On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model

Michele Gianfelice; Marco Isopi; Michele Gianfelice; Marco Isopi

doi:10.3934/nhm.2011.6.127

Networks and Heterogeneous Media

2011, Volume 6, Issue 1: 127-144. doi: 10.3934/nhm.2011.6.127

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On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model

Michele Gianfelice ^{1
,},
Marco Isopi ^{2
,}

1.
Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci, cubo 30B, 87036 Arcavacata di Rende
2.
Dipartimento di Matematica, "Sapienza" Università di Roma, Piazzale Aldo Moro 5, 00185 Roma

Received: 01 November 2009 Revised: 01 November 2010
Primary: 82B44; Secondary: 60K35.

We analyse the lower non trivial part of the spectrum of the generator of the Glauber dynamics for a $d$-dimensional nearest neighbour Ising model with a bounded random potential. We prove conjecture 1 in [1]: for sufficiently large values of the temperature, the first band of the spectrum of the generator of the process coincides with a closed non random segment of the real line.
- Glauber dynamics,
- spectral gap,
- disordered Ising model
Citation: Michele Gianfelice, Marco Isopi. On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model[J]. Networks and Heterogeneous Media, 2011, 6(1): 127-144. doi: 10.3934/nhm.2011.6.127

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Abstract

We analyse the lower non trivial part of the spectrum of the generator of the Glauber dynamics for a $d$-dimensional nearest neighbour Ising model with a bounded random potential. We prove conjecture 1 in [1]: for sufficiently large values of the temperature, the first band of the spectrum of the generator of the process coincides with a closed non random segment of the real line.

References

[1]	S. Albeverio, R. Minlos, E. Scacciatelli and E. Zhizhina, Spectral analysis of the disordered stochastic 1-D Ising model, Comm. Math. Phys., 204 (1999), 651-668. doi: i:10.1007/s002200050660
[2]	F. Cesi, C. Maes and F. Martinelli, Relaxation in disordered magnets in the the Griffiths' regime, Comm. Math. Phys., 188 (1997), 135-173. doi: 10.1007/s002200050160
[3]	M. Gianfelice and M. Isopi, Quantum methods for interacting particle systems. II. Glauber dynamics for Ising spin systems, Markov Processes and Relat. Fields, 4 (1998), 411-428.
[4]	M. Gianfelice and M. Isopi, Erratum and addenda to: "Quantum methods for interacting particle systems II, Glauber dynamics for Ising spin systems'', Markov Processes and Relat. Fields, 9 (2003), 513-516.
[5]	A. Guionnet and B. Zegarlinski, Decay to equilibrium in random spin systems on a lattice, Comm. Math. Phys., 181 (1996), 703-732. doi: 10.1007/BF02101294
[6]	R. Holley, "On the Asymptotics of the Spin-Spin Autocorrelation Function In Stochastic Ising Models Near the Critical Temperature," Progress in Probability n. 19, Spatial Stochastic Processes Birkäuser Boston, 1991.
[7]	T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, 1966.
[8]	T. Liggett, "Interacting Particle Systems," Springer-Verlag, 1985.
[9]	R. A. Minlos, Invariant subspaces of the stochastic ising high temperature dynamics, Markov Processes Relat. Fields, 2 (1996), 263-284.
[10]	C. J. Preston, "Gibbs States on Countable Sets," Cambridge Tracts in Mathematics, No. 68. Cambridge University Press, London-New York, 1974.
[11]	L. Pastur and A. Figotin, "Spectra of Random and Almost Periodic Operators," Springer-Verlag, 1992.
[12]	H. Spohn and E. Zhizhina, Long-time behavior for the 1-D stochastic Ising model with unbounded random couplings, J. Statist. Phys., 111 (2003), 419-431. doi: 10.1023/A:1022225612366
[13]	B. Zegarlinski, Strong decay to equilibrium in one dimensional random spin systems, J. Stat. Phys., 77 (1994), 717-732. doi: 10.1007/BF02179458
[14]	E. Zhizhina, The Lifshitz tail and relaxation to equilibrium in the one-dimensional disordered Ising model, J. Statist. Phys., 98 (2000), 701-721. doi: 10.1023/A:1018623424891

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