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On the Ising model with random boundary condition
 J. Stat. Phys
"... ABSTRACT. The infinitevolume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic sizedependence at low temperatures and we prove that the ‘+ ’ and ‘ ’ phases are the only almost sure limit Gibbs measures, assuming that the l ..."
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Cited by 6 (2 self)
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ABSTRACT. The infinitevolume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic sizedependence at low temperatures and we prove that the ‘+ ’ and ‘ ’ phases are the only almost sure limit Gibbs measures, assuming that the limit is taken along a sparse enough sequence of squares. In particular, we give a multiscale perturbative argument to show that in a sufficiently large volume typical spin configuration under a typical boundary condition contains no interfaces. 1.
A Combinatorial Proof of Tree Decay of Semiinvariants
"... We consider nite range Gibbs elds and provide a purely combinatorial proof of the exponential tree decay of semi{invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely ..."
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We consider nite range Gibbs elds and provide a purely combinatorial proof of the exponential tree decay of semi{invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely analytical Gibbs elds; in this context the tree decay of semi{invariants has been proven via analyticity arguments. However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called Griths' phase when analyticity arguments fail.
Gibbsian properties and convergence of the iterates for the BlockAveraging Transformation
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Renormalization Group, NonGibbsian states, their relationship and further developments
, 2005
"... We review what we have learned about the “Renormalization Group peculiarities ” which were discovered more than twentyfive years ago by Griffiths and Pearce. We discuss which of the questions they asked have been answered and which ones are still widely open. The problems they raised have led to the ..."
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We review what we have learned about the “Renormalization Group peculiarities ” which were discovered more than twentyfive years ago by Griffiths and Pearce. We discuss which of the questions they asked have been answered and which ones are still widely open. The problems they raised have led to the study of nonGibbsian states (probability measures). We also mention some further related developments, which find applications in nonequilibrium questions and disordered models.
Perturbative analysis of disordered Ising models close to criticality
"... Abstract We consider a twodimensional Ising model with random i.i.d. nearestneighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a convergent cluster expansion with probability one. The asso ..."
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Abstract We consider a twodimensional Ising model with random i.i.d. nearestneighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a convergent cluster expansion with probability one. The associated polymers are defined on a sequence of increasing scales; in particular the convergence of the above expansion implies the infinite differentiability of the free energy but not its analyticity. The basic tool in the proof are a general theory of graded cluster expansion and a stochastic domination of the disorder. MSC2000. Primary 82B44, 60K35.