The Parisi Variational Problem

Transcription

1 The Parisi Variational Problem Aukosh S. Jagannath 1 joint work with Ian Tobasco 1 1 New York University Courant Institute of Mathematical Sciences September 28, 2015 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

2 Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Introduction Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

3 Introduction Mean Field Spin Glasses An Intro to Mean Field Spin glasses Introduced to study magnetic properties of disordered media Posed Fundamental Problem for Physicists: - Difficult to analyze: classical static assumptions (LLN s, CLTs) fail - Difficult to simulate: Bad mixing properties in the glassy phase Introduced Replica Symmetry Breaking and Cavity Method Key Idea is a break down of symmetry called the Replica Symmetry in the glassy phase Techniques developed here later used to study problems in other fields, e.g. - Biology: Hopfield and Perceptron model - Combinatorial Optimization (CO): MAX-CUT and ksat Idea for CO: the hardness glassy transition Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

4 Introduction Mean Field Spin Glasses The Sherrington-Kirkpatrick Model and Mixed p-spin glasses Models around which theory was originally developed N spins, p-way interaction with N (0, 1) coupling coefficients Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

5 Introduction Mean Field Spin Glasses The Sherrington-Kirkpatrick Model and Mixed p-spin glasses Configuration space: Σ N = { 1, 1} N Hamiltonian: H N (σ) = β p 1 β p N (p 1)/2 N i 1,..,i p=1 g i1,...,i p σ i1...σ ip + h i σ i coupling coefficients g i1,...,i p N (0, 1) iid Choice of model: ξ 0 (t) = βpt 2 p and h 0. Free Energy: F = lim 1 N log e H N(σ) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

6 Introduction Mean Field Spin Glasses The Sherrington-Kirkpatrick Model and Mixed p-spin glasses Configuration space: Σ N = { 1, 1} N Hamiltonian: H N (σ) = β 2N N i 1,i 2 =1 g i1,i 2 σ i1 σ i2 + h i σ i coupling coefficients g i1,...,i p N (0, 1) iid Choice of model: ξ 0 (t) = 1 2 t2 and h 0. Free Energy: F = lim 1 N log e H N(σ) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

7 Parisi Formula Introduction Mean Field Spin Glasses Theorem (Guerra 03, Talagrand 06, Panchenko 14) For any choice of (β, h, ξ 0 ), F = inf P(µ; β, h, ξ 0) a.s. µ Pr[0,1] Glassy phase minimizer is not a dirac mass Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

8 The Parisi Functional Introduction The Parisi Variational Problem Parameters: 1 The model: ξ 0 (t) = p 1 β2 pt p. 2 External field: h 0 3 (inverse) temperature: β Parisi functional: P : Pr([0, 1]) R P(µ; β, h, ξ 0 ) = u µ (0, h) β2 2 u µ solves the Parisi PDE: ˆ 1 0 ξ 0(s)µ[0, s]s ds. ( { t u µ (t, x) + β2 ξ 0 (t) 2 xx u µ (t, x) + µ [0, t] ( x u µ (t, x)) 2) = 0 u µ (1, x) = log cosh(x) (t, x) [0, 1) R Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

9 The Parisi Functional Introduction The Parisi Variational Problem Parisi functional: P : Pr([0, 1]) R P(µ; β, h, ξ 0 ) = u µ (0, h) β2 2 ˆ 1 0 ξ 0(s)µ[0, s]s ds. ( { t u µ (t, x) + β2 ξ 0 (t) 2 xx u µ (t, x) + µ [0, t] ( x u µ (t, x)) 2) = 0 u µ (1, x) = log cosh(x) (t, x) [0, 1) R Question (Parisi Variational Problem) Characterize the minimizer of F (β, h; ξ 0 ) = for fixed ξ 0 and varying β, h 0. inf P(µ; β, h, ξ 0) µ Pr([0,1]) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

10 Main Questions Introduction Main Questions Parisi variational problem: Minimizer called Parisi measure F (β, h; ξ 0 ) = min P(µ; β, h, ξ 0 ) Question Can you characterize the minimizer? Where are the transitions? Question When is it: 1 1-atomic (Replica Symmetric, RS), 2 (k+1)-atomic (k-replica Symmetry Breaking (k-rsb) 3 many atoms or interval in support (Full RSB) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

