Statistical Physics on Sparse Random Graphs: Mathematical Perspective

Transcription

1 Statistical Physics on Sparse Random Graphs: Mathematical Perspective Amir Dembo Stanford University Northwestern, July 19, 2016

2 x 5 x 6 Factor model [DM10, Eqn. (1.4)] x 1 x 2 x 3 x 4 x 9 x8 x 7 x 10 x 11 x 12 G = (V, E), V = [n], x = (x 1,..., x n ), x i X (finite set). µ(x) = 1 Z G (ij) E ψ(x i, x j ) i V ψ(x i ).

3 Ising models

4 Ising models [DM10, Eqn. (1.20)] G n = (V n [n], E n ), finite, non-directed graphs. x i {+1, 1} µ(x) = 1 Z n (β, B) exp β x i x j + B (ij) E n n i=1 x i β 0 Ferromagnetic, β < 0 Anti-ferromagnetic, B 0 wlog

5 Potts models [DMS13, Eqn. (1.3)] G n = (V n [n], E n ), finite, non-directed graphs. x i [q] {1, 2,..., q} µ(x) = 1 Z n (β, B) (ij) E n e β1{xi =xj } n i=1 distribution on spin configurations x [q] V. Interaction strength parametrized by β; e B1{x i =1} May add external field B towards distinguished spin 1 Natural generalization of Ising models (q = 2);

6 Potts models [DMS13, Eqn. (1.3)] G n = (V n [n], E n ), finite, non-directed graphs. x i [q] {1, 2,..., q} µ(x) = 1 Z n (β, B) (ij) E n e β1{xi =xj } n i=1 distribution on spin configurations x [q] V. Interaction strength parametrized by β; e B1{x i =1} May add external field B towards distinguished spin 1 Natural generalization of Ising models (q = 2);

7 Potts models [DMS13, Eqn. (1.3)] G n = (V n [n], E n ), finite, non-directed graphs. x i [q] {1, 2,..., q} µ(x) = 1 Z n (β, B) (ij) E n e β1{xi =xj } n i=1 distribution on spin configurations x [q] V. Interaction strength parametrized by β; e B1{x i =1} May add external field B towards distinguished spin 1 Natural generalization of Ising models (q = 2);

8 Potts models [DMS13, Eqn. (1.3)] G n = (V n [n], E n ), finite, non-directed graphs. x i [q] {1, 2,..., q} µ(x) = 1 Z n (β, B) (ij) E n e β1{xi =xj } n i=1 distribution on spin configurations x [q] V. Interaction strength parametrized by β; e B1{x i =1} May add external field B towards distinguished spin 1 Natural generalization of Ising models (q = 2);

9 Constraint Satisfaction Problem: q-coloring G = (V, E) graph. x = (x 1, x 2,..., x n ), x i {1,..., q} variables

10 Constraint Satisfaction Problem: q-coloring G = (V, E) graph. x = (x 1, x 2,..., x n ), x i {1,..., q} variables

11 Uniform measure over proper colorings [DM10, Eqn. (1.22)] µ(x) = 1 Z G (i,j) E ψ(x i, x j ), ψ(x, y) = I(x y). Z G counts number of proper q-colorings of G. AF Potts at β =, B = 0

12 The hard-core model [SS14] µ(x) = 1 Z G (λ) (ij) E(1 x i x j ) i V supported on x {0, 1} V encoding independent sets (no neighbors occupied) Occupied vertices are weighted by a fugacity λ Z G (1) counts independent sets of G; λ = for largest indepednent set of G. λ x i

13 The hard-core model [SS14] µ(x) = 1 Z G (λ) (ij) E(1 x i x j ) i V supported on x {0, 1} V encoding independent sets (no neighbors occupied) Occupied vertices are weighted by a fugacity λ Z G (1) counts independent sets of G; λ = for largest indepednent set of G. λ x i

14 The hard-core model [SS14] µ(x) = 1 Z G (λ) (ij) E(1 x i x j ) i V supported on x {0, 1} V encoding independent sets (no neighbors occupied) Occupied vertices are weighted by a fugacity λ Z G (1) counts independent sets of G; λ = for largest indepednent set of G. λ x i

