arXiv subjects

Christian Hirsch

Publications and source records attributed to Christian Hirsch.

At least 19 recordsLinked to original sources

arXiv2026

The voter model on the hyperbolic graph

We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $\alpha\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/\alpha}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^\varphi$, where we also establish the consensus time orders for all $\varphi\geq 0$. These orders have 3 regimes, with a phase transition at $\varphi=2-2\alpha$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.

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arXiv2026

Local large deviations for linear-region growth in random piecewise-linear networks

We study a random compositional model for the growth of affine regions in deep piecewise-linear networks. The model is generated by i.i.d.\ perturbations of the symmetric height-one tent map, and the main observable is the number $N_n$ of affine pieces after $n$ layers. We prove the existence of a submultiplicative pressure for $N_n$, yielding exponential upper bounds for both tails of $n^{-1}\log N_n$. The same argument applies to abstract submultiplicative complexity observables and gives higher-dimensional extensions for convex-polytopal affine-cover counts and worst-line affine-piece counts. Since the true branch count has no matching supermultiplicative inequality, lower bounds require a separate certified construction. We introduce a finite-state defect process that records branches whose future splitting can be guaranteed, and use bridge words to obtain constructive upper-tail lower bounds. In a uniformly favorable small-noise regime, this process is governed by a companion matrix whose Perron root tends to $2$, implying eventual exclusion of lower tails below $\log 2-\xi$.

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arXiv2026

A mathematical analysis of hierarchical Hopfield models

The central question that we address is: How can structured information be stored in a hierarchical Hopfield model involving hidden layers? To this end, we develop a formalism of strokes and concepts that allows us to appropriately structure information: initial features are first classified into strokes, which in a second step are aggregated into concepts. We rigorously derive criteria under which concepts can be retrieved from noisy input data. A remarkable effect is that we do not require a perfect retrieval at the level of strokes, as the second-layer retrieval procedure compensates for first-layer errors. We treat separately the cases of fixed and variable-sized concepts.

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arXiv2026

The contact process on a bipartite spatial network

We study the contact process on a random bipartite connection hypergraph generated from two Poisson point processes, with mark-dependent connection thresholds. For asymmetric infection rates and asymmetric power law tail decays of the two degree distributions, we determine the dominant survival strategies in all parameter regimes and provide asymptotics for the epidemic probability up to logarithmic factors.

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arXiv2026

Central limit theorem for linear eigenvalue statistics of random geometric graphs

Random spatial networks-that is, graphs whose connectivity is governed by geometric proximity-have emerged as fundamental models for systems constrained by an underlying spatial structure. A prototypical example is the random geometric graph, obtained by placing vertices according to a Poisson point process and connecting two vertices whenever their Euclidean distance is less than a certain threshold. Despite their broad applicability, the spectral properties of such spatial models remain far less understood than those of classical random graph models, such as Erd\H{o}s-R\'enyi graphs and Wigner matrices. The main obstacle is the presence of spatial constraints, which induce highly nontrivial dependencies among edges, placing these models outside the scope of techniques developed for purely combinatorial random graphs. In this paper, we provide the first rigorous analysis of Gaussian fluctuations for linear eigenvalue statistics of random geometric graphs. Specifically, we establish central limit theorems for $\text{Tr}[\phi(A)]$, where $A$ is the adjacency matrix and $\phi$ ranges over a broad class of suitable (possibly non-polynomial) test functions. In the polynomial setting, we moreover obtain a quantitative central limit theorem, including an explicit convergence rate to the limiting Gaussian law. We further obtain polynomial-test-function CLTs for other canonical random spatial networks, including $k$-nearest neighbor graphs and relative neighborhood graphs. Our results open new avenues for the study of spectral fluctuations in spatially embedded random structures and underscore the delicate interplay between geometry, local dependence, and spectral behavior.

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arXiv2026

Normal and Poisson approximation for Gibbs point processes with pair potentials

We provide a Poisson approximation result for dependent thinnings of Gibbs point processes as well as qualitative and quantitative central limit theorems for geometric functionals of Gibbs point processes in increasing observation windows. The present paper extends prior work on finite-range Gibbs processes to processes with repulsive pairwise interaction of unbounded interaction range as well as processes on marked Euclidean space. The proofs rely on coupling different Gibbs processes using the disagreement coupling technique, which we generalize to infinite-volume domains under a suitable non-percolation condition. For the case of repulsive pairwise interactions, we introduce a version of disagreement coupling that constructs the Gibbs process by thinning a random connection model thus making previous approximation methods more flexible.

