# Alexander Fribergh's research while affiliated with Université de Montréal and other places

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## Publications (21)

We prove that the rescaled historical processes associated to critical spread-out lattice trees in dimensions d>8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>8$$\e...

We consider a biased random walk in positive random conductances on $\mathbb{Z}^d$ for $d\geq 5$. In the sub-ballistic regime, we prove the quenched convergence of the properly rescaled random walk towards a Fractional Kinetics.

We present a simple model of a random walk with partial memory, which we call the 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, an...

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 hig...

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 nat...

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 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 moti...

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...

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 polynomial.

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.

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 Fractiona...

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 grap...

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 tim...

The speed v(β) of a β-biased random walk on a Galton-Watson tree without leaves is increasing for β ≥ 1160. © 2013 Wiley Periodicals, Inc.

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 z...

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...

We consider a biased random walk Xn on a Galton–Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |Xn| is of order nγ. Denoting Δn the hitting time of level n, we prove that Δn/n1/γ is tight. Moreover, we show that Δn/n1/γ does not converge in l...

The speed $v(\beta)$ of a $\beta$-biased random walk on a Galton-Watson tree
without leaves is increasing for $\beta \geq 717$.

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. Thi...

We study the speed of a biased random walk on a percolation cluster on $\Z^d$ in function of the percolation parameter $p$. We obtain a first order expansion of the speed at $p=1$ which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice.

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n
κ
) from the origin, \({\kappa \in (0, 1)}\) . We investigate the probabilities of moderate deviations from this behaviou...

A mistake has been pointed out to us. The announced result does not hold. We withdraw for the moment and apologize.

## Citations

... This convergence follows neither from the notions of weak convergence above, nor from the weak convergence of the so-called historical processes (see e.g. [8,27,5]). ...

... First, the behavior of two-dimensional percolation at criticality and near criticality are very closely related via scaling or hyperscaling relations (first observed by Kesten [24]) which relate several key quantities of interest. Second, critical percolation on the triangular lattice exhibits conformal invariance, as shown by Smirnov [40], which has been used to show that SLE 6 is the scaling limit of interfaces in the model. Finally, many power laws can be exactly computed via the connection to SLE [32,33]. ...

... We mention that although we will rely on Corollary 3.3 for the next two sections, Theorem 3.1 is a stronger result that might prove to be useful in the future. This is the case in the related model of random interlacements, where a conditional decoupling inequality akin to Theorem 3.1 is proved in [4] and used as pivotal ingredient in [22] to study the biased random walk on the interlacement set. ...

... There is a substantial amount of past work on the heights and widths of random Bienaymé trees and random combinatorial trees [1-3, 14, 15], and bounds on these quantities, particularly the height, often feature in scaling limit theorems for random trees and associated objects [4,5,8,17,19]. The works [1][2][3] all bound the height via the study of the depth-first exploration process of the tree. ...

... And even in high dimensions, where the picture is perhaps most complete, there are still many big open problems. For instance, it is currently unknown what the scaling limit of random walk on large critical clusters is (although there are good conjectures [HS00a,Sla02], on which much progress has been made recently [BACF16a,BACF16b]). ...

... We would further like to mention the works [ESZ09b; ESZ09a;ESTZ13] that are conceptually important. Let us also acknowledge the works [BS19; BS20;Zin09] and [Fri13;FK18;FLS22], which deal with a biased version of the random walks on random conductances in dimension one and in higher dimensions, respectively. Now, as noted above, the main goal of our paper is to study the aging phenomenon for these random walks, when in the presence of walls, or of walls and traps, in dimension one. ...

... The most prominent examples are biased random walk on supercritical percolation clusters, introduced in [3] and biased random walk on supercritical Galton-Watson tree, introduced in [23]. We refer to [5] for a survey. Another specific model which has found a lot of recent interest in the physics literature is the random comb graph, see [2,13,21,26]. ...

... It can actually be shown that whenever the law of the jump probabilities at a single site is asymptotically independent at small values, it is also a sharp condition. Nevertheless, as explained in [4], condition (E ) 1 is not a sharp condition in general. There, the authors present a condition expressed in terms if the exit time from a hypercube of the random walk. ...

... Monotonicity of the speed for biased random walks on supercritical Galton-Watson trees without leaves is a famous open question, see [24]. We refer to [1] and [7] for recent results on Galton-Watson trees and [8] for a counterexample to monotonicity in the random conductance model. The Normal Transport regime for biased random walks on supercritical percolation clusters has been established in [9,14,25]. ...

... has a positive speed [LPP95]. Since then, much attention has been devoted to biased random walks, i.e. the SRW with a bias towards (or away from) the root of the tree, see for instance [Aïd14,BFGH12,Bow18,CHK18,CFK13,LPP96]. ...