Courant Institute (NYU) Wilhelm Lenz Introduced by Wilhelm Lenz in 1920 1888-1957 as a model of ferromagnetism: Place iron in a magnetic field: increase field to maximum , then slowly reduce it to zero.
There is a critical temperature 푇푐 (the Curie point) below which the iron retains residual magnetism. Magnetism caused by charged particles spinning or moving in orbit in alignment with each other. How do local interactions between nearby particles affect the global behavior at different temperatures?
Eyal Lubetzky, Courant Institute Gives random binary values (spins) to vertices accounting for nearest-neighbor interactions. Initially thought to be over-simplified Sociology Biology to capture ferromagnetism, but Chemistry Economics turned out to have a crucial role in Physics … understanding phase transitions & critical phenomena. One of the most studied models in Math. Phys.: more than 10,000 papers About 13,300 results over the last 30 years…
Eyal Lubetzky, Courant Institute Cyril Domb Proposed in 1951 by C. Domb to his 1920-2012 Ph.D. student R. Potts. Generalization of the Ising model to allow 푞 > 2 states per site. Special case 푞 = 4 was first studied in 1943 by Ashkin and Teller. Renfrey Potts Rich critical phenomena: 1925-2005 first/second order phase transitions depending on 푞 and the dimension.
figure taken from: The Potts model FY Wu – Reviews of modern physics, 1982 Cited by 2964
Eyal Lubetzky, Courant Institute Underlying geometry: Λ = finite 2D grid. Set of possible configurations: + + + - Λ Ω = ±1 - + + + (each site receives a plus/minus spin) - + - + Probability of a configuration 휎 ∈ Ω given by the Gibbs distribution: + - + + 1 휇 휎 = exp 1훽 휎 푥 휎(푦) I 푍 훽 2 푥∼푦
Inverse Partition Josiah W. Gibbs temperature 1839-1903 function 훽 ≥ 0
Eyal Lubetzky, Courant Institute Underlying geometry: Λ = finite 2D grid. Set of possible configurations: Ω = 1, … , 푞 Λ (each site receives a color) Probability of a configuration 휎 ∈ Ω given by the Gibbs distribution: 1 휇 휎 = exp 훽 핝 P 푍 훽 휎 푥 =휎 푦 푥∼푦
Inverse Partition Josiah W. Gibbs temperature 1839-1903 function 훽 ≥ 0
Eyal Lubetzky, Courant Institute MCMC sampler introduced in 1963 by Roy Glauber (Nobel in Physics 2005). Time-dependent statistics of the Ising model RJ Glauber – Journal of mathematical physics, 1963 Cited by 3186 R.J. Glauber One of the most commonly used
samplers for the Ising/Potts model: + + Update sites via IID Poisson(1) clocks - + - Each update replaces a spin at 푥 ∈ 푉 - - by a new spin ∼ 휇 given spins at 푉 ∖ {푥}. (heat-bath version; famous other flavor: Metropolis) + + + + - + - - +- - Time it takes to converge to 휇 ? - - - -
Eyal Lubetzky, Courant Institute 1 σ Λ 휇I 휎 ∝ exp 2훽 푥∼푦 휎 푥 휎(푦) for 휎 ∈ Ω = ±1 Larger 훽 favors configurations with + + + - aligned spins at neighboring sites. - + + + Spin interactions: local, justified by rapid - + - + decay of magnetic force with distance. + - + +
The magnetization is the (normalized) sum of spins: 1 푀 휎 = 휎(푥) Λ 푥∈Λ Distinguishes between disorder (푀 ≈ 0) and order. Symmetry: 피 푀 휎 = 0. What if we break the symmetry?
Eyal Lubetzky, Courant Institute 1 [recall 푀 휎 = σ 휎(푥)] Ferromagnetism in this setting: Λ Condition on the boundary sites + + + + all having plus spins. + + + + + + - + Let the system size Λ tend → ∞ + + + + (≈ a magnetic field with effect → 0). What is the typical 푀(휎) for large Λ ? Does the effect of plus boundary vanish in the limit?
