# WILBERT

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• 1
Electronic Resource
Springer
Journal of statistical physics 33 (1983), S. 437-476
ISSN: 1572-9613
Keywords: One-dimensional Gibbs systems ; transfer matrix ; Markov chains ; renormalization group ; decimation procedure ; cluster expansion
Source: Springer Online Journal Archives 1860-2000
Topics: Physics
Notes: Abstract 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.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
Communications in mathematical physics 108 (1987), S. 241-255
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract One-dimensional Ising spin systems interacting via a two-body random potential are considered; a decay with the distance like 1/r 1+ε is assumed. We consider only boundary conditions independent of the random realization of the interactions and prove uniqueness and cluster properties of Gibbs states with probability one.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
Communications in mathematical physics 111 (1987), S. 555-577
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract 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 andJ ij are independent, identically distributed random variables with distributiondF(x) such that i) $$\int {xdF\left( x \right) = 0}$$ , 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.
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• 4
Electronic Resource
Springer
Communications in mathematical physics 70 (1979), S. 125-132
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We extend the validity of the implication of a local limit theorem from an integral one. Our extension eliminates the finite range assumption present in the previous works by using the cluster expansion to analyze the contribution from the tail of the potential.
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• 5
Electronic Resource
Springer
Communications in mathematical physics 79 (1981), S. 261-302
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We give a proof of the existence of aC 2, even solution of Feigenbaum's functional equation $$g{\text{(}}x) = - \lambda _0^{ - 1} g{\text{(}}g( - \lambda _0 x)),g{\text{(0) = 1,}}$$ whereg is a map of [−1, 1] into itself. It extends to a real analytic function over ℝ.
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