11 An Energy Balance Introduction Main Questions P(µ; β, h, ξ 0 ) = u µ (0, h) β2 2 ˆ 1 0 ξ 0(s)µ[0, s]s ds. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

12 An Energy Balance Introduction Main Questions P(µ; β, h, ξ 0 ) = u µ (0, h) β2 2 ˆ 1 0 ξ 0(s)µ[0, s]s ds. Linear term wants µ = δ 0 β2 2 ˆ 1 0 ξ 0(s)µ[0, s]s ds. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

13 An Energy Balance Introduction Main Questions P(µ; β, h, ξ 0 ) = u µ (0, h) β2 2 ˆ 1 0 ξ 0(s)µ[0, s]s ds. Linear term wants µ = δ 0 PDE term: u µ solves t u µ (t, x) + β2 ξ 0 (t) ( xx u µ (t, x) + µ [0, t] ( x u µ (t, x)) 2) = 0 2 Comparison Principle: wants µ = δ 1 µ[0, t] ν[0, t] t = u µ (t, x) u ν (t, x) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

14 An Energy Balance Introduction Main Questions P(µ; β, h, ξ 0 ) = u µ (0, h) β2 2 ˆ 1 0 ξ 0(s)µ[0, s]s ds. Linear term wants µ = δ 0 PDE term: u µ has comparison principle = wants µ = δ 1. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

15 Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Main Results Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

16 Main questions Main Results Question 1 Can we characterize the minimizer? 2 Where is the transition between RS and RSB? Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

17 Preliminary definitions Main Results Optimality conditions Recall the Parisi variational problem inf µ Pr[0,1] P(µ; ξ, h) = inf u µ(0, h) 1 2 ˆ 1 0 ξ (s)µ[0, s]sds Here ξ = β 2 ξ 0 Let X s solve the SDE (optimal path for HJB): { dx s = ξ (s)µ[0, s]u x (s, X s )ds + ξ (s)dw s X 0 = h Let G µ (s) be given by G µ (s) = ˆ 1 t ξ (s) 2 ( [ EX0 =h u 2 x (s, X s ) ] s ) ds Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

18 Main Results 1 Main Results Optimality conditions Theorem ((J-T 15) Optimality Conditions) µ is a Parisi measure iff µ (G µ (s) = min G µ (t)) = 1 Furthermore, the following self-consistency conditions must be solved { Eux(q, 2 X q ) = q ξ (s)e h u 2 xx(q, X q ) 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

19 Main Results AT line de Almeida-Thouless Line: The Glassy transition Question Where is the glassy phase for the mixed p-spin glass? Studied for SK by de Almeida-Thouless in 70s Glassy phase = RSB phase Conjectured function α(t, h) that determined validity of RS anzatz Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

20 Main Results AT line de Almeida-Thouless Line: The Glassy transition Consider self-consistency equations { Eu 2 x(q, X q ) = q ξ (s)e h u 2 xx(q, X q ) 1 take µ = δ q = AT conditions Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

21 Main Results AT line de Almeida-Thouless Line: The Glassy transition Consider self-consistency equations { E tanh 2 (h + ξ (q) ξ (0)Z) = q ξ (s)e h sech 4 (h + ξ (q) ξ (0)Z) 1 take µ = δ q = AT conditions Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

22 Main Results AT line de Almeida-Thouless Line: The Glassy transition define α(β, h) = inf q Q α(q, β, h) q (β, h) = sup{q Q : α(β, h) = α(q, β, h)}. In general, card(q ) > 1. For the SK model, the set α = 1 is called the AT-line Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

23 Main Results AT line de Almeida-Thouless Line: The Glassy transition define α(β, h) = inf q Q α(q, β, h) q (β, h) = sup{q Q : α(β, h) = α(q, β, h)}. Conjecture For all mixed p-spin glass models, ξ 0 the RS anzatz, µ = δ q, is valid if and only if, α(t, h) 1 for all (T, h). Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