Eyal Lubetzky, Courant Institute 1 [recall 푀 휎 = σ 휎(푥)] Ferromagnetism in this setting: Λ Condition on the boundary sites + + + + all having plus spins. + + + + + + - + Let the system size Λ tend → ∞ + + + +
Expect: phase-transition at some critical 훽푐:
0 if 훽 < 훽푐 lim 피+ 푀(휎) = ቐ Λ →∞ 푚훽 > 0 if 훽 > 훽푐
all-plus spontaneous boundary magnetization
Eyal Lubetzky, Courant Institute The magnetization +1 phase-transition at 훽푐 : 0 훽푐 훽 Replace magnetization ↭ price to find this diagram in “Why Stock Markets Crash” / D. Sornette (2001) [Chapter 5 “Modeling bubbles and crashes”]
Such applications of the Ising Model emphasize the dimension of time: D. Sornette How does the system evolve? From a given starting state, how long does it take for certain configurations to appear?
Eyal Lubetzky, Courant Institute Bacteria growth with mutation modelled by an evolution of a Potts-type interface.
K.S. Korolev, O.M. Avlund, O. Hallatschek, and D.R. Nelson, Genetic demixing and evolution in linear stepping stone models. Reviews of Modern Physics 82(2) (2010), 1691–1718.
Eyal Lubetzky, Courant Institute Behavior for the Ising distribution:
+ + 피 푀(휎) 0 피 푀(휎) 푐훽 > 0 Λ →∞ Λ →∞
Expected behavior for the inverse-gap of dynamics:
power constant exponential Torus/free law Symmetry breaking: + power law Plus b.c. + + + + + + + + + + - + + + + + Eyal Lubetzky, Courant Institute Static behavior for the Potts distribution:
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Expected behavior for the inverse-gap of dynamics: 훽푐(푞) exponential
Torus
constant power law 푞
Eyal Lubetzky, Courant Institute Ph.D. in Physics in 1924 from U. Hamburg under the supervision of Lenz. Studied the 1D model of Lenz in his thesis:
Exact solution for the 1D model. Ernst Ising 1900-1998 Unfortunately: no phase-transition… . 피+ 푀(휎) 0 for any 훽 ≥ 0. Λ →∞ . Intuition: ( + + + + + + + + ) similar to (+ + + + - - - - ) Heuristic arguments why there would not be a phase-transition in higher dimensions either.
Eyal Lubetzky, Courant Institute Ising [letter to S. Brush in 1967]: ”...I discussed the result of my paper widely with Prof. Lenz and with Dr. Wolfgang Pauli, who at that time was teaching in Hamburg. There was some disappointment that the linear model did not show the expected ferromagnetic properties...” Left research after a few years at the German General Electric Co. and turned to teaching in public schools. Survived WW2 in Luxembourg isolated from scientific life. Came to the US in 1947 and only then “...did I learn that the idea had been expanded.”
Eyal Lubetzky, Courant Institute Heisenberg (1928) proposed his own theory of ferromagnetism, motivated by Ising’s result. Followed by other models attempting to W. Heisenberg explain order/disorder in metallic alloys. 1901-1976 In 1936 Rudolf Peierls published the paper
R. Peierls 1907-1995 arguing that the 2D and 3D Ising models do have spontaneous magnetization at low enough temperature (contrary to Ising’s prediction).
Eyal Lubetzky, Courant Institute Peierls’ combinatorial argument is simple and robust.
Key idea: represent Ising configurations as contours in the dual graph: the edges are dual to disagreeing edges.
Eyal Lubetzky, Courant Institute When all boundary spins are + ‘s the Peierls contours are all closed [marking “islands” containing of - ‘s ].
The proof will follow a first moment argument on the number of sites inside such a - component.
Eyal Lubetzky, Courant Institute Setting: Λ ⊂ ℤ2 is an 푛 × 푛 box with all-plus boundary. Fix a contour 퐶 of length ℓ. For any 휎 containing 퐶 flip all spins in the interior of 퐶 :
bijection
1 1 ℙ 휎 = 푒퐵 ℙ 휎′ = 푒퐵+훽ℓ 푍 훽 푍 훽 ⇒ ℙ C belongs to contours ≤ 푒−훽ℓ.
Eyal Lubetzky, Courant Institute For a fixed contour 퐶 of length ℓ : ⇒ ℙ C belongs to contours ≤ 푒−훽ℓ. 퐶 can contain ≤ ℓ2 sites (isoperimetric).