15 Free energy density [DM10, Eqn. (2.1)] Graph sequences G n = ([n], E n ) Z n x ψ(x i, x j ) (ij) E n n i=1 ψ(x i ) Find asymptotic free energy density φ lim n n 1 log Z n (if limit exists);

16 Free energy density [DM10, Eqn. (2.1)] Graph sequences G n = ([n], E n ) Z n x ψ(x i, x j ) (ij) E n n i=1 ψ(x i ) Find asymptotic free energy density φ lim n n 1 log Z n (if limit exists);

17 Free energy density [DM10, Eqn. (2.1)] Graph sequences G n = ([n], E n ) Z n x ψ(x i, x j ) (ij) E n n i=1 ψ(x i ) Find asymptotic free energy density φ lim n n 1 log Z n (if limit exists);

18 Random graph models [DM10, Sec ] Erdős-Renyi Random graph G = (V, E) G n,m V = n vertices Uniform among graphs of m = E edges (or edges independently present with probability p = m/ ( n 2) ) Average degree d = 2m/n Random regular graph G = (V, E) G n,d V = n vertices Uniformly random among graphs with i = d i V. d = O(1) for (uniformly) sparse graphs

19 Random graph models [DM10, Sec ] Erdős-Renyi Random graph G = (V, E) G n,m V = n vertices Uniform among graphs of m = E edges (or edges independently present with probability p = m/ ( n 2) ) Average degree d = 2m/n Random regular graph G = (V, E) G n,d V = n vertices Uniformly random among graphs with i = d i V. d = O(1) for (uniformly) sparse graphs

20 Random graph models [DM10, Sec ] Erdős-Renyi Random graph G = (V, E) G n,m V = n vertices Uniform among graphs of m = E edges (or edges independently present with probability p = m/ ( n 2) ) Average degree d = 2m/n Random regular graph G = (V, E) G n,d V = n vertices Uniformly random among graphs with i = d i V. d = O(1) for (uniformly) sparse graphs

21 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

22 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

23 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

24 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

25 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

26 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

27 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

28 Locally tree-like graphs: regular limit Consider graphs that are sparse ( E n V n ) and locally tree-like the local neighborhood of a uniformly random vertex converges in distribution to a (random) rooted tree the random d-regular graph G n,d converges (locally) to the infinite d-regular tree T d, denoted G n,d T d

29 UGW T(P, ρ, t) P k ρ k+1

30 UGW T(P, ρ, t) P k ρ k+1

31 UGW T(P, ρ, t) P k ρ k+1

32 UGW T(P, ρ, t) P k ρ k+1

33 UGW T(P, ρ, t) P k ρ k+1

34 Uniformly sparse, locally tree-like graphs: GW limit The Unimodular Galton-Watson tree: P {P k } k 0 Degree distribution, law of L (of finite mean P) ρ { k P P k} k 0 Law of K (degree of uniform edge) T(P, ρ, t) t-generations UGW tree (root degree P, else ρ) B i (t) Ball of radius t in G n centered at node i Definition (DM10, Dfn. 2.1) {G n } converges locally to T(P, ρ, ) if for uniformly random I n [n] and fixed t, law of B In (t) converges as n to T(P, ρ, t). {G n } uniformly sparse if { I n } is uniformly integrable. see Benjamini Schramm 01, Aldous Lyons 07

35 Tree-like graphs: examples Random regular graph G n,d regular tree T d Sparse Erdős Rényi graph G n,nd/2 T(P, ρ, ) with P = ρ Poisson(d) [DM10, Prop. 2.6] Graphs of fixed degree distribution P T(P, ρ, ) [DM10, Prop. 2.5]

36 Tree-like graphs: examples Random regular graph G n,d regular tree T d Sparse Erdős Rényi graph G n,nd/2 T(P, ρ, ) with P = ρ Poisson(d) [DM10, Prop. 2.6] Graphs of fixed degree distribution P T(P, ρ, ) [DM10, Prop. 2.5]