24 Main Results 2 Main Results AT line Theorem ( J-T 15) For all mixed p-spin glass models, ξ 0, (and h 0 > 0), the RS anzatz is valid if and only if, α(t, h) 1 for (T, h) in R \ K where K [0, T u ] [0, h u ]. Furthermore, if the model is uniformly elliptic, ξ (0) > 0, then K [T l, T u ] [0, h u ] Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

25 Main Results AT line Relation to What is known: SK and uniformly elliptic models Aizenman-Lebowitz-Ruelle 87, Toninelli 02 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

26 Main Results AT line Relation to What is known: SK and uniformly elliptic models Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

27 Main Results AT line Relation to What is known: General models Chen 15, JT 15 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

28 Main Results AT line Relation to What is known: General models Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

29 Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Theorem 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

30 Theorem 1 A Shift of Methodology functional involves analysis of Hamilton-Jacobi-Bellman PDE Avoid explicit formulas and Ruelle Probability Cascades Study functional for all measures using: PDE techniques Stochastic Analysis/Optimal Control Avoid finite dimensional optimization = Compute variational derivative. Functional is strictly convex [Auffinger-Chen 14] = use Optimality Conditions to our advantage. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

31 Theorem 1 Formal Sketch Thm 1 Part 1 Formal Sketch Thm 1 Part 1 Theorem ((J-T 15) Optimality Conditions) µ is a Parisi measure iff µ (G µ (s) = min G µ (t)) = 1 Furthermore, the following self-consistency conditions must be solved { Eux(q, 2 X q ) = q ξ (s)e h u 2 xx(q, X q ) 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

32 Calculating δp Theorem 1 Formal Sketch Thm 1 Part 1 To obtain the formula for δp µ : P(µ) = u µ (0, h) 1 2 ˆ 1 1 Linear term easy: t ξ (s)sds, µ 2 PDE term is the difficulty. 3 u µ is wellposed. [JT 15] 4 u µ has differentiable dependence on µ 5 To see: formally differentiate in µ 0 ξ (s)µ[0, s]sds Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

33 Calculating δp Theorem 1 Formal Sketch Thm 1 Part 1 To obtain the formula for δp µ : P(µ) = u µ (0, h) 1 2 ˆ 1 1 Linear term easy: t ξ (s)sds, µ 2 PDE term is the difficulty. 3 u µ is wellposed. [JT 15] 4 u µ has differentiable dependence on µ 5 To see: formally differentiate in µ 0 ξ (s)µ[0, s]sds Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

34 Calculating δp Theorem 1 Formal Sketch Thm 1 Part 1 To obtain the formula for δp µ : P(µ) = u µ (0, h) 1 2 ˆ 1 1 Linear term easy: t ξ (s)sds, µ 2 PDE term is the difficulty. 3 u µ is wellposed. [JT 15] 4 u µ has differentiable dependence on µ 5 To see: formally differentiate in µ ( t +L µ )δu = ξ (s) 2 δµu2 x IBP = 0 ξ (s)µ[0, s]sds Itô = δu(0, h) = 1 ˆ 2 ˆ ξ (s)e h ux, 2 δµ t ξ (s)δµ E h u 2 x(s, X s )ds Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

35 Calculating δp Theorem 1 Formal Sketch Thm 1 Part 1 To obtain the formula for δp µ : 1 Linear term easy: P(µ) = u µ (0, h) ˆ 1 0 t ξ (s)sds, µ 2 PDE term is the difficulty. 3 u µ is wellposed. [JT 15] 4 u µ has differentiable dependence on µ 5 To see: formally differentiate in µ Combine to get G µ δp(s) = G µ (s) = ˆ 1 t ξ (s) 2 1 t ξ (s)µ[0, s]sds 1 2 ξ (s)e h ux, 2 δµ ( Eh u 2 x(s, X s ) s ) ds Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

