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Sander Dommers

Publications and source records attributed to Sander Dommers.

11 recordsLinked to original sources
arXiv2019

Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field

We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.

math.PR↗
arXiv2016

Continuous spin models on annealed generalized random graphs

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution. First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case. We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in \cite{DomGiaGibHofPri16}. On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

math.PR↗
arXiv2016

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

We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant $J_{ij}(β)$ for the edge $ij$ on the complete graph is given by $J_{ij}(β)=βw_iw_j/(\sum_{k\in[N]}w_k)$. We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises (with inverse temperature $β$ replaced by $\sinh(β)$) 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 $τ$ with $τ\in(3,5)$, then the critical exponents depend sensitively on $τ$. In addition, at criticality, the total spin $S_{N}$ satisfies that $S_{N}/N^{(τ-1)/(τ-2)}$ converges in law to some limiting random variable whose distribution we explicitly characterize.

math.PR↗
arXiv2016

Metastability in the reversible inclusion process

We study the condensation regime of the finite reversible inclusion process, i.e., the inclusion process on a finite graph $S$ with an underlying random walk that admits a reversible measure. We assume that the random walk kernel is irreducible and its reversible measure takes maximum value on a subset of vertices $S_\star\subset S$. We consider initial conditions corresponding to a single condensate that is localized on one of those vertices and study the metastable (or tunneling) dynamics. We find that, if the random walk restricted to $S_\star$ is irreducible, then there exists a single time-scale for the condensate motion. In this case we compute this typical time-scale and characterize the law of the (properly rescaled) limiting process. If the restriction of the random walk to $S_\star$ has several connected components, a metastability scenario with multiple time-scales emerges. We prove such a scenario, involving two additional time-scales, in a one-dimensional setting with two metastable states and nearest-neighbor jumps.

math.PR↗
arXiv2016

Metastability for Glauber dynamics on random graphs

In this paper we study metastable behaviour at low temperature of Glauber spin-flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the Configuration Model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover. In the limit as the temperature tends to zero, and subject to a certain \emph{key hypothesis} implying metastable behaviour, the average crossover time follows the classical \emph{Arrhenius law}, with an exponent and a prefactor that are controlled by the \emph{energy} and the \emph{entropy} of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the Configuration Model are expander graphs.

math.PR↗
arXiv2015

Metastability of the Ising model on random regular graphs at zero temperature

We study the metastability of the ferromagnetic Ising model on a random $r$-regular graph in the zero temperature limit. We prove that in the presence of a small positive external field the time that it takes to go from the all minus state to the all plus state behaves like ${\exp(β(r/2+ \mathcal{O}(\sqrt{r}))n)}$ when the inverse temperature $β\rightarrow\infty$ and the number of vertices $n$ is large enough but fixed. The proof is based on the so-called pathwise approach and bounds on the isoperimetric number of random regular graphs.

math.PR↗
arXiv2012

Ising critical exponents on random trees and graphs

We study the critical behavior of the ferromagnetic Ising model on random trees as well as so-called locally tree-like random graphs. We pay special attention to trees and graphs with a power-law offspring or degree distribution whose tail behavior is characterized by its power-law exponent $τ>2$. We show that the critical temperature of the Ising model equals the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree distribution. In particular, the inverse critical temperature equals zero when $τ\in(2,3]$ where this mean equals infinity. We further study the critical exponents $δ, β$ and $γ$, describing how the (root) magnetization behaves close to criticality. We rigorously identify these critical exponents and show that they take the values as predicted by Dorogovstev, et al. and Leone et al. These values depend on the power-law exponent $τ$, taking the mean-field values for $τ>5$, but different values for $τ\in(3,5)$.

math.PR↗
arXiv2012

Antiferromagnetic Potts model on the Erdos-Renyi random graph

We study the antiferromagnetic Potts model on the Poissonian Erdös-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.

math.PR↗
arXiv2010

Ising models on power-law random graphs

We 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 adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($τ>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy.

math.PR↗
arXiv2010

Diameters in preferential attachment models

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent τ>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law exponent τexceeds 3, then we prove that \log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for τ>3, distances are of the order \log{t}. For τ\in (2,3), we improve the upper bound to a constant times \log\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \log\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \log\log{t} when τ\in (2,3), and of order \log{t} when τ>3.

math.PR↗