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  • 1
    Electronic Resource
    Electronic Resource
    Infinite differentiability for one-dimensional spin system with long range random interaction (1982)
    Springer
    Communications in mathematical physics 87 (1982), S. 229-252 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider one-dimensional spin systems with Hamiltonian: $$H\left( {\sigma _\Lambda } \right) = - \sum\limits_{t,t' \in \Lambda } {\frac{{\varepsilon _{tt'} }}{{\left| {t - t'} \right|^\alpha }}\sigma _t \sigma _{t'} - h\sum\limits_{t \in \Lambda } {\sigma _t } } $$ , where ɛ tt′ are independent random variables and, using decimation and the cluster expansion, we show that, when α〉3/2 andE(ɛ tt′ )=0, for any magnetic fieldh and inverse temperature β, the correlation functions and the free energy areC ∞ both inh and β. Moreover we discuss an example, obtained by a particular choice of the probability distribution of the ɛ tt′ 's, where the quenched magnetization isC ∞ but fails to be analytic inh for suitableh and β.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01218562
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  • 2
    Electronic Resource
    Electronic Resource
    Some rigorous results on the phase diagram of the dilute Ising model
    Amsterdam : Elsevier
    Physics Letters A 94 (1983), S. 309-311 
    Details
    ISSN: 0375-9601
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
    Type of Medium: Electronic Resource
    URL: http://linkinghub.elsevier.com/retrieve/pii/0375-9601(83)90726-0
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  • 3
    Electronic Resource
    Electronic Resource
    Approach to equilibrium of Glauber dynamics in the one phase region (1994)
    Springer
    Communications in mathematical physics 161 (1994), S. 447-486 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed. In particular somefinite size conditions are discussed, together with their implications for the Gibbs measures and for the approach to equilibrium of Glauber dynamics inarbitrarily large volumes. It is shown that Dobrushin-Shlosman's theory ofcomplete analyticity and its dynamical counterpart due to Stroock and Zegarlinski, cannot be applied, in general, to the whole one phase region since it requires mixing properties for regions ofarbitrary shape. An alternative approach, based on previous ideas of Oliveri, and Picco, is developed, which allows to establish results on rapid approach to equilibrium deeply inside the one phase region. In particular, in the ferromagnetic case, we considerably improve some previous results by Holley and Aizenman and Holley. Our results are optimal in the sene that, for example, they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed-dimensional Ising model with or without an external field. In part II we extensively consider the general case (not necessarily attractive) and we develop a new method, based on renormalizations group ideas and on an assumption of strong mixing in a finite cube, to prove hypercontractivity of the Markov semigroup of the Glauber dynamics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02101929
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  • 4
    Electronic Resource
    Electronic Resource
    For 2-D lattice spin systems weak mixing implies strong mixing (1994)
    Springer
    Communications in mathematical physics 165 (1994), S. 33-47 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove that for finite range discrete spin systems on the two dimensional latticeZ 2, the (weak) mixing condition which follows, for instance, from the Dobrushin-Shlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analyticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the associated Glauber dynamics on nice subsets ofZ 2, including the full lattice.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02099735
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  • 5
    Electronic Resource
    Electronic Resource
    Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem (1981)
    Springer
    Communications in mathematical physics 80 (1981), S. 255-269 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters. This result is not new (Dobrushin, 1973) but our proof is simpler and physically more transparent. We show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks (decimation procedure), the resulting effective interaction is such that the system can always be dealt with as a high temperature system.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01213013
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  • 6
    Electronic Resource
    Electronic Resource
    One dimensional spin glasses with potential decay 1/r 1+g . Absence of phase transitions and cluster properties (1987)
    Springer
    Communications in mathematical physics 108 (1987), S. 241-255 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01210614
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  • 7
    Electronic Resource
    Electronic Resource
    One-dimensional random Ising systems with interaction decayr −(1+ɛ): A convergent cluster expansion (1987)
    Springer
    Communications in mathematical physics 111 (1987), S. 555-577 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01219074
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  • 8
    Electronic Resource
    Electronic Resource
    Approach to equilibrium of Glauber dynamics in the one phase region (1994)
    Springer
    Communications in mathematical physics 161 (1994), S. 487-514 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube Λ0, a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the Markov semigroup of the associated Glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes Λ “multiples” of the cube Λ0.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02101930
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  • 9
    Electronic Resource
    Electronic Resource
    Van der Waals-Maxwell theory, Lebowitz-Penrose limit and superstable interactions (1973)
    Springer
    Communications in mathematical physics 29 (1973), S. 219-231 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We obtain rigorous results about the liquid-vapour phase transition. We prove that in the Lebowitz-Penrose limit the hypothesis of hard core for the interaction is not essential to reproduce the Van der Waals-Maxwell theory. It can be replaced by the superstability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01645248
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  • 10
    Electronic Resource
    Electronic Resource
    Ultraviolet stability in Euclidean scalar field theories (1980)
    Springer
    Communications in mathematical physics 71 (1980), S. 95-130 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We develop a technique for reducing the problem of the ultraviolet divergences and their removal to a free field problem. This work is an example of a problem to which a rather general method can be applied. It can be thought as an attempt towards a rigorous version (in 2 or 3 space-time dimensions) of the analysis of the structure of the functional integrals developed in [9], the underlying mechanism being essentially the same as in [11,3].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01197916
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