arXiv subjects

Johannes Bäumler

Publications and source records attributed to Johannes Bäumler.

17 recordsLinked to original sources

arXiv2026

On the largest common subtree of uniform attachment trees

We study the largest common subtree of two independent unlabeled uniform attachment trees (also known as random recursive trees). Our main result shows that, when the two trees have $n$ vertices each, their largest common subtree has at least $n^{0.83}$ vertices with high probability. This is obtained by starting with the common subtree induced by the Ulam--Harris labels in the two trees and improving using local optimization steps. We also give some upper bounds and bounds for general random tree growth models. We leave as an intriguing open question to understand the magnitude of the size of the largest common subtree.

math.PR↗

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

History estimation in random recursive trees: Pointwise approach via iterated Jordan centralities

We study the problem of estimating the arrival times of vertices in a uniform random recursive tree from its unlabeled structure. We adopt a pointwise perspective and analyze the distribution of the relative estimation error, and derive tail bounds that are uniform in both the vertex and the tree size. For the ranking induced by Jordan centrality, the probability that the estimate exceeds the true arrival time by a factor $S$ decays on the order of $1/S$, while the probability of underestimating the arrival time by a factor $1/S$ decays exponentially in $S$. We introduce a refined centrality measure whose overestimation tail decays on the order of $(\log S)/S^{2}$, at the cost of a heavier lower tail of order $1/S^{2}$. These results reveal a tradeoff between upper- and lower-tail performance in arrival-time estimation that is invisible to the previously studied risk functional. Nevertheless, the refined centrality measure attains the optimal order of the risk for all its parameter values.

math.PR↗

arXiv2026

Correlated uniform attachment trees

We introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. In this model, two UA trees are grown in parallel, and at each time step a new node is added to each tree, with an edge between it and a uniformly chosen existing vertex in the respective tree. The two choices of attachment are correlated: with probability $α$, the edges attach to nodes with the same time label in both trees, and with probability $1-α$, the choices are made independently. We study fundamental detection and estimation questions for this model, given two \emph{unlabeled} trees. In our main result, we construct a consistent estimator of the correlation parameter $α$, as the size of the trees goes to infinity. The construction of our statistic relies on two key ideas. First, we use Jordan centrality to identify subsets of vertices of each tree whose intersection has a sufficient number of common early vertices. The second idea is that, across multiple time scales, it is possible to approximately determine the labels of vertices that have attached to these early vertices, using the sizes of fringe subtrees. Our analysis includes novel quantitative bounds on the fraction of early vertices that remain central, which are of independent interest in the network archaeology literature.

math.PR↗

arXiv2026

Continuity of the critical value and a shape theorem for long-range percolation

Consider supercritical long-range percolation on $\Z^d$ where two vertices $x,y \in \Z^d$ are connected with probability asymptotic to $\|x-y\|^{-s}$ for some $s>2d$. Conditioned that the origin is in the infinite cluster, we prove a shape theorem for the set of points that can be reached within $n$ steps from the origin. As part of the proof, we show that for long-range percolation with polynomially decaying connection probabilities in dimensions $d\geq 2$, the critical value depends continuously on the precise specifications of the model.

math.PR↗

arXiv2026

Invariance principles for rough walks in random conductances

We establish annealed and quenched invariance principles for random walks in random conductances lifted to the p-variation rough path topology, allowing for degenerate environments and long-range jumps. Our proof is based on a unified structural strategy where pathwise convergence is viewed as a natural upgrade of the classical theory. This approach decouples the martingale lift from terms involving the integrals with respect to the corrector and the quadratic covariations. In the quenched regime, we show that the existence of a stationary potential for the corrector with $2+ε$ moments is sufficient to ensure the vanishing of the corrector in $p$-variation for any $p>2$. This input, combined with our structural framework, provides a direct and modular pathway to rough path convergence. We further provide a transfer lemma to construct this potential from spatial moment bounds. While presently verified in the literature primarily for nearest-neighbor settings, our formulation isolates the exact analytic input required for pathwise convergence in more general environments.

math.PR↗

arXiv2026

Local criteria for global connectivity comparisons: beyond stochastic domination

We introduce a site-wise domination criterion for local percolation models, which enables the comparison of one-arm probabilities even in the absence of stochastic domination. The method relies on a local-to-global principle: if, at each site, one model is more likely than the other to connect to a subset of its neighbors, for all nontrivial such subsets, then this advantage propagates to connectivity events at all scales. In this way, we obtain a robust alternative to stochastic domination, applicable in all cases where the latter works and in many where it does not. As a main application, we compare classical Bernoulli bond percolation with degree-constrained models, showing that degree constraints enhance percolation, and obtain asymptotically optimal bounds on critical parameters for degree-constrained models.

