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Lukas Lüchtrath

Publications and source records attributed to Lukas Lüchtrath.

At least 19 recordsLinked to original sources
arXiv2026

Recurrence of strong-decay inhomogeneous long-range percolation clusters

We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional spatial scale-free graphs whenever the standard strong-decay long-edge estimate holds. In dimension two, we combine the linear chemical-distance estimate of Lüchtrath with an area-order bound on the degree measure. Graph-distance layers in exponentially separated bands then give the required Nash-Williams cutsets for planar random geometric graphs satisfying the polynomial mixing and long-edge estimates [J. Theoret. Probab. 39 (2026), Paper No. 12]. As a concrete consequence, every connected component of the two-dimensional weight-dependent random connection model with interpolation kernel is recurrent throughout the strong-decay region $δ>2$, $γ<1-\frac{1}δ$, and $α<1-γ$.

math.PR↗
arXiv2026

Phase transitions for contact processes on sparse random graphs via metastability and local limits

We propose a new perspective on the asymptotic regimes of fast and slow extinction in the contact process on locally converging sequences of sparse finite graphs. We characterise the phase boundary by the existence of a metastable density, which makes the study of the phase transition particularly amenable to local-convergence techniques. We use this approach to derive general conditions for the coincidence of the critical threshold with the survival/extinction threshold in the local limit. We further argue that the correct time scale to separate fast extinction from slow extinction in sparse graphs is, in general, the exponential scale, by showing that fast extinction may occur on stretched exponential time scales in sparse scale-free spatial networks. Together with {the results of} Nam, Nguyen and Sly (Trans.\ Am.\ Math.\ Soc.\ 375, 2022), our methods can be applied to deduce that the fast/slow threshold in sparse configuration models coincides with the survival/extinction threshold on the limiting Galton-Watson tree.

math.PR↗
arXiv2026

Cluster sizes in subcritical soft Boolean models

We consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Pareto-distributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine path-counting arguments identifying the precise order of decay of the probability that far-away vertices are connected.

math.PR↗
arXiv2026

Convex order and faster transmission in first contact percolation

Inspired by strict-monotonicity criteria for the time constant in first passage percolation, we investigate convex ordering of point processes in relation to the time constant in first contact percolation. In a nutshell, first contact percolation models the spread of an infection as a contact process without recovery based on a generalized graphical representation, where the usual homogeneous Poisson point processes on the edges are replaced by general simple point processes. Based on a notion of convex ordering for point processes, we prove monotonicity in the number and existence of infection paths. We argue that this convex ordering is however not enough to ensure strict monotonicities in the asymptotic speed of the infection. Instead, we propose a criterion based on an ordering of void probabilities and prove a speed-up for one-dimensional systems based on $\mathbb{Z}$-stationary point processes.

math.PR↗
arXiv2026

Maximising homomorphism counts between digraphs

We prove a Sidorenko-type inequality for directed trees: for every oriented tree $T$ on $k$ vertices and every finite directed graph $G$, the homomorphism count hom$(T,G)$ is bounded above by the maximum of the two pure star counts hom$(S_{0,k-1},G)$ and hom$(S_{k-1,0},G)$. In other words, among all directed trees on $k$ vertices, the pure in- and out-stars maximise the homomorphism count into host digraphs. The proof is purely combinatorial, based on an iterative leaf-reallocation scheme combined with Hölder's inequality. We further investigate the corresponding homomorphism order on directed trees, discuss refinements via tail-truncation and pointwise bounds for rooted host graphs, and record several consequences, e.g. for random directed graph models and local weak limits, where the inequality reduces tree statistics to controlled pure in- and out-degree moments.

math.CO↗
arXiv2026

Age-dependent random connection models with arc reciprocity: clustering and connectivity

We introduce a model for directed spatial networks. Starting from an age-based preferential attachment model in which all arcs point from younger to older vertices, we add \emph{reciprocal} connections whose probabilities depend on the age difference between their end-vertices. This yields a directed graph with reciprocal correlations, a power-law indegree distribution, and a tunable outdegree distribution. We consider two versions of the model: an infinite version embedded in $\mathbb{R}^d$, which can be constructed as a weight-dependent random connection model with a non-symmetric kernel, and a growing sequence of graphs on the unit torus that converges locally to the infinite model. Besides establishing the local limit result linking the two models, we investigate degree distributions, various directed clustering metrics, and directed percolation.

