van der Rw Remco Hofstad

Bio: van der Rw Remco Hofstad is an academic researcher. The author has contributed to research in topics: Central limit theorem & Random graph. The author has an hindex of 1, co-authored 1 publications receiving 22 citations.

Annealed central limit theorems for the ising model on random graphs

Abstract: The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by √N 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 0≤ βan n 0 and B ≠ 0. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters β and B, 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.

Quenched Central Limit Theorems for the Ising Model on Random Graphs

Abstract: The main goal of the paper is to prove central limit theorems for the magnetization rescaled by $$\sqrt{N}$$ 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 $$\beta >\beta _c$$ or $$\beta >0$$ and $$B e 0$$ , where $$\beta $$ is the inverse temperature, $$\beta _c$$ 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 $$\mathbb {Z}^d$$ . 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 models.

Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs

Abstract: We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant $J_{ij}(\beta)$ for the edge $ij$ on the complete graph is given by $J_{ij}(\beta)=\beta w_iw_j/(\sum_{k\in[N]}w_k)$. We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises (with inverse temperature $\beta$ replaced by $\sinh(\beta)$) from the annealed Ising model on the generalized random graph. We assume that the vertex weights $(w_i)_{i\in[N]}$ are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie-Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent $\tau$ with $\tau\in(3,5)$, then the critical exponents depend sensitively on $\tau$. In addition, at criticality, the total spin $S_{N}$ satisfies that $S_{N}/N^{(\tau-1)/(\tau-2)}$ converges in law to some limiting random variable whose distribution we explicitly characterize.

Fluctuations of the Magnetization for Ising Models on Dense Erdős-Rényi Random Graphs

Abstract: We analyze Ising/Curie–Weiss models on the (directed) Erdős–Renyi random graph on N vertices in which every edge is present with probability p. These models were introduced by Bovier and Gayrard (J Stat Phys 72(3–4):643–664, 1993). We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime $$\beta <1$$ when $$p=p(N)$$ satisfies $$p^3N^2\rightarrow +\,\infty $$ .

Ising Critical Behavior of Inhomogeneous Curie-Weiss Models and Annealed Random Graphs

Abstract: We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant \({J_{ij}(\beta)}\) for the edge \({ij}\) on the complete graph is given by \({J_{ij}(\beta)=\beta w_iw_j/( {\sum_{k\in[N]}w_k})}\). We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises [with inverse temperature \({\beta}\) replaced by \({\sinh(\beta)}\) ] from the annealed Ising model on the generalized random graph. We assume that the vertex weights \({(w_i)_{i\in[N]}}\) are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie-Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent \({\tau}\) with \({\tau\in(3,5)}\), then the critical exponents depend sensitively on \({\tau}\). In addition, at criticality, the total spin \({S_N}\) satisfies that \({S_N/N^{(\tau-2)/(\tau-1)}}\) converges in law to some limiting random variable whose distribution we explicitly characterize.

Fluctuations in Mean-Field Ising models

Abstract: In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately $d_N$ regular graph $G_N$ on $N$ vertices. In particular, if $G_N$ is \enquote{well connected}, we show that whenever $d_N\gg \sqrt{N}$, the fluctuations are universal and same as that of the Curie-Weiss model in the entire Ferro-magnetic parameter regime. We give a counterexample to demonstrate that the condition $d_N\gg \sqrt{N}$ is tight, in the sense that the limiting distribution changes if $d_N\sim \sqrt{N}$ except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to $d_N\gg N^{1/3}$. Our results conclude universal fluctuations of the average magnetization in Ising models on regular graphs, Erdős-Renyi graphs (directed and undirected), stochastic block models, and sparse regular graphons. In fact, our results apply to general matrices with non-negative entries, including Ising models on a Wigner matrix, and the block spin Ising model. As a by-product of our proof technique, we obtain Berry-Esseen bounds for these fluctuations, exponential concentration for the average of spins, and tight error bounds for the Mean-Field approximation of the partition function.