Julia Komjathy

Microscopic concavity and fluctuation bounds in a class of deposition processes

We prove fluctuation bounds for the particle current in totally asymmetric zero range processes in one dimension with nondecreasing, concave jump rates whose slope decays exponentially. Fluctuations in the characteristic directions have order of magnitude t 1/3 . This is in agreement with the expectation that these systems lie in the same KPZ universality class as the asymmetric simple exclusion process. The result is via a robust argument formulated for a broad class of deposition-type processes. Besides this class of zero range processes, hypotheses of this argument have also been verified in the authors’ earlier papers for the asymmetric simple exclusion and the constant rate zero range processes, and are currently under development for a bricklayers process with exponentially increasing jump rates.

Generating hierarchial scale-free graphs from fractals

Abstract Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabasi, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal Λ . With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale-free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. We point out a connection between the power law exponent of the degree distribution and some intrinsic geometric measure theoretical properties of the underlying fractal. Using our (deterministic) fractal Λ we generate random graph sequence sharing similar properties.

Order of current variance and diffusivity in the rate one totally asymmetric zero range process

We prove that the variance of the current across a characteristic is of order t2/3 in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t1/3. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed in Balázs and Seppäläinen (Order of current variance and diffusivity in the asymmetric simple exclusion process, 2006) for simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t2/3-order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion.

First passage percolation on inhomogeneous random graphs

In this paper we investigate first passage percolation on an inhomogeneous random graph model introduced by Bollobás et al. (2007). Each vertex in the graph has a type from a type space, and edge probabilities are independent, but depend on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal-weight path, properly normalized, follows a central limit theorem. We handle the cases where the average number of neighbors λ̃ n of a vertex tends to a finite λ̃ in full generality and consider λ̃ = ∞ under mild assumptions. This paper is a generalization of the paper of Bhamidi et al. (2011), where first passage percolation is explored on the Erdős-Rényi graphs.

Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent τ ∈ (2, 3). In this model two colors spread with a fixed but not necessarily equal speed on the unweighted random graph. We show that if the speeds are not equal, then the faster color paints almost all vertices, while the slower color can paint only a random subpolynomial fraction of the vertices. We investigate the case when the speeds are equal and typical distances in a follow-up paper.

Mixing and relaxation time for random walk on wreath product graphs

Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H\wr G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z for z different of x,y remains unchanged. We estimate the mixing time of X* in terms of the parameters of H and G. Further, we show that the relaxation time of X* is the same order as the maximal expected hitting time of G plus |G| times the relaxation time of the chain on H.

Nonuniversality of weighted random graphs with infinite variance degree

We prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n , as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).

Degree distribution of shortest path trees and bias of network sampling algorithms

In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large degrees of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance), as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random r-regular graphs for large r, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean-field model of distance. We use our results to shed light on an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [ Ann. Appl. Probab. 20 (2010) 1907–1965], [ Combin. Probab. Comput. 20 (2011) 683–707], [ Adv. in Appl. Probab. 42 (2010) 706–738] of analyzing the effect of attaching random edge lengths on the geometry of random network models.

The front of the epidemic spread and first passage percolation

We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments as n → ∞, with an appropriate X 2 log + X condition. We also study the epidemic trail between the source and typical vertices in the graph.

Uniform mixing time for random walk on lamplighter graphs

Suppose that G is a finite, connected graph and X is a lazy random walk on G . The lamplighter chain X ? associated with X is the random walk on the wreath product G ? =Z 2 ?G , the graph whose vertices consist of pairs (f - ,x) where f is a labeling of the vertices of G by elements of Z 2 ={0,1} and x is a vertex in G . There is an edge between (f - ,x) and (g - ,y) in G ? if and only if x is adjacent to y in G and f z =g z for all z?x,y . In each step, X ? moves from a configuration (f - ,x) by updating x to y using the transition rule of X and then sampling both f x and f y according to the uniform distribution on Z 2 ; f z for z?x,y remains unchanged. We give matching upper and lower bounds on the uniform mixing time of X ? provided G satisfies mild hypotheses. In particular, when G is the hypercube Z d 2 , we show that the uniform mixing time of X ? is T(d2 d ) . More generally, we show that when G is a torus Z d n for d=3 , the uniform mixing time of X ? is T(dn d ) uniformly in n and d . A critical ingredient for our proof is a concentration estimate for the local time of the random walk in a subset of vertices.