# Spectral gap for an unrestricted Kawasaki type dynamics

• Volume: 1, page 145-181
• ISSN: 1292-8100

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## Abstract

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We give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain number of charged particles moves on the segment [1,L] according to a Markovian law. One unitary charge, positive or negative, jumps from a site k to another site k'=k+1 or k'=k-1 at a rate which depends on the charge at site k and at site k'. The total charge of the system is preserved by the dynamics, in this sense our dynamics is similar to the Kawasaki dynamics, but in our case there is no restriction on the maximum charge allowed per site. The model is equivalent to an interface dynamics connected with the stochastic Ising model at very low temperature: the “unrestricted solid on solid model”. Thus the results we obtain may be read as results for this model. We give necessary and sufficient conditions to ensure that the spectral gap tends towards zero as the inverse of the square of L, independently of the total charge. We follow the method outlined in some papers by Yau (Lu, Yau (1993), Yau (1994)) where a similar spectral gap is proved for the original Kawasaki dynamics.

## How to cite

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Posta, Gustavo. "Spectral gap for an unrestricted Kawasaki type dynamics ." ESAIM: Probability and Statistics 1 (2010): 145-181. <http://eudml.org/doc/197732>.

@article{Posta2010,
abstract = { We give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain number of charged particles moves on the segment [1,L] according to a Markovian law. One unitary charge, positive or negative, jumps from a site k to another site k'=k+1 or k'=k-1 at a rate which depends on the charge at site k and at site k'. The total charge of the system is preserved by the dynamics, in this sense our dynamics is similar to the Kawasaki dynamics, but in our case there is no restriction on the maximum charge allowed per site. The model is equivalent to an interface dynamics connected with the stochastic Ising model at very low temperature: the “unrestricted solid on solid model”. Thus the results we obtain may be read as results for this model. We give necessary and sufficient conditions to ensure that the spectral gap tends towards zero as the inverse of the square of L, independently of the total charge. We follow the method outlined in some papers by Yau (Lu, Yau (1993), Yau (1994)) where a similar spectral gap is proved for the original Kawasaki dynamics. },
author = {Posta, Gustavo},
journal = {ESAIM: Probability and Statistics},
keywords = {Kawasaki dynamics / SOS model / spectral gap.; Kawasaki dynamics; SOS model; spectral gap; Ising model; interacting particle system},
language = {eng},
month = {3},
pages = {145-181},
publisher = {EDP Sciences},
title = {Spectral gap for an unrestricted Kawasaki type dynamics },
url = {http://eudml.org/doc/197732},
volume = {1},
year = {2010},
}

TY - JOUR
AU - Posta, Gustavo
TI - Spectral gap for an unrestricted Kawasaki type dynamics
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 1
SP - 145
EP - 181
AB - We give an accurate asymptotic estimate for the gap of the generator of a particular interacting particle system. The model we consider may be informally described as follows. A certain number of charged particles moves on the segment [1,L] according to a Markovian law. One unitary charge, positive or negative, jumps from a site k to another site k'=k+1 or k'=k-1 at a rate which depends on the charge at site k and at site k'. The total charge of the system is preserved by the dynamics, in this sense our dynamics is similar to the Kawasaki dynamics, but in our case there is no restriction on the maximum charge allowed per site. The model is equivalent to an interface dynamics connected with the stochastic Ising model at very low temperature: the “unrestricted solid on solid model”. Thus the results we obtain may be read as results for this model. We give necessary and sufficient conditions to ensure that the spectral gap tends towards zero as the inverse of the square of L, independently of the total charge. We follow the method outlined in some papers by Yau (Lu, Yau (1993), Yau (1994)) where a similar spectral gap is proved for the original Kawasaki dynamics.
LA - eng
KW - Kawasaki dynamics / SOS model / spectral gap.; Kawasaki dynamics; SOS model; spectral gap; Ising model; interacting particle system
UR - http://eudml.org/doc/197732
ER -

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