37 Tree-like graphs: examples Random regular graph G n,d regular tree T d Sparse Erdős Rényi graph G n,nd/2 T(P, ρ, ) with P = ρ Poisson(d) [DM10, Prop. 2.6] Graphs of fixed degree distribution P T(P, ρ, ) [DM10, Prop. 2.5]

38 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

39 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

40 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

41 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

42 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

43 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

44 Tree-like graphs: motivation Encompasses natural random graph ensembles such as sparse Erdős Rényi graphs or random regular graphs; Natural randomized computational problems also described by a locally tree-like constraint structure (random hard-core, q-coloring) Trees are amenable to exact analysis (recursive equations) physicists predict many exact asymptotic results for the setting G n T, based on comparisons between models on G n and on T The Bethe replica symmetric prediction for φ in graphs G n T : φ = Φ Bethe explicit which depends only on the limit tree T

45 Mean field equations µ i j ( ) Marginal of x i on without edge (i, j) replacing factor ψ(x i, x j ) by 1 Belief propagation, Bethe-Peierls (BP) equations (Replica Symmetric cavity method): µ i j (x i ) 1 z i j ψ(x i ) l i\j x l ψ(x i, x l )µ l i (x l ) Exact if G = T a tree; approximate marginals of µ( ) by measures on trees. [DM10, Dfn. 3.4 & Prop. 3.7]

46 Mean field equations µ i j ( ) Marginal of x i on without edge (i, j) replacing factor ψ(x i, x j ) by 1 Belief propagation, Bethe-Peierls (BP) equations (Replica Symmetric cavity method): µ i j (x i ) 1 z i j ψ(x i ) l i\j x l ψ(x i, x l )µ l i (x l ) Exact if G = T a tree; approximate marginals of µ( ) by measures on trees. [DM10, Dfn. 3.4 & Prop. 3.7]

47 Mean field equations µ i j ( ) Marginal of x i on without edge (i, j) replacing factor ψ(x i, x j ) by 1 Belief propagation, Bethe-Peierls (BP) equations (Replica Symmetric cavity method): µ i j (x i ) 1 z i j ψ(x i ) l i\j x l ψ(x i, x l )µ l i (x l ) Exact if G = T a tree; approximate marginals of µ( ) by measures on trees. [DM10, Dfn. 3.4 & Prop. 3.7]

48 Bethe-Peierls approximation [DM10, Dfn. 3.10] u(i) i A tree T = (U, E U ) G with i U or i U = {u(i)} µ U (x U ) ν U (x U ) = 1 Z U ψ(x i, x j ) ψ(x i ) h i u(i) (x i ). (i,j) E U i / U i U

49 Bethe Gibbs measures [DMS13, Eqn. (1.6)-(1.12)] On G n T, Φ Bethe = Φ(ν ), with Φ an explicit function, and ν a certain infinite-volume Gibbs measure on T represents proposed local limit µ n ν of models µ n on G n. The replica symmetric prediction assumes special structure for ν : marginal on any finite subgraph U T given by ν (x U ) ( weight of x ) = U h (x within U v ) v U Bethe Gibbs measure with entrance law h Consistency mandates fixed-point relations for h (BP equations) Φ Bethe is equivalently defined in terms of BP solutions h

50 Bethe Gibbs measures [DMS13, Eqn. (1.6)-(1.12)] On G n T, Φ Bethe = Φ(ν ), with Φ an explicit function, and ν a certain infinite-volume Gibbs measure on T represents proposed local limit µ n ν of models µ n on G n. The replica symmetric prediction assumes special structure for ν : marginal on any finite subgraph U T given by ν (x U ) ( weight of x ) = U h (x within U v ) v U Bethe Gibbs measure with entrance law h Consistency mandates fixed-point relations for h (BP equations) Φ Bethe is equivalently defined in terms of BP solutions h