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arXiv2026

Stable and Fr\'echet limit theorem for subgraph functionals in the hyperbolic random geometric graph

We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincar\'e ball in the heterogeneous, heavy-tailed degree regime. In a hyperbolic random geometric graph whose vertices are given by a Poisson point process on a growing hyperbolic ball, we consider two basic families of subgraphs: star shape counts and clique counts, and we analyze their global counts and maxima over the vertex set. Working in the parameter regime where a small number of vertices close to the center of the Poincar\'e ball carry very large degrees and act as hubs, we establish joint functional limit theorems for suitably normalized star shape and clique count processes together with the associated maxima processes. The limits are given by a two-dimensional dependent process whose components are a stable L\'evy process and an extremal Fr\'echet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, we derive fluctuation results for the global clustering coefficient, showing that its asymptotic behavior is described by the ratio of the components of a bivariate L\'evy process with perfectly dependent stable jumps.

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arXiv2025

Normal approximation of stabilizing Poisson pair functionals with column-type dependence

In this paper, we study two specific types of $d$-dimensional Poisson functionals: a double-sum type and a sum-log-sum type, both over pairs of Poisson points. On these functionals, we impose column-type dependence, i.e., local behavior in the first $k$ directions and allow non-local, yet stabilizing behavior in the remaining $d-k$ directions. The main contribution of the paper is to establish sufficient conditions for Normal approximation for sequences of such functionals over growing regions. Specifically, for any fixed region, we provide an upper bound on the Wasserstein distance between each functional and the standard Normal distribution. We then apply these results to several examples. Inspired by problems in computer science, we prove a Normal approximation for the rectilinear crossing number, arising from projections of certain random graphs onto a 2-dimensional plane. From the field of topological data analysis, we examine two types of barcode summaries, the inversion count and the tree realization number, and establish Normal approximations for both summaries under suitable models of the topological lifetimes.

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arXiv2025

Functional limit theorems for edge counts in dynamic random connection hypergraphs

We introduce a dynamic random hypergraph model constructed from a bipartite graph. In this model, both vertex sets of the bipartite graph are generated by marked Poisson point processes. Vertices of both vertex sets are equipped with marks representing their weight that influence their connection radii. Additionally, we also assign the vertices of the first vertex set a birth-death process with exponential lifetimes and the vertices of the second vertex set a time instant representing the occurrence of the corresponding vertices. Connections between vertices are established based on the marks and the birth-death processes, leading to a weighted dynamic hypergraph model featuring power-law degree distributions. We analyze the edge-count process in two distinct regimes. In the case of finite fourth moments, we establish a functional central limit theorem for the normalized edge count, showing convergence to a Gaussian AR(2)-type process as the observation window grows. In the challenging case of the heavy-tailed regime with infinite variance, we prove convergence to a novel stable process that is not L\'evy and not even Markov.

math.PR↗

arXiv2025

Normal approximation for subgraph counts in age-dependent random connection models

We study normal approximation of subgraph counts in a model of spatial scale-free random networks known as the age-dependent random connection model. In the light-tailed regime where only moments of order $(2 + \varepsilon)$ are finite, we study the asymptotic normality of both clique and subtree counts. For clique counts, we establish a multivariate quantitative normal approximation result through the Malliavin-Stein method. In the more delicate case of subtree counts, we obtain distributional convergence based on a central limit theorem for sequences of associated random variables.

math.PR↗

arXiv2025

LDP for the covariance process in fully connected neural networks

In this work, we study large deviation properties of the covariance process in fully connected Gaussian deep neural networks. More precisely, we establish a large deviation principle (LDP) for the covariance process in a functional framework, viewing it as a process in the space of continuous functions. As key applications of our main results, we obtain posterior LDPs under Gaussian likelihood in both the infinite-width and mean-field regimes. The proof is based on an LDP for the covariance process as a Markov process valued in the space of non-negative, symmetric trace-class operators equipped with the trace norm.

