event
ARC Colloquium: Andreas Galanis  University of Oxford
Primary tabs
Algorithms & Randomness Center (ARC) Colloquium
Andreas Galanis – University of Oxford
Wednesday, August 26, 2015
Klaus 2447  2:00 pm
Title:
SwendsenWang Algorithm on the MeanField Potts Model
Abstract:
This talk will focus on the qstate ferromagnetic Potts model on the nvertex complete graph known as the meanfield (CurieWeiss) model. We analyze the SwendsenWang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit singlesite Glauber dynamics.
The case q=2 (the SwendsenWang algorithm for the ferromagnetic Ising model) undergoes a slowdown at a critical temperature beta=betac (Long et al., 2014), but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000).
In contrast, for q>=3 there are two critical temperatures 0<betau<betarc that are relevant (these correspond to phase transitions on the infinite tree). We prove that the mixing time of the SwendsenWang algorithm for the ferromagnetic Potts model on the nvertex complete graph satisfies:
(i) O(log n) for beta<betau, (ii) O(n^{1/3}) for beta=betau, (iii) exp(n^(Omega(1))) for betau<beta<betarc, and (iv) O(log n) for beta>=betarc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.
The most interesting aspect of our analysis is at the critical temperature beta=betau, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over.
Joint work with Daniel Stefankovic and Eric Vigoda.
Status

Workflow Status:
Published 
Created By:
Dani Denton 
Created:
08/18/2015 
Modified By:
Fletcher Moore 
Modified:
04/13/2017
Categories
Keywords
Target Audience