exploiting this, we describe possible covariance and variance asymptotics for Sobolev and indicator statistics. In the smooth case, we show that the order of variance asymptotics drops by even powers and give a simple example of a random measure exhibiting such a variance reduction. In the case of indicator statistics of C1-smooth sets, we derive covariance asymptotics at surface-order scale with the limiting constant depending on the intersection of the boundaries of the two sets. We complement this with a lower bound for random measures with a non-trivial atomic part. Restricting to hyperuniform simple point processes, we prove a central limit theorem for Sobolev and Holder continuous statistics of simple point processes satisfying a certain integral identity for higher-order truncated correlation functions.

math.PR↗

arXiv2024

Spectra of Poisson functionals and applications in continuum percolation

Let $η$ be a Poisson random measure (defined on some Polish space), and let $F(η)$ be a square-integrable functional of $η$. In this paper we define and study a new notion of {\it spectral point process} associated with $F(η)$, and use such an object to study sharp noise instability and sensitivity properties of planar critical continuum percolation models under spatial birth-death (Ornstein-Uhlenbeck) dynamics -- the notion of sharp noise instability being a natural strengthening of the absence of noise stability. The concept of spectral point process is defined by exploiting the Wiener-Itô chaos expansion of $F$, and represents a natural continuum counterpart to the notion of {\it spectral sample}, as introduced in Garban, Pete and Schramm (2010), in the context of discrete percolation models. In the particular case where $η$ is a marked Poisson measure, we use Hoeffding-ANOVA decompositions to establish an explicit connection with the notion of {\it annealed spectral sample}, introduced in Vanneuville (2021) in the context of Poisson-Voronoi percolation. We also relate spectral processes with an appropriate notion of {\it pivotal processes}. As applications, we show sharp noise instability of crossing events in the critical Poisson Boolean model with unit-radius balls and, using an observation of Vanneuville, we obtain sharp noise sensitivity (as well as sharp noise instability) for crossing events in the Poisson Voronoi percolation model. As an important ingredient, we prove quasi-multiplicativity of the $4$-arm probabilities in the critical Poisson Boolean percolation model.

math.PR↗

arXiv2024

Normal approximation for statistics of randomly weighted complexes

We prove normal approximation bounds for statistics of randomly weighted (simplicial) complexes. In particular, we consider the complete $d$-dimensional complex on $n$ vertices with $d$-simplices equipped with i.i.d. weights. Our normal approximation bounds are quantified in terms of stabilization of difference operators, i.e., the effect on the statistic under addition/deletion of simplices. Our proof is based on Chatterjee's normal approximation bound and is a higher-dimensional analogue of the work of Cao on sparse Erdős-Rényi random graphs but our bounds are more in the spirit of `quantitative two-scale stabilization' bounds by Lachièze-Rey, Peccati, and Yang. As applications, we prove a CLT for nearest face-weights in randomly weighted $d$-complexes and give a normal approximation bound for local statistics of random $d$-complexes.

math.PR↗

arXiv2024

Hyperuniformity and optimal transport of point processes

We examine optimal matchings or transport between two stationary random measures. It covers allocation from the Lebesgue measure to a point process and matching a point process to a regular (shifted) lattice. The main focus of the article is the impact of hyperuniformity(reduced variance fluctuations in point processes) to optimal transport: in dimension 2, we show that the typical matching cost has finite second moment under a mild logarithmic integrability condition on the reduced pair correlation measure, showing that most planar hyperuniform point processes are L2-perturbed lattices. Our method also retrieves known sharp bounds in finite windows for neutral integrable systems such as Poisson processes, and also applies to hyperfluctuating systems. Further, in three dimensions onwards, all point processes with an integrable pair correlation measure are L2-perturbed lattices without requiring hyperuniformity.

math.PR↗

arXiv2023

Limit theorems for critical faces above the vanishing threshold

We investigate convergence of point processes associated with critical faces for a Čech filtration built over a homogeneous Poisson point process in the $d$-dimensional flat torus. The convergence of our point process is established in terms of the $\mathcal M_0$-topology, when the connecting radius of a Čech complex decays to $0$, so slowly that critical faces are even less likely to occur than those in the regime of threshold for homological connectivity. We also obtain a series of limit theorems for positive and negative critical faces, all of which are considerably analogous to those for critical faces.