36 Theorem 1 Formal Sketch Thm 1 Part 1 Formal Sketch Thm 1 Part 1 Theorem ((J-T 15) Optimality Conditions) µ is a Parisi measure iff µ (G µ (s) = min G µ (t)) = 1 Furthermore, the following self-consistency conditions must be solved { Eux(q, 2 X q ) = q ξ (s)e h u 2 xx(q, X q ) 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

37 Theorem 1 Formal Sketch Thm 1 Part 1 Formal Sketch Thm 1 Part 1 Strictly Convex Functional on (Pr[0, 1], τ w ) First order optimality conditions: µ is optimal if and only if G µ, σ 0 σ D T µ Pr([0, 1]) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

38 Theorem 1 Formal Sketch Thm 1 Part 1 Formal Sketch Thm 1 Part 1 Strictly Convex Functional on (Pr[0, 1], τ w ) First order optimality conditions: µ is optimal if and only if G µ, σ 0 σ D T µ Pr([0, 1]) Check on σ = δ s µ = d dθ ((1 θ)µ + θδ s) G µ, δ s µ 0 = G µ (s) G µ, µ s µ minimizes µ(g µ (s) = min G µ ) = 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

39 Sketch Thm 1 Part 2 Theorem 1 Part 2 Theorem ((J-T 15) Optimality Conditions) µ is a Parisi measure iff µ (G µ (s) = min G µ (t)) = 1 Furthermore, the following self-consistency conditions must be solved { Eux(q, 2 X q ) = q ξ (s)e h u 2 xx(q, X q ) 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

40 Sketch Thm 1 Part 2 Theorem 1 Part 2 G µ (x) = ˆ 1 G µ is minimized on suppµ. t ξ (s) 2 ( Eh u 2 x(s, X s ) s ) ds Regularity of u+itô s lemma = G C 2 Derivative tests: G (q) = 0 = Eu 2 x(q, X q ) = q G (q) 0 = ξ (s)eu 2 xx(q, X q ) 1 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

41 Theorem 1 Part 2 Application: RSB below the AT line Corollary If α > 1 then the RS anzatz fails. Proof. Suppose µ = δ q G C 2, G(q) = min G = q Q α > 1 = G δ q (q) < 0 Contradiction Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

42 Plan of Attack for Main Thm 2 Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

43 Plan of Attack for Main Thm 2 de Almeida-Thouless Line Main Idea Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

44 Plan of Attack for Main Thm 2 de Almeida-Thouless Line Main Idea Consider self-consistency equations for 1-atomic measure { E tanh 2 (h + ξ (q) ξ (0)Z) = q ξ (s)e h sech 4 (h + ξ (q) ξ (0)Z) 1 define α(β, h) = inf q Q α(q, β, h) q (β, h) = sup{q Q : α(β, h) = α(q, β, h)}. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

45 Plan of Attack for Main Thm 2 de Almeida-Thouless Line Main Idea Conjecture (Rephrased) If (T, h) satisfy the stability conditon α 1, then is the optimizer 1 atomic? Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

46 Plan of Attack for Main Thm 2 Main Idea A Tale of Two Topologies: AT conjecture as Local-to-Global stability Take best guess location, q, for minimizer in class of 1 atomic measures α < 1 = Derivative in direction of measures supported in ball around q is positive = Derivative in direction of all measures nonnegative If you do worse in a neighborhood of q, you do worse everywhere. Derivative in direction of µ = µ-average of potential differences of G. G q Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

47 Plan of Attack for Main Thm 2 Main Idea: A naive move Main Idea Focus on RS µ = δ q Need to show G δq is minimized at q Easy part: α 1 = G δq Goal: show G is convex decreasing for y q G q Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

48 A Shift of Viewpoints Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

49 A Shift of Viewpoints A Shift of Viewpoints Change of coordinates Expect RS for large h and small β Difficulty near phase boundary for large β IDEA: Studying region α 1 = work on curves α = const. Provides scaling relation between β and h probe large beta up to α = 1. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

