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Register a new userDecimation of the Dyson-Ising Ferromagnet
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Aernout Coert Daniel van Enter
Publication date
2016
fields
Physics
and research's language is
English
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We study the decimation to a sublattice of half the sites, of the one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair interactions of the form $frac{1}{{|i-j|}^{alpha}}$, in the phase transition region (1< $alpha leq$ 2, and low temperature). We prove non-Gibbsianness of the decimated measure at low enough temperatures by exhibiting a point of essential discontinuity for the finite-volume conditional probabilities of decimated Gibbs measures. Thus result complements previous work proving conservation of Gibbsianness for fastly decaying potentials ($alpha$ > 2) and provides an example of a standard non-Gibbsian result in one dimension, in the vein of similar resuts in higher dimensions for short-range models. We also discuss how these measures could fit within a generalized (almost vs. weak) Gibbsian framework. Moreover we comment on the possibility of similar results for some other transformations.
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We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.
The dynamic evolution at zero temperature of a uniform Ising ferromagnet on a square lattice is followed by Monte Carlo computer simulations. The system always eventually reaches a final, absorbing state, which sometimes coincides with a ground state (all spins parallel), and sometimes does not (parallel stripes of spins up and down). We initiate here the numerical study of ``Chaotic Time Dependence (CTD) by seeing how much information about the final state is predictable from the randomly generated quenched initial state. CTD was originally proposed to explain how nonequilibrium spin glasses could manifest equilibrium pure state structure, but in simpler systems such as homogeneous ferromagnets it is closely related to long-term predictability and our results suggest that CTD might indeed occur in the infinite volume limit.
We study the Ising model two-point diagonal correlation function $ C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $lambda$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding $ lambda$ extension of the two-point diagonal correlation function, $ C(N,N; lambda)$, is shown, for arbitrary $lambda$, to be a solution of the sigma form of the Painlev{e} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $ f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $ E$. Each $ f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $ E$ and $ K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll structure. The previous $ lambda$-extensions, $ C(N,N; lambda)$ are, for singled-out values $ lambda= cos(pi m/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painleve VI are actually algebraic functions, being associated with modular curves.
In this paper, we study the product of two complex Ginibre matrices and the loop equations satisfied by their resolvents (i.e. the Stieltjes transform of the correlation functions). We obtain using Schwinger-Dyson equation (SDE) techniques the general loop equations satisfied by the resolvents. In order to deal with the product structure of the random matrix of interest, we consider SDEs involving the integral of higher derivatives. One of the advantage of this technique is that it bypasses the reformulation of the problem in terms of singular values. As a byproduct of this study we obtain the large $N$ limit of the Stieltjes transform of the $2$-point correlation function, as well as the first correction to the Stieltjes transform of the density, giving us access to corrections to the smoothed density. In order to pave the way for the establishment of a topological recursion formula we also study the geometry of the corresponding spectral curve. This paper also contains explicit results for different resolvents and their corrections.
We consider families of multiple and simple integrals of the ``Ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODEs and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behavior of these integrals is obtained in the self-dual variable $w= s/2/(1+s^2)$, with $s= sinh(2K)$, where $K=J/kT$ is the usual Ising model coupling constant. Switching to the variable $s$, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers-Wannier duality. We revisit the singular behavior (J. Phys. A {bf 38} (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model $chi^{(3)}$ at the points $1+3w+4w^2= 0$ and show that $chi^{(3)}(s)$ is not singular at the corresponding points inside the unit circle $| s |=1$, while its analytical continuation in the variable $s$ is actually singular at the corresponding points $ 2+s+s^2=0$ oustside the unit circle ($| s | > 1$).
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