math.PR↗

arXiv2025

A discontinuous percolation phase transition on the hierarchical lattice

For long-range percolation on $\mathbb{Z}$ with translation-invariant edge kernel $J$, it is a classical theorem of Aizenman and Newman (1986) that the phase transition is discontinuous when $J(x,y)$ is of order $|x-y|^{-2}$ and that there is no phase transition at all when $J(x,y)=o(|x-y|^{-2})$. We prove a strengthened version of this theorem for the hierarchical lattice, where the relevant threshold is at $|x-y|^{-2d} \log\log |x-y|$ rather than $|x-y|^{-2}$: There is a continuous phase transition for kernels of larger order, a discontinuous phase transition for kernels of exactly this order, and no phase transition at all for kernels of smaller order. As such, $|x-y|^{-2d} \log\log |x-y|$ is essentially the \emph{only} kernel that produces a discontinuous phase transition. We also prove a hierarchical analogue of the ``$M^2β=1$'' conjecture of Imbrie and Newman (1988), which gives an exact formula for the density of the infinite cluster at the point of discontinuous phase transition and remains open in the Euclidean setting.

math.PR↗

arXiv2025

Superiority of Instantaneous Decisions in Thin Dynamic Matching Markets

We study a dynamic matching setting where homogeneous agents arrive at random according to a Poisson process and randomly form edges yielding a sparse market. Agents stay in the market according to a certain sojourn time and wait to be matched with a compatible agent by a matching algorithm. When their maximum sojourn time is reached, they perish unmatched. The primary objective is to maximize the number of matched agents. Our main result is to show that a uniformly guaranteed sojourn time suffices to get almost optimal performance of instantaneous matching. Interestingly, this matching policy essentially keeps as few agents in the market as possible. Hence, in contrast to the common paradigm that market thickness is the crucial property for obtaining strong matching performance, we show that the agents' sojourn behavior can be an equally powerful factor. In addition, instantaneous matching is close to optimal with respect to minimizing waiting time. We develop new techniques for proving our results going beyond commonly adopted methods for Markov processes.

cs.DS↗

arXiv2025

Behavior of the distance exponent for $\frac{1}{|x-y|^{2d}}$ long-range percolation

We study independent long-range percolation on $\mathbb{Z}^d$ where the vertices $u$ and $v$ are connected with probability asymptotic to $\fracβ{\|u-v\|^{2d}}$ for $\|u-v\|_\infty\geq 2$ and with probability 1 for $\|u-v\|_\infty=1$, where $β\geq 0$ is a parameter. It is proven in [5] that there exists an exponent $θ=θ(d,β) \in \left(0,1\right]$ such that the graph distance between the origin $\mathbf{0}$ and $x \in \mathbb{Z}^d$ scales like $\|x\|^θ$. We prove that this exponent $θ(d,β)$ is continuous and strictly decreasing as a function in $β$. Furthermore, we show that $θ(d,β)=1-β+o(β)$ for small $β$ in dimension $d=1$.

math.PR↗

arXiv2025

Permutations in competing growth processes and balls-in-bins

Consider a model of $N$ independent, increasing $\mathbb{N}_0$-valued processes, with random, independent waiting times between jumps. It is known that there is either an emergent `leader', in which a single process possesses the maximal value for all sufficiently large times, or every pair of processes alternates leadership infinitely often. We show that in the latter regime, almost surely, one sees every possible permutation of rankings of processes infinitely often. In the case that the waiting times are exponentially distributed, this proves a conjecture from Spencer (appearing in a paper from Oliveira) on the `balls-in-bins' process with feedback [Conjecture 1, Combin. Probab. Comput. 17(1):87-110, (2008)].

math.PR↗

arXiv2025

Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$

Consider independent long-range percolation on $\mathbb{Z}^d$ for $d\geq 3$. Assuming that the expected degree of the origin is infinite, we show that there exists an $N\in \mathbb{N}$ such that an infinite open cluster remains after deleting all edges of length at least $N$. For the isotropic case in dimensions $d\geq 3$, we show that if the expected degree of the origin is at least $10^{400}$, then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range $q$-states Potts model.

math.PR↗

arXiv2025

The polynomial growth of the infinite long-range percolation cluster

We study independent long-range percolation on $\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\|x-y\|>1$ are connected by an edge is proportional to $\fracβ{\|x-y\|^s}$, where $β>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the probability that the origin and $ne_1$ are connected by a path of length $k$ in the critical case $s=2d$. We use these results to determine the Hausdorff dimension of the critical long-range percolation metric that was recently constructed by Ding, Fan, and Huang [14].