math.PR↗

arXiv2025

Efficient Rare-Event Simulation for Random Geometric Graphs via Importance Sampling

Random geometric graphs defined on Euclidean subspaces, also called Gilbert graphs, are widely used to model spatially embedded networks across various domains. In such graphs, nodes are located at random in Euclidean space, and any two nodes are connected by an edge if they lie within a certain distance threshold. Accurately estimating rare-event probabilities related to key properties of these graphs, such as the number of edges and the size of the largest connected component, is important in the assessment of risk associated with catastrophic incidents, for example. However, this task is computationally challenging, especially for large networks. Importance sampling offers a viable solution by concentrating computational efforts on significant regions of the graph. This paper explores the application of an importance sampling method to estimate rare-event probabilities, highlighting its advantages in reducing variance and enhancing accuracy. Through asymptotic analysis and numerical studies, we demonstrate the effectiveness of our methodology, contributing to improved analysis of Gilbert graphs and showcasing the broader applicability of importance sampling in complex network analysis.

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arXiv2024

Poisson approximation of large-lifetime cycles

In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime features on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions $d \geq 2$ under suitable model conditions.

math.PR↗

arXiv2024

Large Deviation Analysis for Canonical Gibbs Measures

In this paper, we present a large-deviation theory developed for functionals of canonical Gibbs processes, i.e., Gibbs processes with respect to the binomial point process. We study the regime of a fixed intensity in a sequence of increasing windows. Our method relies on the traditional large-deviation result for local bounded functionals of Poisson point processes noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions allowing us to handle delicate and unlikely pathological events. The presented results cover three types of Gibbs models - a model given by a bounded local interaction, a model given by a non-negative possibly unbounded increasing local interaction and the hard-core interaction model. The derived large deviation principle is formulated for the distributions of individual empirical fields driven by canonical Gibbs processes, with its special case being a large deviation principle for local bounded observables of the canonical Gibbs processes. We also consider unbounded non-negative increasing local observables, but the price for treating this more general case is that we only get large-deviation bounds for the tails of such observables. Our primary setting is the one with periodic boundary condition, however, we also discuss generalizations for different choices of the boundary condition.

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arXiv2024

Functional central limit theorem for topological functionals of Gaussian critical points

We consider Betti numbers of the excursion of a smooth Euclidean Gaussian field restricted to a rectangular window, in the asymptotics where the window grows to R^d . With motivations coming from Topological Data Analysis, we derive a functional Central Limit Theorem where the varying argument is the thresholding parameter, under assumptions of regularity and covariance decay for the field and its derivatives. We also show fixed-level CLTs coming from martingale based techniques inspired from the theory of geometric stabilisation, and limiting non-degenerate variance.

math.PR↗

arXiv2024

Limit theorems under heavy-tailed scenario in the age dependent random connection models

This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable assumptions on the shape of trees. The proof relies on an intermediate result on weak convergence of associated point processes towards a Poisson point process. Additionally, we prove the same type of results for the clique counts. Here, a crucial ingredient includes the expectation asymptotics for clique counts, which itself is a result of independent interest.

math.PR↗

arXiv2024

Random Connection Hypergraphs

In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes, and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of various graph-theoretic and topological characteristics, including higher-order degree distributions, Betti numbers of the associated Dowker complex, and simplex counts. In particular, for the latter quantity, we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation with a simulation study and an application to the collaboration network extracted from the arXiv dataset.

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arXiv2024

Planar reinforced $k$-out percolation

We investigate the percolation properties of a planar reinforced network model. In this model, at every time step, every vertex chooses $k \ge 1$ incident edges, whose weight is then increased by 1. The choice of this $k$-tuple occurs proportionally to the product of the corresponding edge weights raised to some power $\alpha > 0$. Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for $k = 2$ and $\alpha \gg 1$. First, we study the case $\alpha = \infty$, where we show the percolation for $k = 2$ after adding arbitrarily sparse independent sprinkling and also allowing dual connectivities. We also derive a finite-size criterion for percolation without sprinkling. Then, we extend this finite-size criterion to the $\alpha < \infty$ case. Finally, we verify these conditions numerically.

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