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Register a new userAsymptotics of the mean-field Heisenberg model
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We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramer- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Steins method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.
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We study mean-field classical $N$-vector models, for integers $Nge 2$. We use the theory of large deviations and Steins method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in cite{KM} but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.
We derive mean-field equations for a general class of ferromagnetic spin systems with an explicit error bound in finite volumes. The proof is based on a link between the mean-field equation and the free convolution formalism of random matrix theory, which we exploit in terms of a dynamical method. We present three sample applications of our results to Ka{c} interactions, randomly diluted models, and models with an asymptotically vanishing external field.
The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions $2n times 2n$ with arbitrary boundary configuration,and we show that the mean square displacement of particles near the center of the box is bounded from below by $c log n$. The result generalizes to a large class of models with fairly arbitrary interaction.
The 1-arm exponent $rho$ for the ferromagnetic Ising model on $mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Suppose that the spin-spin coupling $J$ is translation-invariant, $mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $eta=0$, we show that the optimal mean-field bound $rhole1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.
We introduce and study the following model for random resonances: we take a collection of point interactions $Upsilon_j$ generated by a simple finite point process in the 3-D space and consider the resonances of associated random Schrodinger Hamiltonians $H_Upsilon = -Delta + ``sum mathfrak{m}(alpha) delta (x - Upsilon_j)``$. These resonances are zeroes of a random exponential polynomial, and so form a point process $Sigma (H_Upsilon)$ in the complex plane $mathbb{C}$. We show that the counting function for the set of random resonances $Sigma (H_Upsilon)$ in $mathbb{C}$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $Upsilon$, and obtain an explicit formula for the limiting distribution as $m to infty$ of the leading parameter of the asymptotic chain of `most narrow resonances generated by a sequence of uniform binomial processes $Upsilon^m$ with $m$ points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of $m$ samples of a random point in the 3-D space and related statistics of extreme values.
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