math.PR↗

arXiv2023

Distances in $\frac{1}{|x-y|^{2d}}$ percolation models for all dimensions

We study independent long-range percolation on $\mathbb{Z}^d$ for all dimensions $d$, where the vertices $u$ and $v$ are connected with probability 1 for $\|u-v\|_\infty=1$ and with probability $p(β,\{u,v\})=1-e^{-β\int_{u+\left[0,1\right)^d} \int_{v+\left[0,1\right)^d} \frac{1}{\|x-y\|_2^{2d}}d x d y } \approx \fracβ{\|u-v\|_2^{2d}}$ for $\|u-v\|_\infty \geq 2$. Let $u \in \mathbb{Z}^d$ be a point with $\|u\|_\infty=n$. We show that both the graph distance $D(\mathbf{0},u)$ between the origin $\mathbf{0}$ and $u$ and the diameter of the box $\{0 ,\ldots, n\}^d$ grow like $n^{θ(β)}$, where $0<θ(β) < 1$. We also show that the graph distance and the diameter of boxes have the same asymptotic growth when two vertices $u,v$ with $\|u-v\|_2 > 1$ are connected with a probability that is close enough to $p(β,\{u,v\})$. Furthermore, we determine the asymptotic behavior of $θ(β)$ for large $β$, and we discuss the tail behavior of $\frac{D(\mathbf{0},u)}{\|u\|_2^{θ(β)}}$.

math.PR↗

arXiv2023

Isoperimetric lower bounds for critical exponents for long-range percolation

We study independent long-range percolation on $\mathbb{Z}^d$ where the vertices $x$ and $y$ are connected with probability $1-e^{-β\|x-y\|^{-d-α}}$ for $α> 0$. Provided the critical exponents $δ$ and $2-η$ defined by $δ= \lim_{n\to \infty} \frac{-\log(n)}{\log\left(\mathbb{P}_{β_c}\left(|K_0|\geq n\right)\right)}$ and $2-η= \lim_{x \to \infty} \frac{\log\left(\mathbb{P}_{β_c}\left(0\leftrightarrow x\right)\right)}{\log(\|x\|)} + d$ exist, where $K_0$ is the cluster containing the origin, we show that \begin{equation*} δ\geq \frac{d+(α\wedge 1)}{d-(α\wedge 1)} \ \text{ and } \ 2-η\geq α\wedge 1 \text. \end{equation*} The lower bound on $δ$ is believed to be sharp for $d = 1, α\in \left[\frac{1}{3},1\right)$ and for $d = 2, α\in \left[\frac{2}{3},1\right]$, whereas the lower bound on $2-η$ is sharp for $d=1, α\in (0,1)$, and for $α\in \left(0,1\right]$ for $d>1$, and is not believed to be sharp otherwise. Our main tool is a connection between the critical exponents and the isoperimetry of cubes inside $\mathbb{Z}^d$.

math.PR↗

arXiv2023

Recurrence and transience of symmetric random walks with long-range jumps

Let $X_1, X_2, \ldots$ be i.i.d. random variables with values in $\mathbb{Z}^d$ satisfying $\mathbb{P} \left(X_1=x\right) = \mathbb{P} \left(X_1=-x\right) = Θ\left(\|x\|^{-s}\right)$ for some $s>d$. We show that the random walk defined by $S_n = \sum_{k=1}^{n} X_k$ is recurrent for $d\in \{1,2\}$ and $s \geq 2d$, and transient otherwise. This also shows that for an electric network in dimension $d\in \{1,2\}$ the condition $c_{\{x,y\}} \leq C \|x-y\|^{-2d}$ implies recurrence, whereas $c_{\{x,y\}} \geq c \|x-y\|^{-s}$ for some $c>0$ and $s<2d$ implies transience. This fact was already previously known, but we give a new proof of it that uses only electric networks. We also use these results to show the recurrence of random walks on certain long-range percolation clusters. In particular, we show recurrence for several cases of the two-dimensional weight-dependent random connection model, which was previously studied by Gracar et al. [Electron. J. Probab. 27. 1-31 (2022)].

math.PR↗

arXiv2019

Uniqueness and Non-Uniqueness for Spin-Glass Ground States on Trees

We consider a Spin Glass at temperature $T = 0$ where the underlying graph is a locally finite tree. We prove for a wide range of coupling distributions that uniqueness of ground states is equivalent to the maximal flow from any vertex to $\infty$ (where each edge $e$ has capacity $|J_{e}|$) being equal to zero which is equivalent to recurrence of the simple random walk on the tree.

math.PR↗