random cluster dynamics for the ising model is rapidly
play

Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng - PowerPoint PPT Presentation

Feb 25, 2023 •340 likes •1.35k views

Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng Guo Queen Mary, University of London Joint work with Mark Jerrum Oxford Nov 03 2016 Heng Guo (QMUL) Random Cluster 2016/11/03 1 / 41 The model and its dynamics Heng Guo

  1. Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng Guo Queen Mary, University of London Joint work with Mark Jerrum Oxford Nov 03 2016 Heng Guo (QMUL) Random Cluster 2016/11/03 1 / 41

  2. The model and its dynamics Heng Guo (QMUL) Random Cluster 2016/11/03 2 / 41

  3. The random cluster model [Fortuin, Kasteleyn 1969] Parameters 0 ⩽ p ⩽ 1 (edge weight), q ⩾ 0 (cluster weight). Given graph G = ( V , E ) , the measure on subgraph r ⊆ E is defined as π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) , where κ ( r ) is the number of connected components in ( V , r ) . ( 1 − p ) 4 q 4 p 2 ( 1 − p ) 2 q 2 p 4 q Heng Guo (QMUL) Random Cluster 2016/11/03 3 / 41

  4. The random cluster model [Fortuin, Kasteleyn 1969] The partition function (normalizing factor): ∑ p | r | ( 1 − p ) | E \ r | q κ ( r ) . Z RC ( p , q ) = r ⊆ E Equivalent to the Tutte polynomial Z Tutte ( x , y ) : p = 1 − 1 q = ( x − 1 )( y − 1 ) y Heng Guo (QMUL) Random Cluster 2016/11/03 4 / 41

  5. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model Potts model Bond percolation Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  6. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model Bond percolation Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  7. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  8. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On K n , Erd˝ os-Rényi random graph) Electrical network Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  9. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On K n , Erd˝ os-Rényi random graph) q → 0 (Spanning trees if p → 0 and q Electrical network p → 0) Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  10. The random cluster model [Fortuin, Kasteleyn 1969] π RC ( r ) ∝ p | r | ( 1 − p ) | E \ r | q κ ( r ) The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On K n , Erd˝ os-Rényi random graph) q → 0 (Spanning trees if p → 0 and q Electrical network p → 0) Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

  11. Glauber dynamics Glauber dynamics (single edge update) P RC (Metropolis): Current state x ⊆ E With prob. 1 / 2 do nothing. (Lazy) 1 Otherwise, choose an edge e u.a.r. 2 { } 1, π RC ( y ) Move to y = x ⊕ { e } with prob. min . 3 π RC ( x ) Detailed balance: π ( x ) P ( x , y ) = π ( y ) P ( y , x ) = min { π ( x ) , π ( y ) } Heng Guo (QMUL) Random Cluster 2016/11/03 6 / 41

  12. Glauber dynamics Glauber dynamics (single edge update) P RC (Metropolis):  { } 1 1, π RC ( y ) 2 m min if | x ⊕ y | = 1;  π RC ( x )     { } P RC ( x , y ) = 1 1, π RC ( x ⊕ { e } ) 1 − e ∈ E min if x = y ; ∑ 2 m π RC ( x )    0 otherwise.   We are interested in the mixing time τ ϵ ( P RC ) : τ ϵ ( P RC ) = min { t : || P t RC ( x 0 , · ) − π || TV ⩽ ϵ } . Heng Guo (QMUL) Random Cluster 2016/11/03 7 / 41

  13. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  14. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  15. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  16. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  17. A simple example Let p < 1 / 2. { } 1, π RC ( x ∪ { e } ) min π RC ( x )  p if e is not a cut edge   1 − p = p if e is a cut edge   q ( 1 − p ) Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

  18. Brief History Studied extensively for special graphs, such as the complete graph (mean-field) and the lattice Z 2 . Mean-field: [Gore, Jerrum 1999] [Blanca, Sinclair 2015] Z 2 : [Borgs et al. 1999] [Blanca, Sinclair 2016] [Gheissari, Lubetzky 2016] q > 2: Slow mixing for the complete graph. 0 ⩽ q ⩽ 2: No known fast mixing bound for general graphs. Heng Guo (QMUL) Random Cluster 2016/11/03 9 / 41