푂(푁 3ℓ) possible such contours, where 푁 = Λ = 푛2.
Summing we get: ℓ 피 # 푥 ∶ 휎 푥 = −1 ≲ 푁 ℓ2 3푒−훽 < 휀푁 ℓ where 휀 < 1/2 for a suitably large 훽.
Eyal Lubetzky, Courant Institute Critical point candidate 훽푐 = log 1 + 2 ≈ 0.88 found by Kramers and Wannier in 1941 via duality. 2D Ising model exactly solved in 1944 in seminal work of Lars Onsager (Nobel in Chemistry 1968). Proof analyzed the 2푛 × 2푛 transfer matrix using the theory of Lie algebras. L. Onsager 1903-1976 Understanding of critical geometry boosted by advent of SLE [Schramm ’00] and breakthrough results of Smirnov. O. Schramm S. Smirnov 1961-2008 In parallel: extensive study of the dynamical model...
Eyal Lubetzky, Courant Institute Glauber dynamics dynamics (1963): + + Update sites via IID Poisson(1) clocks - + - Each update replaces the state of 푥 ∈ 푉 by a new 푆 ∼ 휇P(⋅ ∣ other states of 푉 ∖ 푥 ). - - Local: ℙ S = 푠 ∝ exp 훽 # 푦 ∼ 푥: 휎 푦 = 푠 Advantage: easy to simulate. + + + + Drawback: susceptible to - + - - +- - bottlenecks in energy landscape. - - - - 1 1 푒퐵+ퟐ휷 푒퐵+ퟒ휷 푍 훽 푍 훽 Swendsen–Wang dynamics (1987) : Draw percolation clusters as per the Fortuin–Kasteleyn representation (1969) of the Potts model. Let each cluster choose its color i.i.d. uniformly. Global (able to treat the Ising model double-well).
Eyal Lubetzky, Courant Institute Spectral gap in the spectrum of the generator:
gap = smallest positive eigenvalue of the heat-kernel 퐻푡 of the dynamics. ℰ(푓,푓) = inf for ℰ 푓, 푓 the Dirichlet form. Var휇(푓)
Mixing time (according to a given metric): Standard choice: 퐿1 (total-variation) mixing time to within distance 휀 is defined as
푡 푡mix 휀 = inf 푡 ∶ max 푝 푥0,⋅ − 휇 ≤ 휀 푥0 tv
where 휇 − 휈 tv = sup 휇 퐴 − 휈(퐴) 퐴⊂Ω
Eyal Lubetzky, Courant Institute For some critical inverse-temperature 훽푐: free or periodic b.c.
푑−1 gap−1: 푂(1) ≍ 푛푧 ≍ 푒휏훽 푛 universal
Analogous picture verified for: Ising model on complete graph [Ding, L., Peres ‘09a, ‘09b], [Levin, Luczak, Peres ’10] Regular tree [Berger, Kenyon, Mossel, Peres ‘05], [Ding, L., Peres ’10]
Potts model on complete graph [Cuff, Ding, L., Louidor, Peres, Sly ‘12], [Galanis, Stefankovic, Vigoda ‘15] , [Blanca, Sinclair ‘15]
Eyal Lubetzky, Courant Institute Intuition: low temperature Exponential mixing due to a bottleneck between the “mostly-plus” and the “mostly-minus” states Intuition: high temperature Stationary magnetization At 훽 = 0 there is complete independence. For very small 훽 > 0 a spin is likely to choose the same update given 2 very different neighborhoods (weak “communication” between sites). States can be coupled quickly, hence rapid mixing. Intuition: critical power-law Doubling the box incurs a constant factor in mixing...
