14 research outputs found
Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field
We study the inhomogeneous Curie-Weiss model with external field, where the inhomo-geneity 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
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 . 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
where this mean equals infinity.
We further study the critical exponents and ,
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 , taking the mean-field values for , but
different values for
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⋆⊆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⋆ 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⋆ 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
Large deviations for the annealed Ising model on inhomogeneous random graphs: spins and degrees
We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs
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 ), 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 (). We further
identify the thermodynamic limits of various physical quantities, such as the
magnetization and the internal energy
Antiferromagnetic Potts model on the Erdos-Renyi random graph
We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi
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.Comment: 36 pages, revisions to improve resul
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+O(r√))n) when the inverse temperature β→∞ 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
Metastability for Glauber dynamics on random graphs
\u3cp\u3eIn 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 key hypothesis implying metastable behaviour, the average crossover time follows the classical Arrhenius law, with an exponent and a prefactor that are controlled by the energy and the 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.\u3c/p\u3