  19. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  20. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  21. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  22. Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2 , τ ϵ ( P RC ) ⩽ 10 n 4 m 2 ( ln π RC ( x 0 ) − 1 + ln ϵ − 1 ) . For q > 2, there exists p such that P RC is slow mixing on complete graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015] For q > 2 and 0 < p < 1, it is #BIS-hard to approximate Z RC ( p , q ) . [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle. Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

  23. Swandsen-Wang algorithm Heng Guo (QMUL) Random Cluster 2016/11/03 11 / 41

  24. Ferromagnetic Ising model [Ising, Lenz 1925] Parameter β > 1. A configuration σ : V → { + , − } . π Ising ( σ ) ∝ β mono ( σ ) = β m − cut ( σ ) Partition function Z Ising ( β ) = ∑ σ β mono ( σ ) β 0 β 2 β 4 Heng Guo (QMUL) Random Cluster 2016/11/03 12 / 41

  25. Equivalence at q = 2 1 Let β = 1 − p . Z Ising ( β ) = β | E | Z RC ( p , 2 ) Heng Guo (QMUL) Random Cluster 2016/11/03 13 / 41

  26. Swendsen-Wang algorithm [Swendsen, Wang 1987] A global Markov chain to sample Ising configurations. Current configuration σ Mark all monochromatic edges under σ as M 1 Remove each edge in M with probability β − 1 (Recall β − 1 = 1 − p ) 2 Assign a random spin to each component of ( V , M ) 3 Practically very fast for the Ising model, but difficult to analyze. Conjectured to be rapidly mixing for all graphs. (Open problem since 90s.) Heng Guo (QMUL) Random Cluster 2016/11/03 14 / 41

  27. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  28. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  29. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  30. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  31. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

  32. Another simple example Activate mono edges 1 Re-randomize mono edges 2 Color components 3 Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

Recommend

Ising model with random boundary condition A. C. D. van Enter # , K. Neto y &amp; , and H. G.

Ising model with random boundary condition A. C. D. van Enter # , K. Neto y &amp; , and H. G.

Equilibrium and Dynamics of Spin Glasses Monte Verita, April 18-23, 2004 Ising model with random boundary condition A. C. D. van Enter # , K. Neto y &amp; , and H. G. Schaap # cn # Institute for Theoretical Physics R.U.G. Groningen, The

473 views • 9 slides
A power-law upper bound on the correlations in the two-dimensional random-field Ising model Ron

A power-law upper bound on the correlations in the two-dimensional random-field Ising model Ron

A power-law upper bound on the correlations in the two-dimensional random-field Ising model Ron Peled, Tel Aviv University Based on a joint work with Michael Aizenman, Princeton ICMP 2018, Montreal Random-field Ising model Standard Ising

447 views • 12 slides
Cluster algorithm for the Ising model Define bond index corresponding to pair of interacting spins

Cluster algorithm for the Ising model Define bond index corresponding to pair of interacting spins

Cluster algorithm for the Ising model Define bond index corresponding to pair of interacting spins Number of bonds N b = dN for a d-dimensional cubic lattice Write the energy of the Ising ferromagnet as Write the partition function as Define

303 views • 14 slides
Quantum Integer Programming 47-779 Ising Model 1 William Larimer Mellon, Founder Agenda o

Quantum Integer Programming 47-779 Ising Model 1 William Larimer Mellon, Founder Agenda o

Quantum Integer Programming 47-779 Ising Model 1 William Larimer Mellon, Founder Agenda o Ising Model: Background, Physics o Ising Model: Solutions o Ising Model and Combinatorial Optimization o Ising Model and Integer Programming o Solving

669 views • 19 slides
Random triangulations coupled with an Ising model Laurent M enard (Paris Nanterre) joint work

Random triangulations coupled with an Ising model Laurent M enard (Paris Nanterre) joint work

