We prove that Gibbs states for the Hamiltonian
, with thesx varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if
withJxy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions. 相似文献
WE consider a one-dimensional random Ising model with Hamiltonian $$H = \sum\limits_{i\ddag j} {\frac{{J_{ij} }}{{\left| {i - j} \right|^{1 + \varepsilon } }}S_i S_j } + h\sum\limits_i {S_i } $$ , where ε>0 andJij are independent, identically distributed random variables with distributiondF(x) such thati) $$\int {xdF\left( x \right) = 0} $$ ,ii) $$\int {e^{tx} dF\left( x \right)< \infty \forall t \in \mathbb{R}} $$ . We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables areC∞ functions of the temperature and of the magnetic fieldh. Moreover we can estimate the decay of truncated correlation functions. In particular for every ε′>0 there exists a random variablec(ω)m, finite almost everywhere, such that $$\left| {\left\langle {s_0 s_j } \right\rangle _H - \left\langle {s_0 } \right\rangle _H \left\langle {s_j } \right\rangle _H } \right| \leqq \frac{{c\left( \omega \right)}}{{\left| j \right|^{1 + \varepsilon - \varepsilon '} }}$$ , where 〈 〉H denotes the Gibbs average with respect to the HamiltonianH. 相似文献
We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on ${\mathbb{Z}}^{d},\,d\geq2$ in the limit where their distances tend to infinity. 相似文献
Summary We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp
c: (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL–(d–1)/2. (2) The correlation length, (p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x1=qL,0, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>pc, the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.Work supported in part by G.N.A.F.A. (C.N.R.)Work supported in part by NSF Grant No. DMS-88-06552 相似文献
We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Zd, σ1, σ3 are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Zd} are independent identically distributed random variables. We consider the ground state correlation function 〈σ3(x)σ3(y)〉 and prove:
Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Zd, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Zd withCxh<∞.
Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Zd.
Let
and
be independent sets of nonnegative i.i.d.r.v.'s, <x,y> denoting a pair of nearest neighbors inZd; let , >0. We consider the random systems: 1. A bond Bernoulli percolation model onZd+1 with random occupation probabilities 相似文献
We show that at the special energiesE=2cosp/q, the invariant measure, the Lyapunov exponent, and the density of states can be extended to zero disorder as C functions in the disorder parameter. In particular, we obtain asymptotic series in the disorder parameter. This gives a rigorous proof of the existence of the anomalies originally discovered by Kappus and Wegner and studied by Derrida and Gardner and by Bovier and Klein.Partially supported by NSF grant DMS 87-02301 相似文献
LetH=–+V onl2(), whereV(x),x, are i.i.d.r.v.'s with common probability distributionv. Leth(t)=e–itvdv(v) and letk(E) be the integrated density of states. It is proven: (i) Ifh isn-times differentiable withh(j)(t)=O((1+|t|)–) for some >0,j=0, 1, ...,n, thenk(E) is aCn function. In particular, ifv has compact support andh(t)=O((1+|t|)–) with >0, thenk(E) isC. This allowsv to be singular continuous. (ii) Ifh(t)=O(e–|t|) for some >0 thenk(E) is analytic in a strip about the real axis.The proof uses the supersymmetric replica trick to rewrite the averaged Green's function as a two-point function of a one-dimensional supersymmetric field theory which is studied by the transfer matrix method.Research partially supported by the NSF under grant MC-8301889 相似文献
We consider unbounded spin systems and classical continuous particle systems in one dimension. We assume that the interaction is described by a superstable two-body potential with a decay at large distances at least asr?2(lnr)?(2+ε), ε > 0. We prove the analyticity of the free energy and of the correlations as functions of the interaction parameters. This is done by using a “renormalization group technique” to transform the original model into another, physically equivalent, model which is in the high-temperature (small-coupling) region. 相似文献