51 Bethe Gibbs measures [DMS13, Eqn. (1.6)-(1.12)] On G n T, Φ Bethe = Φ(ν ), with Φ an explicit function, and ν a certain infinite-volume Gibbs measure on T represents proposed local limit µ n ν of models µ n on G n. The replica symmetric prediction assumes special structure for ν : marginal on any finite subgraph U T given by ν (x U ) ( weight of x ) = U h (x within U v ) v U Bethe Gibbs measure with entrance law h Consistency mandates fixed-point relations for h (BP equations) Φ Bethe is equivalently defined in terms of BP solutions h

52 Bethe Gibbs measures [DMS13, Eqn. (1.6)-(1.12)] On G n T, Φ Bethe = Φ(ν ), with Φ an explicit function, and ν a certain infinite-volume Gibbs measure on T represents proposed local limit µ n ν of models µ n on G n. The replica symmetric prediction assumes special structure for ν : marginal on any finite subgraph U T given by ν (x U ) ( weight of x ) = U h (x within U v ) v U Bethe Gibbs measure with entrance law h Consistency mandates fixed-point relations for h (BP equations) Φ Bethe is equivalently defined in terms of BP solutions h

53 Bethe Gibbs measures [DMS13, Eqn. (1.6)-(1.12)] On G n T, Φ Bethe = Φ(ν ), with Φ an explicit function, and ν a certain infinite-volume Gibbs measure on T represents proposed local limit µ n ν of models µ n on G n. The replica symmetric prediction assumes special structure for ν : marginal on any finite subgraph U T given by ν (x U ) ( weight of x ) = U h (x within U v ) v U Bethe Gibbs measure with entrance law h Consistency mandates fixed-point relations for h (BP equations) Φ Bethe is equivalently defined in terms of BP solutions h

54 Bethe Gibbs measures [DMS13, Eqn. (1.6)-(1.12)] On G n T, Φ Bethe = Φ(ν ), with Φ an explicit function, and ν a certain infinite-volume Gibbs measure on T represents proposed local limit µ n ν of models µ n on G n. The replica symmetric prediction assumes special structure for ν : marginal on any finite subgraph U T given by ν (x U ) ( weight of x ) = U h (x within U v ) v U Bethe Gibbs measure with entrance law h Consistency mandates fixed-point relations for h (BP equations) Φ Bethe is equivalently defined in terms of BP solutions h

55 Bethe free energy density for T d [DMSS14, Dfn ] If G n T d entrance law h in simplex H of prob. measures on X, with h H set of fixed points of mapping BP : H H: (BPh)(x) 1 ( d 1, ψ(x) ψ(x, y)h(y)) x X z h (z h the normalizing constant) y The Bethe functional { ( ) } d Φ(h) = log ψ(x) ψ(x, y)h(y) d { 2 log x y x,y }{{} vertex term Φ vx (h) Φ Bethe sup{φ(h) : h H }. } ψ(x, y)h(x)h(y) } {{ } edge term Φ e (h).

56 Interpolation in phase-space: Ferromagnetic Ising [DM10, Thm. 2.15] For β 0, B > 0 if G n T then φ = Φ Bethe (h + ). Proof outline. 0. B = 0 given by limit B 0, so fix B > Reduce to expectations of local quantities d db log Z n(β, B) = n x i n i=1 ( n denote expectation under Ising on G n ). 2. Prove convergence of local expectations to tree values.

57 2. Convergence to tree values o T(t) t z 1 z 2 z 3 T infinite tree with max degree d max T(t) first t generations µ t,z ( ) Ising model on T(t) boundary condition z root spin expectation µ t,z o

58 2. Convergence to tree values o T(t) t z 1 z 2 z 3 T infinite tree with max degree d max T(t) first t generations µ t,z ( ) Ising model on T(t) boundary condition z root spin expectation µ t,z o

59 2. Convergence to tree values o T(t) t z 1 z 2 z 3 T infinite tree with max degree d max T(t) first t generations µ t,z ( ) Ising model on T(t) boundary condition z root spin expectation µ t,z o