math.PR↗
arXiv2026

Subcritical annulus crossing in spatial random graphs

We consider general continuum percolation models obeying sparseness, translation invariance, and spatial decorrelation. In particular, this includes models constructed on general point sets other than the standard Poisson point process or the Bernoulli-percolated lattice. Moreover, in our setting the existence of an edge may depend not only on the two end vertices but also on a surrounding vertex set and models are included that are not monotone in some of their parameters. We study the critical annulus-crossing intensity $\widehatλ_{c}$, which is smaller or equal to the classical critical percolation intensity $λ_{c}$ and derive a condition for $\widehatλ_{c}>0$ by relating the crossing of annuli to the occurrence of long edges. This condition is sharp for models that have a modicum of independence. In a nutshell, our result states that annuli are either not crossed for small intensities or crossed by a single edge. Our proof rests on a multiscale argument that further allows us to directly describe the decay of the annulus-crossing probability with the decay of long edges probabilities. We apply our result to a number of examples from the literature. Most importantly, we extensively discuss the weight-dependent random connection model in a generalised version, for which we derive sufficient conditions for the presence or absence of long edges that are typically easy to check. These conditions are built on a decay coefficient $ζ$ that has recently seen some attention due to its importance for various proofs of global graph properties.

math.PR↗
arXiv2026

Detection, coverage and percolation in dynamic Boolean models with random radii based on $α$-stable processes

We consider a dynamic network in continuum time and space in which nodes, with initial locations given by a Poisson point process, move according to i.i.d. isotropic $α$-stable processes. Each node is additionally equipped with an i.i.d. detection radius. Inspired by corresponding results by Peres et. al. on mobile networks based on Brownian sausages with fixed width, we investigate the tail behaviour of three stopping times: The detection time of the first discovery of a designated node, the first coverage of an entire set, and the first discovery of a node by the infinite connected component of the system. Broadly speaking, we discover that the stability index as well as the random radii manifest themselves only in constants in the otherwise exponential decay rates. The proofs rest on heat-kernel bounds for the underlying Lévy processes and a detailed multiscale analysis allowing us to control the space-time correlations of the system.

math.PR↗
arXiv2025

All spatial random graphs with weak long-range effects have chemical distance comparable to Euclidean distance

This note provides a sufficient condition for linear lower bounds on chemical distances (compared to the Euclidean distance) in general spatial random graphs. The condition is based on the scarceness of long edges in the graph and weak correlations at large distances and is valid for all translation invariant and locally finite graphs that fulfil these conditions. The proof is based on a renormalisation scheme introduced by Berger [arXiv: 0409021 (2004)].

math.PR↗
arXiv2025

Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension

We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study of models with long-range effects and heavy-tailed degree distributions. We introduce a new coefficient $δ_\text{eff}$ which quantifies the influence of heavy-tailed degrees on long-range connections. We show that $δ_\text{eff}<2$ is sufficient for the existence of a supercritical percolation phase in the model and that $δ_\text{eff}>2$ always implies the absence of percolation. In particular, our results complement those in Gracar et al. (Adv. Appl. Prob., 2021), where sufficient conditions were given for the soft Boolean model and the age-dependent random connection model for both the existence and the absence of a subcritical percolation phase. Our results further provide a criterion for the existence or non-existence of a giant component in large finite graphs.

math.PR↗
arXiv2025

First contact percolation

We study a version of first passage percolation on $\mathbb{Z}^d$ where the random passage times on the edges are replaced by contact times represented by random closed sets on $\mathbb{R}$. Similarly to the contact process without recovery, an infection can spread into the system along increasing sequences of contact times. In case of stationary contact times, we can identify associated first passage percolation models, which in turn establish shape theorems also for first contact percolation. In case of periodic contact times that reflect some reoccurring daily pattern, we also present shape theorems with limiting shapes that are universal with respect to the within-one-day contact distribution. In this case, we also prove a Poisson approximation for increasing numbers of within-one-day contacts. Finally, we present a comparison of the limiting speeds of three models -- all calibrated to have one expected contact per day -- that suggests that less randomness is beneficial for the speed of the infection. The proofs rest on coupling and subergodicity arguments.

