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Alexander Fribergh

Publications and source records attributed to Alexander Fribergh.

At least 19 recordsLinked to original sources
arXiv2025

Bi-infinite incipient cluster in high dimensions

We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x'$, and subsequently take the limit as $|x|$, $|x'|$ as well as $|x-x'|$ diverge to infinity. This limiting procedure gives rise to a new percolation measure that locally resembles critical percolation but is concentrated on configurations with two disjoint infinite occupied paths. We coin this the bi-infinite incipient percolation cluster. It is mutually singular with respect to incipient infinite clusters that have been constructed in the literature. We achieve the construction through a double lace expansion of the cluster.

math.PR↗
arXiv2025

Scaling limit for the random walk on critical lattice trees

We prove a scaling limit theorem for the simple random walk on critical lattice trees in $\mathbb{Z}^d$, for $d\geq 8$. The scaling limit is the Brownian motion on the Integrated Super-Brownian Excursion (BISE) which is the same one that we have identified earlier for other simpler models of anomalous diffusion on critical graphs in large enough dimension. The proof of this theorem is based on a combination of the tools of lace-expansion (contained in the articles \cite{CFHP} and \cite{CFHP2}), and a new and general convergence theorem.

math.PR↗
arXiv2025

Random skeletons in high-dimensional lattice trees

We study the behaviour of the rescaled minimal subtree containing the origin and K random vertices selected from a random critical (sufficiently spread-out, and in dimensions d > 8) lattice tree conditioned to survive until time ns, in the limit as n goes to infinity. We prove joint weak convergence of various quantities associated with these subtrees under this sequence of conditional measures to their counterparts for historical Brownian motion. We also show that when K is sufficiently large the entire rescaled tree is close to this rescaled skeleton with high probability, uniformly in n. These two results are the key conditions used in [5] to prove that the simple random walk on sufficiently spread-out lattice trees (conditioned to survive for a long time) converges to Brownian motion on a super-Brownian motion (conditioned to survive). The main convergence result is established more generally for a sequence of historical processes converging to historical Brownian motion in the sense of finite dimensional distributions and satisfying a pair of technical conditions. The conditions are readily verified for the lattice trees mentioned above and also for critical branching random walk. We expect that it will also apply with suitable changes to other lattice models in sufficiently high dimensions such as oriented percolation and the voter model. In addition some forms of the second skeleton density result are already established in this generality.

math.PR↗
arXiv2024

Sharp asymptotics for finite point-to-plane connections in supercritical bond percolation in dimension at least three

We consider supercritical bond percolation in $\mathbb{Z}^d$ for $d \geq 3$. The origin lies in a finite open cluster with positive probability, and, when it does, the diameter of this cluster has an exponentially decaying tail. For each unit vector $\bf\ell$, we prove sharp asymptotics for the probability that this cluster contains a vertex $x \in \mathbb{Z}^d$ that satisfies $x \cdot \bf\ell \geq u$. For an axially aligned $\bf\ell$, we find this probability to be of the form $\kappa \exp \{ - \zeta u \}(1+ {\rm err})$ for $u \in \mathbb{N}$, where $\vert {\rm err} \vert$ is at most $C \exp \{ - c u^{1/2} \big\}$; for general $\bf\ell$, the form of the asymptotic depends on whether $\bf\ell$ satisfies a natural lattice condition. To obtain these results, we prove that renewal points in long clusters are abundant, with a renewal block length whose tail is shown to decay as fast as $C \exp \big\{ - c u^{1/2} \big\}$.

math.PR↗
arXiv2020

Random Memory Walk

We present a simple model of a random walk with partial memory, which we call the \emph{random memory walk}. We introduce this model motivated by the belief that it mimics the behavior of the once-reinforced random walk in high dimensions and with small reinforcement. We establish the transience of the random memory walk in dimensions three and higher, and show that its scaling limit is a Brownian motion.

math.PR↗
arXiv2017

Backbone scaling limits for random walks on random critical trees

We prove the existence of scaling limits for the projection on the backbone of the random walks on the Incipient Infinite Cluster and the Invasion Percolation Cluster on a regular tree. We treat these projected random walks as randomly trapped random walks (as defined in [BC\v{C}R15]) and thus describe these scaling limits as spatially subordinated Brownian motions

math.PR↗
arXiv2016

Biased random walk on the interlacement set

We study a biased random walk on the interlacement set of $\mathbb{Z}^d$ for $d\geq 3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.

math.PR↗
arXiv2016

Scaling limit for the ant in high-dimensional labyrinths

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e the behavior of the "ant in the labyrinth". It is natural to conjecture (see [16] and [8]) that the scaling limit for random walks on large critical random graphs exists in high dimensions, and is universal. This scaling limit is simply the natural Brownian Motion on the Integrated Super-Brownian Excursion. We give here a set of four natural sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, spread-out percolation in high enough dimension. In the companion paper [10], we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above, in dimensions larger than 14.

math.PR↗
arXiv2016

Scaling limit for the ant in a simple labyrinth

We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.

math.PR↗
arXiv2016

Scaling limits for sub-ballistic biased random walks in random conductances

We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional Law of Large Numbers for the position of the walker, properly rescaled. Moreover, we state a functional Central Limit Theorem where an atypical process, related to the Fractional Kinetics, appears in the limit.

math.PR↗
arXiv2014

Biased random walks on random graphs

These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our main focus will be on directionally transient and reversible random walks on different types of underlying graph structures, such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.

math.PR↗
arXiv2014

Local trapping for elliptic random walks in random environments in $\mathbb{Z}^d$

We consider elliptic random walks in i.i.d. random environments on $\mathbb{Z}^d$. The main goal of this paper is to study under which ellipticity conditions local trapping occurs. Our main result is to exhibit an ellipticity criterion for ballistic behavior which extends previously known results. We also show that if the annealed expected exit time of a unit hypercube is infinite then the walk has zero asymptotic velocity.

math.PR↗
arXiv2013

Biased random walk in positive random conductances on $\mathbb{Z}^{d}$

We study the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477-510], who proved positivity of the speed in the uniformly elliptic setting.

math.PR↗
arXiv2013

Phase transition for the speed of the biased random walk on the supercritical percolation cluster

We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one-dimensional; we prove such a `dynamic renormalization' statement in a much stronger form than was previously known.

math.PR↗
arXiv2012

Biased random walk on critical Galton-Watson trees conditioned to survive

We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.

math.PR↗
arXiv2010

Biased random walks on a Galton-Watson tree with leaves

We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $γ= γ(β) \in (0,1)$, depending on the bias $β$, such that $X_n$ is of order $n^γ$. Denoting $Δ_n$ the hitting time of level $n$, we prove that $Δ_n/n^{1/γ}$ is tight. Moreover we show that $Δ_n/n^{1/γ}$ does not converge in law (at least for large values of $β$). We prove that along the sequences $n_λ(k)=\lfloor λβ^{γk}\rfloor$, $Δ_n/n^{1/γ}$ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

math.PR↗