Eyal Lubetzky, Courant Institute Fast mixing at high temperatures: . [Aizenman, Holley ’84] . [Dobrushin, Shlosman ’87] . [Holley, Stroock ’87, ’89] . [Holley ’91] . [Stroock, Zegarlinski ’92a, ’92b, ’92c] gap−1 = 푂(1) . [Lu, Yau ’93] . [Martinelli, Olivieri ’94a, ‘94b] 푡mix = 푂(log 푛) . [Martinelli, Olivieri, Schonmann ’94] Slow mixing at low temperatures: free or periodic b.c. . [Schonmann ’87] . [Chayes, Chayes, Schonmann ’87] 푑−1 gap−1 = 푒(휏훽+표 1 )푛 . [Martinelli ’94] . [Cesi, Guadagni, Martinelli, Schonmann ’96] Critical power-law? . lower bound: [Aizenman, Holley ’84], [Holley ’91] gap−1 ≥ 푛푐 . simulations: [Ito ’93], [Wang, Hatano, Suzuki ’95], sim: 푛2.17… [Grassberger ’95], [Nightingale, Blöte ’96], [Wang, Hu ’97],… Eyal Lubetzky, Courant Institute free or periodic b.c. −ퟏ 푐 푑−1 퐠퐚퐩 : 푂(1) ≥ 푛 푒(휏훽+표 1 )푛 sim: 푛2.17…
Further progress ([L., Sly ‘12]): power-law at 훽푐: gap−1 ≤ 푛퐶
analysis used SLE-type behavior of critical interfaces due to [Duminil-Copin, Hongler, Nolin ’09] (cf. [Chelkak, Smirnov ‘09], [Camia, Newman ’09]).
Eyal Lubetzky, Courant Institute Believed to have the same critical slowdown phenomenon as the Ising model in the continuous phase transition regime,
vs. slow critical mixing elsewhere. figure from: 2 The Potts model On ℤ푛, this corresponds to: FY Wu – Reviews of modern physics, 1982 Cited by 2964 Conjectured: 푛푧 inverse-gap at criticality for 1 < 푞 ≤ 4. Conjectured: exp(푐 푛) inverse-gap at criticality for 푞 > 4. For 푞 large enough: [Borgs, Chayes, Frieze, Kim, Tetali, Vu ’99] (refined in [Borgs, Chayes, Tetali ‘12]): gap−1 = exp 푛1+표 1
Eyal Lubetzky, Courant Institute 푞 = 3 푞 = 4 푞 = 5
훽 훽푐(푞)
푞
Eyal Lubetzky, Courant Institute THEOREM: ([Gheissari, L.]) Glauber dynamics and Swendsen–Wang dynamics for the critical Potts model on the torus ℤ/푛ℤ 2 both satisfy: (a) 푞 = 3: gap−1 ≤ 푛퐶. (b) 푞 = 4: gap−1 ≤ 푛퐶 log 푛. 0 1 −1 (c) 푞 ≥ 5, 휋ℤ2 ≠ 휋ℤ2: gap ≥ exp(푐 푛).
Proofs build on ideas from [L., Sly ‘12] and recent results by [Duminil-Copin, Sidoravicius, Tassion ’15].
Eyal Lubetzky, Courant Institute Establish a bottleneck w.r.t. six crossings:
1 use exponential decay of correlations in 휋ℤ2 due to [Duminil-Copin, Sidoravicius, Tassion ‘15].
Eyal Lubetzky, Courant Institute Analysis of Potts model at 훽 = 훽푐 extends to the regime 훽 > 훽푐, where the exponentially slow gap−1 behavior of was established 훽 훽푐(푞) for sufficiently large 푞 by [Borgs,Chayes, Frieze, Kim, Tetali, Vu ’99] [Borgs, Chayes, Tetali ’12]
THEOREM: ([Gheissari, L.]) 푞 Glauber dynamics for the Potts model on the torus ℤ/푛ℤ 2 at 푞 > 1 and 훽 > 훽푐 has gap−1 ≥ exp 푐 푛 .
Eyal Lubetzky, Courant Institute free ≡ periodic ???
Unlike low temperature Ising, free boundary conditions do not induce the same slow mixing behavior as the torus. THEOREM: ([Gheissari, L.])
Fix 푞 large enough. Swendsen–Wang for the critical Potts model on an 푛 × 푛 box with free boundary conditions has gap−1 ≤ exp 푛표 1 , whereas Glauber dynamics has gap−1 ≤ exp 푛1/2+표 1 .
Proofs build on cluster expansion and a powerful multi-scale approach due to [Martinelli, Toninelli ‘11].
Eyal Lubetzky, Courant Institute Free boundary Periodic boundary conditions conditions
Eyal Lubetzky, Courant Institute Eyal Lubetzky, Courant Institute