Abstract
We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Benfatto, E. Marinari, and E. Olivieri, Some numerical results on the block spin transformation for the 2D Ising model at the critical point,J. Stat. Phys. 78:731–757 (1995).
M. Cassandro and E. Olivieri, Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem,Commun. Math. Phys. 80:255–269 (1981).
A. v. Enter, R. Fernandez, and R. Kotecký, Pathological behavior of renormalization group maps at high fields and above the transition temperature, Preprint (1994).
A. v. Enter, R. Fernandez, and A. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations,J. Stat. Phys. 72(5/6):879–1167 (1993).
R. Fernandez and C. E. Pfister, (Non)quasilocality of projections of Gibbs measures, Lausanne preprint (1994).
G. Gallavotti, A. Martin-Lof, and S. Miracle-Sole, InLecture Notes in Physics Vol. 20 (Springer, New York, 1971), pp. 162–202.
E. Grannan and G. Swindle, A particle system with massive destruction,J. Phys. A: Math. Gen. 23:L73 (1990).
O. K. Kozlov, Gibbs description of a system of random variables,Probl. Inform. Transmission 10:258–265 (1974).
F. Martinelli, Dynamical analysis of low temperature Monte Carlo algorithms,J. Stat. Phys. 66(5/6) (1992).
F. Martinelli and E. Olivieri, Some remarks on pathologies of renormalization group transformations for the Ising model,J. Stat. Phys. 72:1169–1177 (1994).
F. Martinelli, E. Olivieri, and R. H. Schonmann, For 2-D lattice spins systems weak mixing implies strong mixing,Commun. Math. Phys. 165:33–47 (1994).
F. Martinelli and E. Scoppola, A simple stochastic cluster dynamics: Rigorous results,J. Phys. A 24:3135–3157 (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martinelli, F., Olivieri, E. Instability of renormalization-group pathologies under decimation. J Stat Phys 79, 25–42 (1995). https://doi.org/10.1007/BF02179382
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02179382