254 research outputs found
Random networking: between order and chaos
Full text linkWith the arrival of the Internet, a good understanding of networks has become important for
everyone. Network theory, which originated in the eighteenth century with Euler, and in the
nineteenth century withMarkov, has until recently concentrated its attentionmainly on regular types of graphs. In his inaugural lecture, Remco van der Hofstad shows us a shift towards highly irregular graphs having vertices with extremely high degrees. He argues that this irregularity is a main characteristic of real life networks such as the Internet, social networks and networks describing biophysical phenomena. On January 1, 2005, Remco van der Hofstad was appointed full professor in Probability at the University of Eindhoven
The constants in the CLT for the Edwards model
Full text linkThe Edwards model in one dimension is a transformed path measure for one-dimensional Brownian motion discouraging selfintersections In van der Hofstad den Hollander and Konig preprint a central limit theorem CLT is proved for the uctuations of the endpoint of the path around its linear asymptotics In the present paper we study the constants appearing in this CLT which represent the mean and the variance and the exponential rate of the normalizing constant We prove that the variance is strictly smaller than which shows that the weak interaction limit is singular Furthermore we give a relation between the normalizing constant in the Edwards model and the normalizing constant in the weakly interacting DombJoyce model The DombJoyce model is the discrete analogue of the Edwards model based on simple random walk and is studied in van der Hofstad den Hollander and Konig preprint
The proofs are based on bounds for the eigenvalues of a certain one-parameter family of SturmLiouville dierential operators These bounds are obtained by using the monotonicity of the zeroes of the eigenfunctions in combination with computer plots of the power series approximation of the eigenfunctions and exact error estimates of the power series approximatio
Unlacing hypercube percolation : a survey
Full text linkThe purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1}^m obtained in the series of papers (Borgs et al. in Random Struct Algorithms 27:137–184, 2005; Borgs et al. in Ann Probab 33:1886–1944, 2005; Borgs et al. in Combinatorica 26:395–410, 2006; van der Hofstad and Nachmias in Hypercube percolation, Preprint 2012). Secondly, we explain how this study can be performed without the use of the so-called lace expansion technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability
Quenched Central Limit Theorems for the Ising Model on Random Graphs
Full text linkThemain goal of the paper is to prove central limit theorems for the magnetization
rescaled by the square root of N for the Ising model on random graphs with N vertices.Both random quenched
and averaged quenched measures are considered.We work in the uniqueness regime β > βc
or β > 0 and B not equal to 0, where β is the inverse temperature, βc is the critical inverse temperature
and B is the external magnetic field. In the random quenched setting our results apply to
general tree-like random graphs (as introduced by Dembo, Montanari and further studied by
Dommers and the first and third author) and our proof follows that of Ellis in Z^d. For the
averaged quenched setting, we specialize to two particular random graph models, namely
the 2-regular configuration model and the configuration model with degrees 1 and 2. In
these cases our proofs are based on explicit computations relying on the solution of the one
dimensional Ising model
The lace expansion approach to ballistic behaviour for one-dimensional weakly self-avoiding walks
No full textWe prove ballistic behaviour in dimension one fora model of weakly self-avoiding walks where loops of length m are penalized by a factor e -ß/mp with p¿ [0, 1] and ß sufficiently large. Furthermore, we prove that the fluctuations around the linear drift satisfy a centrallimit theorem. The proof uses a variant of the lace expansion, together with an inductive analysis of the arising recursion relation. In particular, we derive the law of large numbers, first obtainedby Greven and den Hollander, and the central limit theorem, firstobtained by König, for the weakly self-avoiding walk (p = 0 and ß > 0).Their proofs use large deviation theory for the Markov description of the local times of one-dimensional simple random walk. It is the first time that the lace expansion is used to provebehaviour that is not diffusive. It has previously been used by van der Hofstad, den Hollander and Sladeto prove diffusive behaviour in dimension d for p= 0 such that p > and ß > 0 sufficiently small.The lace expansion presented here compares the above weaklyself-avoiding walk to strictly self-avoiding walk in dimension one, obtained when ß = 8, and shows that the difference in behaviour is small when ß is large
Central limit theorem for the Edwards model
Full text linkThe Edwards model in one dimension is a transformed path measure for standard Brownian motion discouraging self-intersections. We prove a central limit theorem for the endpoint of the path, extending a law of large numbers proved by Westwater. The scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of differential operators, introduced and analyzed by van der Hofstad and den Hollander. Interestingly, the scaled variance turns out to be independent of the strength of self-repellence and to be strictly smaller than one (the value for free Brownian motion)
Expansion in for percolation critical values on the n-cube and : the first three terms
No full textLet and denote the critical values for nearest-neighbour bond percolation on the -cube and on , respectively. Let for and for denote the degree of . We use the lace expansion to prove that for both and , This extends by two terms the result of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for
Ising models on power-law random graphs
Full text linkWe study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent τ>2), for which the random graph has a tree-like structure. For this, we closely follow the analysis by Dembo and Montanari (Ann. Appl. Probab. 20(2):565–592, 2010) which assumes finite variance degrees (τ>3), adapting it when necessary and also simplifying it when possible. Our results also apply in cases where the degree distribution does not obey a power law.We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy
Annealed central limit theorems for the Ising model on random graphs
Full text linkThe aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled
by of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature 0le eta^{mathrm{an}}_c < infty and then prove our results in the uniqueness regime, i.e., the values of inverse temperature and external magnetic field for which either eta<eta^{mathrm{an}}_c and , or eta>0 and . In the case of the configuration model, the central limit theorem holds in the whole region of the parameters and , because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model
Random graph asymptotics on high-dimensional tori II:volume, diameter and mixing time
Full text linkFor critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335–358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551–582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267–1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Rényi random graphs.
Keywords: Percolation – Random graph asymptotics – Mean-field behavior – Critical windo
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