math.PR↗

arXiv2022

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell, Saks, Schramm and Servedio (2005)) and the Schramm-Steif inequality for the Fourier-Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm-Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions - expressed in terms of randomized stopping sets - yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of `stopping set', not requiring any topological property for the underlying measurable space, as well as on the new concept of a `continuous-time decision tree', for which we establish several fundamental properties. We apply our findings to the $k$-percolation of the Poisson Boolean model and to the Poisson-based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one-arm probabilities. This enables us to settle some open problem suggested by Ahlberg, Tassion and Texeira (2018) on noise sensitivity of crossing events for the planar Poisson Boolean model and also planar Confetti percolation model. Further, we also prove that critical probability is $1/2$ in certain planar confetti percolation models. A special case of this result was conjectured by Benjamini and Schramm (1998) and proved by Müller (2017). Other special cases were proven by Hirsch (2015) and Ghosh and Roy (2018).

math.PR↗

arXiv2022

Central Limit Theorem for Euclidean Minimal Spanning Acycles

We investigate asymptotics for the minimal spanning acycles of the (Alpha)-Delaunay complex on a stationary Poisson process on $\mathbb{R}^d, d \geq 2$. Minimal spanning acycles are topological (or higher-dimensional) generalization of minimal spanning trees. We establish a central limit theorem for total weight of the minimal spanning acycle on a Poisson-Delaunay complex. Our approach also allows us to establish central limit theorems for sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of minimal spanning acycles which proceeds by establishing suitable chain maps and uses matroidal properties of minimal spanning acycles. In contrast to the proof of weak-stabilization for Euclidean minimal spanning trees via percolation-theoretic estimates, our weak-stabilization proof is algebraic in nature and provides an alternative proof even in the case of minimal spanning trees.

math.PR↗

arXiv2022

Volume Approximation of Strongly ${\mathbb C}$-Convex Domains by Random Polyhedra

Polyhedral-type approximations of convex-like domains in $\mathbb{C}^d$ have been considered recently by the second author. In particular, the decay rate of the error in optimal volume approximation as a function of the number of facets has been obtained. In this article, we take these studies further by investigating polyhedra constructed using random points (Poisson or binomial process) on the boundary of a strongly $\mathbb{C}$-convex domain. We determine the rate of error in volume approximation of the domain by random polyhedra, and conjecture the precise value of the minimal limiting constant. Analogous to the real case, the exponent appearing in the error rate of random volume approximation coincides with that of optimal volume approximation, and can be interpreted in terms of the Hausdorff dimension of a naturally-occurring metric space. Moreover, the limiting constant is conjectured to depend on the Möbius-Fefferman measure, which is a complex analogue of the Blaschke surface area measure. Finally, we also prove $L^1$-convergence, variance bounds, and normal approximation.

math.PR↗

arXiv2022

An Analysis of Probabilistic Forwarding of Coded Packets on Random Geometric Graphs

We consider the problem of energy-efficient broadcasting on dense ad-hoc networks. Ad-hoc networks are generally modeled using random geometric graphs (RGGs). Here, nodes are deployed uniformly in a square area around the origin, and any two nodes which are within Euclidean distance of $1$ are assumed to be able to receive each other's broadcast. A source node at the origin encodes $k$ data packets of information into $n\ (>k)$ coded packets and transmits them to all its one-hop neighbors. The encoding is such that, any node that receives at least $k$ out of the $n$ coded packets can retrieve the original $k$ data packets. Every other node in the network follows a probabilistic forwarding protocol; upon reception of a previously unreceived packet, the node forwards it with probability $p$ and does nothing with probability $1-p$. We are interested in the minimum forwarding probability which ensures that a large fraction of nodes can decode the information from the source. We deem this a \emph{near-broadcast}. The performance metric of interest is the expected total number of transmissions at this minimum forwarding probability, where the expectation is over both the forwarding protocol as well as the realization of the RGG. In comparison to probabilistic forwarding with no coding, our treatment of the problem indicates that, with a judicious choice of $n$, it is possible to reduce the expected total number of transmissions while ensuring a near-broadcast.

cs.IT↗

arXiv2021

Edge ideals of Erdös-Rényi random graphs : Linear resolution, unmixedness and regularity

