65 research outputs found
Cutoff for the Ising model on the lattice
- Author
- A. Guionnet
- Allan Sly
- B. Zegarliński
- B. Zegarliński
- D. Aldous
- D. Aldous
- D.A. Levin
- D.A. Levin
- D.W. Stroock
- D.W. Stroock
- D.W. Stroock
- E. Lubetzky
- Eyal Lubetzky
- F. Cesi
- F. Martinelli
- F. Martinelli
- F. Martinelli
- F. Martinelli
- F. Martinelli
- G.-Y. Chen
- J. Ding
- J. Ding
- L. Saloff-Coste
- L. Saloff-Coste
- M. Aizenman
- P. Diaconis
- P. Diaconis
- P. Diaconis
- P. Diaconis
- P. Diaconis
- P. Diaconis
- P. Diaconis
- R.A. Holley
- R.A. Holley
- R.A. Holley
- R.J. Glauber
- R.L. Dobrushin
- S.L. Lu
- T.M. Liggett
- T.P. Hayes
- Publication venue
- 'Springer Science and Business Media LLC'
- Publication date
- 23/09/2009
- Field of study
Introduced in 1963, Glauber dynamics is one of the most practiced and
extensively studied methods for sampling the Ising model on lattices. It is
well known that at high temperatures, the time it takes this chain to mix in
L1 on a system of size n is O(logn). Whether in this regime there is
cutoff, i.e. a sharp transition in the L1-convergence to equilibrium, is a
fundamental open problem: If so, as conjectured by Peres, it would imply that
mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing
a rigorous stopping rule for this MCMC sampler. However, obtaining the precise
asymptotics of the mixing and proving cutoff can be extremely challenging even
for fairly simple Markov chains. Already for the one-dimensional Ising model,
showing cutoff is a longstanding open problem.
We settle the above by establishing cutoff and its location at the high
temperature regime of the Ising model on the lattice with periodic boundary
conditions. Our results hold for any dimension and at any temperature where
there is strong spatial mixing: For Z2 this carries all the way to the
critical temperature. Specifically, for fixed d≥1, the continuous-time
Glauber dynamics for the Ising model on (Z/nZ)d with periodic boundary
conditions has cutoff at (d/2λ∞)logn, where λ∞ is
the spectral gap of the dynamics on the infinite-volume lattice. To our
knowledge, this is the first time where cutoff is shown for a Markov chain
where even understanding its stationary distribution is limited.
The proof hinges on a new technique for translating L1 to L2 mixing
which enables the application of log-Sobolev inequalities. The technique is
general and carries to other monotone and anti-monotone spin-systems.Comment: 34 pages, 3 figure
Open data from the third observing run of LIGO, Virgo, KAGRA, and GEO
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- Publication venue
- American Astronomical Society
- Publication date
- 01/01/2023
- Field of study
The global network of gravitational-wave observatories now includes five detectors, namely LIGO Hanford, LIGO Livingston, Virgo, KAGRA, and GEO 600. These detectors collected data during their third observing run, O3, composed of three phases: O3a starting in 2019 April and lasting six months, O3b starting in 2019 November and lasting five months, and O3GK starting in 2020 April and lasting two weeks. In this paper we describe these data and various other science products that can be freely accessed through the Gravitational Wave Open Science Center at https://gwosc.org. The main data set, consisting of the gravitational-wave strain time series that contains the astrophysical signals, is released together with supporting data useful for their analysis and documentation, tutorials, as well as analysis software packages
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