60 Decorrelation Uniform (Gibbs measure uniqueness) µ t,z(1) o µ t,z(2) o µ t,+ o µ t, o 0. Easier True only at high temperature (β < β c )

61 Decorrelation Uniform (Gibbs measure uniqueness) µ t,z(1) o µ t,z(2) o µ t,+ o µ t, o 0. Easier True only at high temperature (β < β c )

62 Decorrelation Uniform (Gibbs measure uniqueness) µ t,z(1) o µ t,z(2) o µ t,+ o µ t, o 0. Easier True only at high temperature (β < β c )

63 Why non-uniform control? Phase transition... For β > β c atanh(1/ρ) lim B 0+ lim n E x I n = lim B 0 lim n E x I n > 0

64 ... and its tree counterpart o T(ρ, t) t z 1 z 2 z 3 z = (+1, +1,..., +1) lim x o t > 0 t z = ( 1, 1,..., 1) lim x o t < 0 t

65 Decorrelation Non uniform µ t,+ o µ t,free o 0. Any temperature

66 Trick 0 µ t,+ o µ t,free o ɛ{µ t,free o µ t 1,free o } 0 (µ t,free o monotone by Griffiths Inequality [DM10, Prop. 2.11]) [Ising specific, but strategy extends to Potts, Hard-core models]

67 Bethe Gibbs measures: Non-uniqueness A major challenge in verifying the Bethe prediction is the non-uniqueness of infinite-volume Gibbs measures Ising already has non-uniqueness, but the correct measure can be isolated by monotonicity considerations Situation is qualitatively different in q 3 Potts models, where monotonicity considerations do not close the gap If multiple Gibbs measures, physics prescription is to select Bethe Gibbs measure achieving highest value of Φ Bethe, Φ Bethe = sup{φ(ν ) : ν Bethe Gibbs measure} (restriction to Bethe Gibbs measures is an assumption) [dmss14] prove such prediction holds (within one non-trivial example).

68 Bethe Gibbs measures: Non-uniqueness A major challenge in verifying the Bethe prediction is the non-uniqueness of infinite-volume Gibbs measures Ising already has non-uniqueness, but the correct measure can be isolated by monotonicity considerations Situation is qualitatively different in q 3 Potts models, where monotonicity considerations do not close the gap If multiple Gibbs measures, physics prescription is to select Bethe Gibbs measure achieving highest value of Φ Bethe, Φ Bethe = sup{φ(ν ) : ν Bethe Gibbs measure} (restriction to Bethe Gibbs measures is an assumption) [dmss14] prove such prediction holds (within one non-trivial example).

69 Bethe Gibbs measures: Non-uniqueness A major challenge in verifying the Bethe prediction is the non-uniqueness of infinite-volume Gibbs measures Ising already has non-uniqueness, but the correct measure can be isolated by monotonicity considerations Situation is qualitatively different in q 3 Potts models, where monotonicity considerations do not close the gap If multiple Gibbs measures, physics prescription is to select Bethe Gibbs measure achieving highest value of Φ Bethe, Φ Bethe = sup{φ(ν ) : ν Bethe Gibbs measure} (restriction to Bethe Gibbs measures is an assumption) [dmss14] prove such prediction holds (within one non-trivial example).

70 Bethe Gibbs measures: Non-uniqueness A major challenge in verifying the Bethe prediction is the non-uniqueness of infinite-volume Gibbs measures Ising already has non-uniqueness, but the correct measure can be isolated by monotonicity considerations Situation is qualitatively different in q 3 Potts models, where monotonicity considerations do not close the gap If multiple Gibbs measures, physics prescription is to select Bethe Gibbs measure achieving highest value of Φ Bethe, Φ Bethe = sup{φ(ν ) : ν Bethe Gibbs measure} (restriction to Bethe Gibbs measures is an assumption) [dmss14] prove such prediction holds (within one non-trivial example).