math.PR↗
arXiv2025

Phase transitions for contact processes on one-dimensional networks

We study the survival/extinction phase transition for contact processes with quenched disorder. The disorder is given by a locally finite random graph with vertices indexed by the integers that is assumed to be invariant under index shifts and augments the nearest-neighbour lattice by additional long-range edges. We provide sufficient conditions that imply the existence of a subcritical phase and therefore the non-triviality of the phase transition. Our results apply to instances of scale-free random geometric graphs with any integrable degree distribution. The present work complements previously developed techniques to establish the existence of a subcritical phase on Poisson--Gilbert graphs and Poisson--Delaunay triangulations (Ménard et al., Ann. Sci. Éc. Norm. Supér., 2016), on Galton--Watson trees (Bhamidi et al., Ann. Probab., 2021) and on locally tree-like random graphs (Nam et al., Trans. Am. Math. Soc., 2022), all of which require exponential decay of the degree distribution. Two applications of our approach are particularly noteworthy: Firstly, for Gilbert graphs derived from stationary point processes on $\mathbb{R}$ marked with i.i.d. random radii, our results are sharp. We show that there is a non-trivial phase transition if and only if the graph is locally finite. Secondly, for independent Bernoulli long-range percolation on $\mathbb{Z}$, with coupling constants $J_{x,y}\asymp |x-y|^{-δ}$, we verify a conjecture of Can (Electron. Commun. Probab., 2015) stating the non-triviality of the phase transition whenever $δ>2$. We believe that the results are indicative of the behaviour of contact processes on spatial random graphs also in dimensions $d > 1$ as long as the degree distribution of the underlying network has at least finite $d$-th moment. We support this by proving that no phase transition exists if the $d$-th moment is infinite.

math.PR↗
arXiv2025

The Erdős-Rényi Random Graph Conditioned on Every Component Being a Clique

Motivated by an application in community detection, we consider an \ER random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. We show that a popular community detection method is equivalent to Bayesian inference with this distribution as prior over the community partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables in this conditional distribution: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability $p$ as the number of vertices $n$ diverges. For $p=\tfrac{1}{2}$, the conditioning yields the uniform distribution over set partitions, which is well-studied, but has not been studied as a graph distribution before. For $p<\tfrac{1}{2}$, we show that the number of cliques is of the order $n/\sqrt{\log n}$, while for $p>\tfrac{1}{2}$, we prove that the graph consists of a single clique with high probability. This shows that there is a phase transition at $p=\tfrac{1}{2}$. We additionally study the near-critical regime $p_n\downarrow\tfrac{1}{2}$, as well as the sparse regime $p_n\downarrow0$. Finally, we discuss the implications of these results for community detection.

math.PR↗
arXiv2024

Existence of subcritical percolation phases for generalised weight-dependent random connection models

We derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space according to an iid-marked stationary Poisson point process. In contrast to many established models, the probability of existence of an edge may not only depend on the distance and the weights of its end vertices but also on a surrounding vertex set. Our results can be applied in particular to models combining heavy-tailed degree distributions and long-range effects, which are typically well connected. More precisely, we study the critical annulus-crossing intensity $\widehatλ_c$ which is smaller or equal to the classical critical percolation intensity $λ_c$ and derive sharp conditions for $\widehatλ_c>0$ by controlling the occurrence of long edges. We further present tail bounds for the Euclidean diameter and number of points of the typical cluster in the subcritical phase and apply our results to several examples including some in which $\widehatλ_c<λ_c$.

math.PR↗
arXiv2021

Preferential attachment with location-based choice: Degree distribution in the noncondensation phase

We consider the preferential attachment model with location-based choice introduced by Haslegrave, Jordan and Yarrow as a model in which condensation phenomena can occur [Haslegrave et al. 2020]. In this model every vertex carries an independent and uniformly drawn location. Starting from an initial tree the model evolves in discrete time. At every time step, a new vertex is added to the tree by selecting $r$ candidate vertices from the graph with replacement according to a sampling probability proportional to these vertices' degrees. The new vertex then connects to one of the candidates according to a given probability associated to the ranking of their locations. In this paper, we introduce a function that describes the phase transition when condensation can occur. Considering the noncondensation phase, we use stochastic approximation methods to investigate bounds for the (asymptotic) proportion of vertices inside a given interval of a given maximum degree. We use these bounds to observe a power law for the asymptotic degree distribution described by the aforementioned function. Hence, this function fully characterises the properties we are interested in. The power law exponent takes the critical value one at the phase transition between the condensation - noncondensation phase.

math.PR↗
arXiv2021

Percolation phase transition in weight-dependent random connection models

We investigate spatial random graphs defined on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a nontrivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

math.PR↗