March  2011, 6(1): 127-144. doi: 10.3934/nhm.2011.6.127

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

1. 

Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci, cubo 30B, 87036 Arcavacata di Rende, Italy

2. 

Dipartimento di Matematica, "Sapienza" Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy

Received  November 2009 Revised  November 2010 Published  March 2011

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.
Citation: Michele Gianfelice, Marco Isopi. On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model. Networks & Heterogeneous Media, 2011, 6 (1) : 127-144. doi: 10.3934/nhm.2011.6.127
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.  doi: i:10.1007/s002200050660.  Google Scholar

[2]

F. Cesi, C. Maes and F. Martinelli, Relaxation in disordered magnets in the the Griffiths' regime,, Comm. Math. Phys., 188 (1997), 135.  doi: 10.1007/s002200050160.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[5]

A. Guionnet and B. Zegarlinski, Decay to equilibrium in random spin systems on a lattice,, Comm. Math. Phys., 181 (1996), 703.  doi: 10.1007/BF02101294.  Google Scholar

[6]

R. Holley, "On the Asymptotics of the Spin-Spin Autocorrelation Function In Stochastic Ising Models Near the Critical Temperature,", Progress in Probability n. \textbf{19}, 19 (1991).   Google Scholar

[7]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1966).   Google Scholar

[8]

T. Liggett, "Interacting Particle Systems,", Springer-Verlag, (1985).   Google Scholar

[9]

R. A. Minlos, Invariant subspaces of the stochastic ising high temperature dynamics,, Markov Processes Relat. Fields, 2 (1996), 263.   Google Scholar

[10]

C. J. Preston, "Gibbs States on Countable Sets,", Cambridge Tracts in Mathematics, (1974).   Google Scholar

[11]

L. Pastur and A. Figotin, "Spectra of Random and Almost Periodic Operators,", Springer-Verlag, (1992).   Google Scholar

[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.  doi: 10.1023/A:1022225612366.  Google Scholar

[13]

B. Zegarlinski, Strong decay to equilibrium in one dimensional random spin systems,, J. Stat. Phys., 77 (1994), 717.  doi: 10.1007/BF02179458.  Google Scholar

[14]

E. Zhizhina, The Lifshitz tail and relaxation to equilibrium in the one-dimensional disordered Ising model,, J. Statist. Phys., 98 (2000), 701.  doi: 10.1023/A:1018623424891.  Google Scholar

show all references

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.  doi: i:10.1007/s002200050660.  Google Scholar

[2]

F. Cesi, C. Maes and F. Martinelli, Relaxation in disordered magnets in the the Griffiths' regime,, Comm. Math. Phys., 188 (1997), 135.  doi: 10.1007/s002200050160.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[5]

A. Guionnet and B. Zegarlinski, Decay to equilibrium in random spin systems on a lattice,, Comm. Math. Phys., 181 (1996), 703.  doi: 10.1007/BF02101294.  Google Scholar

[6]

R. Holley, "On the Asymptotics of the Spin-Spin Autocorrelation Function In Stochastic Ising Models Near the Critical Temperature,", Progress in Probability n. \textbf{19}, 19 (1991).   Google Scholar

[7]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1966).   Google Scholar

[8]

T. Liggett, "Interacting Particle Systems,", Springer-Verlag, (1985).   Google Scholar

[9]

R. A. Minlos, Invariant subspaces of the stochastic ising high temperature dynamics,, Markov Processes Relat. Fields, 2 (1996), 263.   Google Scholar

[10]

C. J. Preston, "Gibbs States on Countable Sets,", Cambridge Tracts in Mathematics, (1974).   Google Scholar

[11]

L. Pastur and A. Figotin, "Spectra of Random and Almost Periodic Operators,", Springer-Verlag, (1992).   Google Scholar

[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.  doi: 10.1023/A:1022225612366.  Google Scholar

[13]

B. Zegarlinski, Strong decay to equilibrium in one dimensional random spin systems,, J. Stat. Phys., 77 (1994), 717.  doi: 10.1007/BF02179458.  Google Scholar

[14]

E. Zhizhina, The Lifshitz tail and relaxation to equilibrium in the one-dimensional disordered Ising model,, J. Statist. Phys., 98 (2000), 701.  doi: 10.1023/A:1018623424891.  Google Scholar

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