Results 1  10
of
14
Non) Gibbsianness and phase transitions in random lattice spin models
 Markov Proc. Relat. Fields
, 1999
"... Abstract: We consider disordered lattice spin models with finite volume Gibbs measures µΛ[η](dσ). Here σ denotes a lattice spinvariable and η a lattice random variable with product distribution IP describing the disorder of the model. We ask: When will the joint measures lim Λ↑Z d IP(dη)µΛ[η](dσ) b ..."
Abstract

Cited by 25 (12 self)
 Add to MetaCart
Abstract: We consider disordered lattice spin models with finite volume Gibbs measures µΛ[η](dσ). Here σ denotes a lattice spinvariable and η a lattice random variable with product distribution IP describing the disorder of the model. We ask: When will the joint measures lim Λ↑Z d IP(dη)µΛ[η](dσ) be [non] Gibbsian measures on the product of spinspace and disorderspace? We obtain general criteria for both Gibbsianness and nonGibbsianness providing an interesting link between phase transitions at a fixed random configuration and Gibbsianness in product space: Loosely speaking, a phase transition can lead to nonGibbsianness, (only) if it can be observed on the spinobservable conjugate to the independent disorder variables. Our main specific example is the random field Ising model in any dimension for which we show almost sure [almost sure non] Gibbsianness for the single [multi] phase region. We also discuss models with disordered couplings, including spinglasses and ferromagnets, where various mechanisms are responsible for [non] Gibbsianness.
Weakly Gibbsian representations for joint measures of quenched lattice spin models, Probab
 Th. Relat. Fields
, 2001
"... Abstract: Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spinspace and disorderspace be represented as (suitably generalized) Gibbs measures of an “annealed system”? We prove that there is always a potential (depending on both spin and diso ..."
Abstract

Cited by 19 (10 self)
 Add to MetaCart
Abstract: Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spinspace and disorderspace be represented as (suitably generalized) Gibbs measures of an “annealed system”? We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive ” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative ” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov’s constructions.
Stochastic symmetrybreaking in a Gaussian Hopfieldmodel
 J. Stat. Phys
, 1999
"... We study a``twopattern'' Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
(Show Context)
We study a``twopattern'' Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables.
Spin glasses
, 2005
"... From a physical point of view, spin glasses, as dilute magnetic alloys, are very interesting systems. They are characterized by such features as exhibiting a new magnetic phase, where magnetic moments are frozen into disordered ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
From a physical point of view, spin glasses, as dilute magnetic alloys, are very interesting systems. They are characterized by such features as exhibiting a new magnetic phase, where magnetic moments are frozen into disordered
Uniqueness of ground states for shortrange spin glasses in the halfplane
 Commun. Math. Phys
, 2010
"... ar ..."
(Show Context)
ShortRange Spin Glasses
, 2006
"... We present some of the results and open problems related to the different behavior of shortrange spin glasses like the EdwardsAnderson (EA) model compared to meanfield spin glasses like the SherringtonKirkpatrick model. In particular, we discuss the possible use of random cluster representations ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We present some of the results and open problems related to the different behavior of shortrange spin glasses like the EdwardsAnderson (EA) model compared to meanfield spin glasses like the SherringtonKirkpatrick model. In particular, we discuss the possible use of random cluster representations to prove the existence of a phase transition in the EA model, open problems concerning minimal spanning trees in dimension d> 2 and their relevance for ground states of the highly disordered spin glass, and partial results on uniqueness of ground state pairs in the d = 2 EA model.
Digital Object Identifier (DOI) 10.1007/s004400000097
, 2000
"... Abstract. Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spinspace and disorderspace be represented as (suitably generalized) Gibbs measures of an “annealed system”? We prove that there is always a potential (depending on both spin and dis ..."
Abstract
 Add to MetaCart
Abstract. Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spinspace and disorderspace be represented as (suitably generalized) Gibbs measures of an “annealed system”? We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive ” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative ” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinitevolume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov’s constructions. 1.
Chapter 1 Frustration and Fluctuations in Systems with Quenched Disorder
"... As Phil Anderson noted long ago, frustration can be generally defined by measuring the fluctuations in the coupling energy across a plane boundary between two large blocks of material. Since that time, a number of groups have studied the free energy fluctuations between (putative) distinct spin gla ..."
Abstract
 Add to MetaCart
As Phil Anderson noted long ago, frustration can be generally defined by measuring the fluctuations in the coupling energy across a plane boundary between two large blocks of material. Since that time, a number of groups have studied the free energy fluctuations between (putative) distinct spin glass thermodynamic states. While upper bounds on such fluctuations have been obtained, useful lower bounds have been more difficult to derive. I present a history of these efforts, and briefly discuss recent work showing that free energy fluctuations between certain classes of distinct thermodynamic states (if they exist) scale as the square root of the volume. The perspective offered here is that the power and generality of the Anderson conception of frustration suggests a potential approach toward resolving some longstanding and central issues in spin glass physics. 1. Phil Anderson and Spin Glass Theory It is a great pleasure, both personally and scientifically, to contribute to this volume in honor of Phil Anderson’s 90th birthday. The importance and influence of Phil’s research in shaping the modern field of condensed matter physics (including coining