71 Bethe Gibbs measures: Non-uniqueness A major challenge in verifying the Bethe prediction is the non-uniqueness of infinite-volume Gibbs measures Ising already has non-uniqueness, but the correct measure can be isolated by monotonicity considerations Situation is qualitatively different in q 3 Potts models, where monotonicity considerations do not close the gap If multiple Gibbs measures, physics prescription is to select Bethe Gibbs measure achieving highest value of Φ Bethe, Φ Bethe = sup{φ(ν ) : ν Bethe Gibbs measure} (restriction to Bethe Gibbs measures is an assumption) [dmss14] prove such prediction holds (within one non-trivial example).

72 Potts: Non-uniqueness regime Non-uniqueness regime for Potts on T d (d = 4, q = 30) in uniqueness regime, ν f = ν 1 on T d in non-uniqueness regime, ν f ν 1 on T d. positive (β, B) quadrant

73 Potts: Non-uniqueness regime Non-uniqueness regime for Potts on T d (d = 4, q = 30) in uniqueness regime, ν f = ν 1 on T d in non-uniqueness regime, ν f ν 1 on T d. positive (β, B) quadrant

74 Potts: Bethe free energy Potts Bethe prediction as function of β (d = 4, q = 30, B = 0.05) Φ(ν f ), with ν f the free-boundary Bethe Gibbs measure nonuniqueness regime Φ Bethe = Φ(ν f ) Φ(ν 1 ) Φ(ν 1 ), with ν 1 the 1-boundary Bethe Gibbs measure

75 Potts: Bethe free energy Potts Bethe prediction as function of β (d = 4, q = 30, B = 0.05) Φ(ν f ), with ν f the free-boundary Bethe Gibbs measure nonuniqueness regime Φ Bethe = Φ(ν f ) Φ(ν 1 ) Φ(ν 1 ), with ν 1 the 1-boundary Bethe Gibbs measure

76 Potts: Bethe free energy Potts Bethe prediction as function of β (d = 4, q = 30, B = 0.05) Φ(ν f ), with ν f the free-boundary Bethe Gibbs measure nonuniqueness regime Φ Bethe = Φ(ν f ) Φ(ν 1 ) Φ(ν 1 ), with ν 1 the 1-boundary Bethe Gibbs measure

77 Potts: Bethe free energy Potts Bethe prediction as function of β (d = 4, q = 30, B = 0.05) Φ(ν f ), with ν f the free-boundary Bethe Gibbs measure nonuniqueness regime Φ Bethe = Φ(ν f ) Φ(ν 1 ) Φ(ν 1 ), with ν 1 the 1-boundary Bethe Gibbs measure

78 Potts: Bethe free energy Potts Bethe prediction as function of β (d = 4, q = 30, B = 0.05) Φ(ν f ), with ν f the free-boundary Bethe Gibbs measure nonuniqueness regime Φ Bethe = Φ(ν f ) Φ(ν 1 ) Φ(ν 1 ), with ν 1 the 1-boundary Bethe Gibbs measure

79 Free energy: other models By interpolation in parameters and other means... mostly for G n T d (regular tree) simpler Bethe prediction. Examples of G n T d include the (uniformly) random d-regular graph, the (uniformly) random bipartite d-regular graph, and any sequence G n of d-regular graphs of divergent girth THEOREM (Sly Sun 14). For the hard-core (independent set) model at any fugacity λ > 0, and for the anti-ferromagnetic Ising model at any B R, the Bethe prediction holds on any bipartite sequence G n T d. (nonbipartite sequence, e.g. G n,d has strictly smaller φ for λ large) THEOREM (D. Montanari Sly Sun 14). For the Potts model at any β, B 0, the Bethe prediction holds on any sequence G n T d with d even. (lower bound by interpolation for tree-like graphs)

80 Free energy: other models By interpolation in parameters and other means... mostly for G n T d (regular tree) simpler Bethe prediction. Examples of G n T d include the (uniformly) random d-regular graph, the (uniformly) random bipartite d-regular graph, and any sequence G n of d-regular graphs of divergent girth THEOREM (Sly Sun 14). For the hard-core (independent set) model at any fugacity λ > 0, and for the anti-ferromagnetic Ising model at any B R, the Bethe prediction holds on any bipartite sequence G n T d. (nonbipartite sequence, e.g. G n,d has strictly smaller φ for λ large) THEOREM (D. Montanari Sly Sun 14). For the Potts model at any β, B 0, the Bethe prediction holds on any sequence G n T d with d even. (lower bound by interpolation for tree-like graphs)

