Wolff algorithm

The Wolff algorithm, named after Ulli Wolff, is an <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> for <a href="/facts/Monte_Carlo_simulation/AkHjY7jc">Monte Carlo simulation</a> of the <a href="/facts/Ising_model/lvBpj2h1">Ising model</a> and <a href="/facts/Potts_model/JlSQMYH1">Potts model</a> in which the unit to be flipped is not a single spin (as in the <a href="/facts/Glauber_dynamics/fRg4wJ9p">heat bath</a> or <a href="/facts/Metropolis%25E2%2580%2593Hastings_algorithm/8K7bz3LN">Metropolis algorithms</a>) but a cluster of them. This cluster is defined as the set of connected spins sharing the same spin states, based on the <a href="/facts/Random_cluster_model/gTUa7XtJ">Fortuin-Kasteleyn representation</a>. 
The Wolff algorithm is similar to the <a href="/facts/Swendsen%25E2%2580%2593Wang_algorithm/aqeJ1GXI">Swendsen–Wang algorithm</a>, but different in that the former only flips one randomly chosen cluster with probability 1, while the latter flip every cluster independently with probability 1/2. It is shown numerically that flipping only one cluster decreases the <a href="/facts/Autocorrelation/dm4MqK0A">autocorrelation</a> time of the spin statistics.
The advantage of Wolff algorithm over other algorithms for magnetic spin simulations like single spin flip is that it allows non-local moves on the energy. One important consequence of this is that in some situations (e.g. ferromagnetic Ising model or fully frustrated Ising model), the scaling of the Multicanonic simulation is 
 
 
 
 
 N
 
 2
 
 
 
 
 {\displaystyle N^{2}}
 
, better than 
 
 
 
 
 N
 
 2
 +
 z
 
 
 
 
 {\displaystyle N^{2+z}}
 
, where z is the exponent associated with the critical slowing down phenomena.

<ul><li>Wolff, Ulli (1989), "Collective Monte Carlo Updating for Spin Systems", Physical Review Letters, 62 (4): 361–364, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1989PhRvL..62..361W">1989PhRvL..62..361W</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevLett.62.361">10.1103/PhysRevLett.62.361</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/10040213">10040213</a></li>
<li>Bae, S.; Ko, S.H.; Coddington, P.D. (1995), "Parallel Wolff cluster algorithms", International Journal of Modern Physics C, 6 (2): 197, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1995IJMPC...6..197B">1995IJMPC...6..197B</a>, <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.138.1448">10.1.1.138.1448</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0129183195000150">10.1142/S0129183195000150</a></li>
<li>Ferrenberg, Alan M.; Landau, D.P.; Wong, Y. Joanna (1992), "Monte Carlo simulations: Hidden errors from good random number generators", Physical Review Letters, 69 (23): 3382–3384, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1992PhRvL..69.3382F">1992PhRvL..69.3382F</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1103%2FPhysRevLett.69.3382">10.1103/PhysRevLett.69.3382</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/10046804">10046804</a></li></ul>

Wolff algorithm open-in-new

Wolff algorithm