Random triangulations coupled with an Ising model Laurent M enard (Paris Nanterre) joint work with Marie Albenque and Gilles Schaeffer (CNRS and LIX) Bordeaux, November 2018 Outline 1. Introduction: 2DQG and planar maps 2. Local weak

1.1k views • 88 slides
Height representation of XOR-Ising loops via bipartite dimers C edric Boutillier (UPMC) B

Height representation of XOR-Ising loops via bipartite dimers C edric Boutillier (UPMC) B

Height representation of XOR-Ising loops via bipartite dimers C edric Boutillier (UPMC) B eatrice de Tili` ere (UPMC) LAGA Universit e Paris Nord March 11, 2015 The Ising model and the XOR-Ising model The Ising model Let G =

797 views • 43 slides
The critical Z -invariant Ising model via dimers B eatrice de Tili` ere University of

The critical Z -invariant Ising model via dimers B eatrice de Tili` ere University of

The critical Z -invariant Ising model via dimers B eatrice de Tili` ere University of Neuch atel joint work with C edric Boutillier, University Paris 6 Ascona, 27 May 2010 The 2 -dimensional Ising model Planar graph, G = ( V ( G )

600 views • 24 slides
1st PRIMA Congress Conformal invariance and universality in the 2D Ising model Stanislav Smirnov

1st PRIMA Congress Conformal invariance and universality in the 2D Ising model Stanislav Smirnov

1st PRIMA Congress Conformal invariance and universality in the 2D Ising model Stanislav Smirnov July 6, 2009 Archetypical example of a phase transition: 2D Ising model Configurations of + and spins (red and blue squares) Prob( config )

669 views • 31 slides
SLE and conformal invariance for critical Ising model Stanislav Smirnov jointly with Dmitry

SLE and conformal invariance for critical Ising model Stanislav Smirnov jointly with Dmitry

3rd LA PIETRA WEEK IN PROBABILITY Stochastic Models in Physics Firenze, June 23-27, 2008 SLE and conformal invariance for critical Ising model Stanislav Smirnov jointly with Dmitry Chelkak 1D Ising model 0 1

355 views • 20 slides
Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg)

Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg)

Crossing probabilities in the critical 2D Ising model Dmitry Chelkak (PDMI Steklov, St.Petersburg) joint work with Stanislav Smirnov (Geneva) arXiv:0910.2045 : Universality in the 2D Ising model and conformal invariance of fermionic

667 views • 42 slides
Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ENS,

Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ENS,

Planar Ising model at criticality: state-of-the-art and perspectives Dmitry Chelkak, ENS, Paris [ &amp; PDMI, St. Petersburg (on leave) ] ICM, Rio de Janeiro, August 4, 2018 Planar Ising model at criticality: outline Combinatorics

791 views • 37 slides
First order phase transition for the Random Cluster model with q &gt; 4 Ioan Manolescu joint work

First order phase transition for the Random Cluster model with q &gt; 4 Ioan Manolescu joint work

First order phase transition for the Random Cluster model with q &gt; 4 Ioan Manolescu joint work with: Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Vincent Tassion University of Fribourg 14th February 2017 Diablerets Ioan Manolescu

830 views • 81 slides
Monte Carlo simulation of the 2D Ising model - tutorial - Zoltn Nda Babe-Bolyai University

Monte Carlo simulation of the 2D Ising model - tutorial - Zoltn Nda Babe-Bolyai University

Monte Carlo simulation of the 2D Ising model - tutorial - Zoltn Nda Babe-Bolyai University Department of Theoretical and Computational Physics . A basic Metropolis Algorithm for simulating the 2D and 3D Ising model on square lattice

462 views • 8 slides
Improved Clustering Algorithms for the Random Cluster Graph Model Ron Shamir Dekel Tsur Tel

Improved Clustering Algorithms for the Random Cluster Graph Model Ron Shamir Dekel Tsur Tel

Improved Clustering Algorithms for the Random Cluster Graph Model Ron Shamir Dekel Tsur Tel Aviv University 1/18 The Clustering Problem Input: A graph G . (edges in G represent similarity between the vertices) Output: A partition of the