81 Free energy: other models By interpolation in parameters and other means... mostly for G n T d (regular tree) simpler Bethe prediction. Examples of G n T d include the (uniformly) random d-regular graph, the (uniformly) random bipartite d-regular graph, and any sequence G n of d-regular graphs of divergent girth THEOREM (Sly Sun 14). For the hard-core (independent set) model at any fugacity λ > 0, and for the anti-ferromagnetic Ising model at any B R, the Bethe prediction holds on any bipartite sequence G n T d. (nonbipartite sequence, e.g. G n,d has strictly smaller φ for λ large) THEOREM (D. Montanari Sly Sun 14). For the Potts model at any β, B 0, the Bethe prediction holds on any sequence G n T d with d even. (lower bound by interpolation for tree-like graphs)

82 Free energy: other models By interpolation in parameters and other means... mostly for G n T d (regular tree) simpler Bethe prediction. Examples of G n T d include the (uniformly) random d-regular graph, the (uniformly) random bipartite d-regular graph, and any sequence G n of d-regular graphs of divergent girth THEOREM (Sly Sun 14). For the hard-core (independent set) model at any fugacity λ > 0, and for the anti-ferromagnetic Ising model at any B R, the Bethe prediction holds on any bipartite sequence G n T d. (nonbipartite sequence, e.g. G n,d has strictly smaller φ for λ large) THEOREM (D. Montanari Sly Sun 14). For the Potts model at any β, B 0, the Bethe prediction holds on any sequence G n T d with d even. (lower bound by interpolation for tree-like graphs)

83 Free energy: other models By interpolation in parameters and other means... mostly for G n T d (regular tree) simpler Bethe prediction. Examples of G n T d include the (uniformly) random d-regular graph, the (uniformly) random bipartite d-regular graph, and any sequence G n of d-regular graphs of divergent girth THEOREM (Sly Sun 14). For the hard-core (independent set) model at any fugacity λ > 0, and for the anti-ferromagnetic Ising model at any B R, the Bethe prediction holds on any bipartite sequence G n T d. (nonbipartite sequence, e.g. G n,d has strictly smaller φ for λ large) THEOREM (D. Montanari Sly Sun 14). For the Potts model at any β, B 0, the Bethe prediction holds on any sequence G n T d with d even. (lower bound by interpolation for tree-like graphs)

84 Free energy: other models By interpolation in parameters and other means... mostly for G n T d (regular tree) simpler Bethe prediction. Examples of G n T d include the (uniformly) random d-regular graph, the (uniformly) random bipartite d-regular graph, and any sequence G n of d-regular graphs of divergent girth THEOREM (Sly Sun 14). For the hard-core (independent set) model at any fugacity λ > 0, and for the anti-ferromagnetic Ising model at any B R, the Bethe prediction holds on any bipartite sequence G n T d. (nonbipartite sequence, e.g. G n,d has strictly smaller φ for λ large) THEOREM (D. Montanari Sly Sun 14). For the Potts model at any β, B 0, the Bethe prediction holds on any sequence G n T d with d even. (lower bound by interpolation for tree-like graphs)

85 References [DM10] Dembo A. & Montanari A. Gibbs measures and phase transitions on sparse random graphs. Brazilian J. Probab. & Stat. 24 (2010), pp [DMS13] Dembo, A., Montanari, A. & Sun, N. Factor models on locally tree-like graphs. Ann. Prob. 41 (2013), pp [DMSS14] Dembo, A., Montanari, A., Sly, A. & Sun, N. The replica symmetric solution for Potts models on d-regular graphs. Comm. Math. Phys. 327 (2014), pp [SS14] Sly, A. & Sun, N. Counting in two-spin models on d-regular graphs. Ann. Probab. 42 (2014), pp