We study the homological algebra of edge ideals of Erdös-Rényi random graphs. These random graphs are generated by deleting edges of a complete graph on $n$ vertices independently of each other with probability $1-p$. We focus on some aspects of these random edge ideals - linear resolution, unmixedness and algebraic invariants like the Castelnuovo-Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., $n,p := p(n)$), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. Next, we study asymptotic behaviour of some algebraic invariants - the Castelnuovo-Mumford regularity, projective dimension and depth - of such random edge ideals in the sparse regime (i.e., $p = \fracλ{n}, λ\in (0,\infty)$). These invariants are studied using local weak convergence (or Benjamini-Schramm convergence) and relating them to invariants on Galton-Watson trees. We also show that when $p \to 0$ or $p \to 1$ fast enough, then with high probability the edge ideals are unmixed and for most other choices of $p$, these ideals are not unmixed with high probability. This is further progress towards the conjecture that random monomial ideals are unlikely to have Cohen-Macaulay property (see De Loera et al. 2019a,2019b) in the setting when the number of variables goes to infinity but the degree is fixed.

math.CO↗

arXiv2021

Poisson process approximation under stabilization and Palm coupling

We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich-Rubinstein distance using the generator approach to Stein's method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework - one is for Morse critical points of the distance function, and the other is for large k-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thale (2016) and Otto (2020).

math.PR↗

arXiv2020

Randomly Weighted $d-$complexes: Minimal Spanning Acycles and Persistence Diagrams

A weighted $d-$complex is a simplicial complex of dimension $d$ in which each face is assigned a real-valued weight. We derive three key results here concerning persistence diagrams and minimal spanning acycles (MSAs) of such complexes. First, we establish an equivalence between the MSA face-weights and \emph{death times} in the persistence diagram. Next, we show a novel stability result for the MSA face-weights which, due to our first result, also holds true for the death and birth times, separately. Our final result concerns a perturbation of a mean-field model of randomly weighted $d-$complexes. The $d-$face weights here are perturbation of some i.i.d. distribution while all the lower-dimensional faces have a weight of $0$. If the perturbations decay sufficiently quickly, we show that suitably scaled extremal nearest face-weights, face-weights of the $d-$MSA, and the associated death times converge to an inhomogeneous Poisson point process. This result completely characterizes the extremal points of persistence diagrams and MSAs. The point process convergence and the asymptotic equivalence of three point processes are new for any weighted random complex model, including even the non-perturbed case. Lastly, as a consequence of our stability result, we show that Frieze's $ζ(3)$ limit for random minimal spanning trees and the recent extension to random MSAs by Hino and Kanazawa also hold in suitable noisy settings.

math.PR↗

arXiv2020

Hyperuniform and rigid stable matchings

We study a stable partial matching $τ$ of the (possibly randomized) $d$-dimensional lattice with a stationary determinantal point process $Ψ$ on $\mathbb{R}^d$ with intensity $α>1$. For instance, $Ψ$ might be a Poisson process. The matched points from $Ψ$ form a stationary and ergodic (under lattice shifts) point process $Ψ^τ$ with intensity $1$ that very much resembles $Ψ$ for $α$ close to $1$. On the other hand $Ψ^τ$ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process $Ψ$, whose so-called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behaviour. For hyperuniformity, we also additionally need to assume some mixing condition on $Ψ$. Further, if $Ψ$ is a Poisson process then $Ψ^τ$ has an exponentially decreasing truncated pair correlation function.

math.PR↗

arXiv2019

Some remarks on associated random fields, random measures and point processes

In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space (POP), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in [Lindqvist 1988]. We use these results to show on Polish spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in [Poinas et al. 2017] and [Lyons 2014] which restrict to point processes in Euclidean spaces and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.

math.PR↗

arXiv2019

Spectral gap bounds for the simplicial Laplacian and an application to random complexes

In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) \cite{ABM} which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erdös-Rényi random graph. We improve the upper bound derived in Kahle (2007) \cite{kahle} by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) \cite{kahle}.

math.CO↗

arXiv2019

Central limit theorem for statistics of subcritical configuration models

We consider sub-critical configuration models and show that the central limit theorem for any additive statistic holds when the statistics satisfies a fourth moment assumption, a variance lower bound and the degree sequence of graph satisfies a growth condition. If the degree sequence is bounded, for well known statistics like component counts, log-partition function, and maximum cut-size which are Lipschitz under addition of an edge or switchings then the assumptions reduce to linear growth condition for the variance of the statistic. Our proof is based on an application of the central limit theorem for martingale-difference arrays due to Mcleish (1974) to a suitable exploration process.

math.PR↗