TY - GEN
T1 - Dyson models under renormalization and in weak fields
AU - Bissacot, Rodrigo
AU - Endo, Eric O.
AU - van Enter, Aernout C.D.
AU - Kimura, Bruno
AU - Le Ny, Arnaud
AU - Ruszel, Wioletta M.
PY - 2019
Y1 - 2019
N2 - We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form (formula presented), mainly focusing on the range of slow decays (formula presented). We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as (formula presented). We study how the critical decay power of the field, (formula presented), for which the phase transition persists and the decay power (formula presented) of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.
AB - We consider one-dimensional long-range spin models (usually called Dyson models), consisting of Ising ferromagnets with slowly decaying long-range pair potentials of the form (formula presented), mainly focusing on the range of slow decays (formula presented). We describe two recent results, one about renormalization and one about the effect of external fields at low temperature. The first result states that a decimated long-range Gibbs measure in one dimension becomes non-Gibbsian, in the same vein as comparable results in higher dimensions for short-range models. The second result addresses the behaviour of such models under inhomogeneous fields, in particular external fields which decay to zero polynomially as (formula presented). We study how the critical decay power of the field, (formula presented), for which the phase transition persists and the decay power (formula presented) of the Dyson model compare, extending recent results for short-range models on lattices and on trees. We also briefly point out some analogies between these results.
KW - Generalized Gibbs measures
KW - Hidden phase transitions
KW - Long-range Ising models
KW - Slowly decaying correlated external fields
UR - http://www.scopus.com/inward/record.url?scp=85077708096&partnerID=8YFLogxK
U2 - 10.1007/978-981-15-0294-1_5
DO - 10.1007/978-981-15-0294-1_5
M3 - Conference contribution
AN - SCOPUS:85077708096
SN - 9789811502934
T3 - Springer Proceedings in Mathematics and Statistics
SP - 123
EP - 137
BT - Sojourns in Probability Theory and Statistical Physics - I - Spin Glasses and Statistical Mechanics, A Festschrift for Charles M. Newman
A2 - Sidoravicius, Vladas
PB - Springer
T2 - International Conference on Probability Theory and Statistical Physics, 2016
Y2 - 25 March 2016 through 27 March 2016
ER -