Marek Biskup

Bio: Marek Biskup is an academic researcher from University of California, Los Angeles. The author has contributed to research in topics: Random walk & Ising model. The author has an hindex of 31, co-authored 123 publications receiving 3083 citations. Previous affiliations of Marek Biskup include Sewanee: The University of the South & University of Southern California.

Quenched invariance principle for simple random walk on percolation clusters

Abstract: We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤ d with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

Recent progress on the Random Conductance Model

Abstract: Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.

On the scaling of the chemical distance in long-range percolation models

Abstract: We consider the (unoriented) long-range percolation on Z d in dimensions d ≥ 1, where distinct sites x, y ∈ Z d get connected with probability p xy ∈ [0, 1]. Assuming p xy = |x - y| -s+o(1) as |x - y| → ∞, where s > 0 and |.| is a norm distance on Z d , and supposing that the resulting random graph contains an infinite connected component C∞, we let D(x, y) be the graph distance between x and y measured on C∞. Our main result is that, for s ∈ (d, 2d), D(x, y) = (log|x - y|) Δ+o(1) , x,y ∈ C∞, |x - y| → ∞, where Δ -1 is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x - y| → ∞. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of small-world phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.

Functional CLT for Random Walk Among Bounded Random Conductances

Abstract: We consider the nearest-neighbor simple random walk on $Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $\omega$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time.

Anomalous heat-kernel decay for random walk among bounded random conductances

Abstract: On considere la marche aleatoire aux plus proches voisins dans ℤd, d≥2, dont les transitions sont donnees par un champ de conductances aleatoires bornees ωxy∈[0, 1]. La loi de conductance est iid sur les aretes, et telle que la probabilite que ωxy>0 soit superieure au seuil de percolation (par aretes) sur ℤd. Pour les environnements dont l’origine est connectee a l’infini a l’aide d’aretes a conductances positives, on etudie l’asymptotique de la probabilite de retour a l’instant 2n : $\mathsf{P}_{\omega}^{2n}(0,0)$. On prouve que $\mathsf{P}_{\omega}^{2n}(0,0)$ est borne par Cn−d/2 pour d=2, 3 (ou C est une constante aleatoire) alors que c’est en o(n−2) pour d≥5 et O(n−2log n) pour d=4. En construisant des exemples dont les noyaux de la chaleur decroissent anormalement en avoisinant 1/n2, on peut prouver que la borne o(n−2) est optimale pour d≥5. On parvient egalement a construire des environnements naturels dependants de n qui presentent le facteur log n supplementaire en dimension d=4.

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Critical phenomena in complex networks

Abstract: The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

Exactly Solved Models in Statistical Mechanics

Abstract: R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.