485 views • 34 slides
Ising Model and Percolation Problem Shahin Rouhani Physics Department Sharif University of

Ising Model and Percolation Problem Shahin Rouhani Physics Department Sharif University of

Statistical Mechanics of Two Dimensional Critical Curves Ising Model and Percolation Problem Shahin Rouhani Physics Department Sharif University of Technology Tehran, Iran. Scale invariance in the critical Ising model Let us illustrate

394 views • 27 slides
Convergence of correlations in the 2D Ising model: primary fields [ and the stress-energy tensor

Convergence of correlations in the 2D Ising model: primary fields [ and the stress-energy tensor

Convergence of correlations in the 2D Ising model: primary fields [ and the stress-energy tensor ] Dmitry Chelkak [ ENS, Paris &amp; Steklov Institute, St. Petersburg ] [ Sample of a critical 2D Ising configuration (with two disorders), c

1.24k views • 54 slides
The Prokofiev-Svistunov-Ising process is rapidly mixing Tim Garoni School of Mathematical

The Prokofiev-Svistunov-Ising process is rapidly mixing Tim Garoni School of Mathematical

Introduction Main Theorem Proof Discussion The Prokofiev-Svistunov-Ising process is rapidly mixing Tim Garoni School of Mathematical Sciences Monash University Introduction Main Theorem Proof Discussion Collaborators Andrea

1.84k views • 147 slides
Metastability in Stochastic Dynamics: Random-Field Curie-Weiss-Potts Model Prag summer school 1

Metastability in Stochastic Dynamics: Random-Field Curie-Weiss-Potts Model Prag summer school 1

05. September 2011 Martin Slowik Metastability in Stochastic Dynamics: Random-Field Curie-Weiss-Potts Model Prag summer school 1 (16) Metastability in stochastic dynamics Metastability: A common phenomenon The paradigm. Related to the

350 views • 21 slides
Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Toronto

Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Toronto

Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Toronto Colloquium, January 7, 2019 Joint work with Pavlo Pylyavskyy arXiv:1807.03282 Pavel Galashin (MIT) Ising model and total positivity U of Toronto,

2.08k views • 176 slides
Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Michigan,

Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Michigan,

Ising model and total positivity Pavel Galashin MIT galashin@mit.edu University of Michigan, October 19, 2018 Joint work with Pavlo Pylyavskyy arXiv:1807.03282 Pavel Galashin (MIT) Ising model and total positivity UMich, 10/19/2018 1 / 32

2.05k views • 186 slides
2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent Charles M.

2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent Charles M.

2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent Charles M. Newman Courant Institute of Mathematical Sciences newman @ courant.nyu.edu Based on joint work with Federico Camia and Christophe Garban. Ising

604 views • 23 slides
Ising models on power-law random graphs Sander Dommers Joint work with: Cristian Giardin

Ising models on power-law random graphs Sander Dommers Joint work with: Cristian Giardin

Ising models on power-law random graphs Sander Dommers Joint work with: Cristian Giardin Remco van der Hofstad Prague School on Mathematical Statistical Physics August 29September 9, 2011 Where innovation starts Introduction 2/18

644 views • 26 slides
The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal &amp; Stephen

The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal &amp; Stephen

The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal &amp; Stephen Sorokanich August 28, 2018 Mini Physics Intro Course Magnets and spins, magnetization Ferromagnetism Spins under an external magnetic field Ising

286 views • 12 slides
WHY? 1 26/01/18 1. Can be used to simulate molecular motion in a viscous medium, without

WHY? 1 26/01/18 1. Can be used to simulate molecular motion in a viscous medium, without

26/01/18 ADVANCED TECHNIQUES (MC/MD) A (seemingly) random selection. Daan Frenkel Beyond Newtonian MD 1. Langevin dynamics 2. Brownian dynamics 3. Stokesian dynamics 4. Dissipative particle dynamics 5. Etc. etc. WHY? 1 26/01/18 1. Can be used

994 views • 48 slides

More recommend