50 A Shift of Viewpoints Dispersive Estimates for Gaussians Change of coordinates Need good large β asymptotics of important quantities, e.g. q q = E tanh 2 (h + ξ (q )Z) Expectations of functions of optimal process X t Need asymptotics of Ef (Y β ) with Y β N (h(β), Σ(β)) R d Σ(β) explodes in some directions Idea x 2 σ e 2σ 2 dx dx 2πσ 2 2π Rescaled large noise limit dual to small noise limit through Fourier inversion Main difficulty: Need uniform estimates as Σ(β) contracts in other direction = Can t just invert Estimates must simultaneously respect large and small noise limits Need sharp constants and correct scaling relation of β and h Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

51 A Shift of Viewpoints Dispersive Estimates for Gaussians Change of coordinates Need asymptotics of Ef (Y β ) with Y β N (h(β), Σ(β)) R d Σ(β) explodes in some directions Idea x 2 σ e 2σ 2 dx dx 2πσ 2 2π Rescaled large noise limit dual to small noise limit through Fourier inversion Main difficulty: Need uniform estimates as Σ(β) contracts in other direction = Can t just invert Estimates must simultaneously respect large and small noise limits Need sharp constants and correct scaling relation of β and h Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

52 A Shift of Viewpoints Dispersive Estimates for Gaussians Change of coordinates Need asymptotics of Ef (Y β ) with Y β N (h(β), Σ(β)) R d Σ(β) explodes in some directions Idea x 2 σ e 2σ 2 dx dx 2πσ 2 2π Rescaled large noise limit dual to small noise limit through Fourier inversion Main difficulty: Need uniform estimates as Σ(β) contracts in other direction = Can t just invert Estimates must simultaneously respect large and small noise limits Need sharp constants and correct scaling relation of β and h Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

53 A Shift of Viewpoints Dispersive Estimates for Gaussians Change of coordinates Need asymptotics of Ef (Y β ) with Y β N (h(β), Σ(β)) R d Σ(β) explodes in some directions Idea x 2 σ e 2σ 2 dx dx 2πσ 2 2π Rescaled large noise limit dual to small noise limit through Fourier inversion Main difficulty: Need uniform estimates as Σ(β) contracts in other direction = Can t just invert Estimates must simultaneously respect large and small noise limits Need sharp constants and correct scaling relation of β and h Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

54 A Shift of Viewpoints Dispersive Estimates for Gaussians Change of coordinates Need asymptotics of Ef (Y β ) with Y β N (h(β), Σ(β)) R d Σ(β) explodes in some directions Idea x 2 σ e 2σ 2 dx dx 2πσ 2 2π Rescaled large noise limit dual to small noise limit through Fourier inversion Main difficulty: Need uniform estimates as Σ(β) contracts in other direction = Can t just invert Estimates must simultaneously respect large and small noise limits Need sharp constants and correct scaling relation of β and h Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

55 A Shift of Viewpoints Dispersive Estimates for Gaussians Change of coordinates Need asymptotics of Ef (Y β ) with Y β N (h(β), Σ(β)) R d Σ(β) explodes in some directions Idea: Rescaled large noise limit dual to small noise limit through Fourier inversion Main difficulty: Need uniform estimates as Σ(β) contracts in other direction = Can t just invert Estimates must simultaneously respect large and small noise limits Need sharp constants and correct scaling relation of β and h Idea: Use estimates on curves with conserved integrals to pick out scaling relations Compare desired integrals to conserved quantities Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

56 Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion 2/3 Arugment Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

57 2/3 Arugment A 1-D dispersive estimate A 1-D dispersive estimate Theorem (1-D Estimate) If f L 2 ((x 2 1)dx) L 1 even, then σ 2 Ef (h + σz) σe 1 2( h σ ) 2 ˆf (0) C σ f 1 Furthermore, if f, g as above, with ˆf (0) 0 and lim σ σ2 Ef (h + σz) <, then lim Eg(h + σz) Ef (h + σz) = ĝ(0) ˆf (0) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

58 2/3 Arugment A 1-D dispersive estimate A 1-D dispersive estimate Theorem (1-D Estimate) If f L 2 ((x 2 1)dx) L 1 even, then σ 2 Ef (h + σz) σe 1 2( h σ ) 2 ˆf (0) C σ f 1 Furthermore, if f, g as above, with ˆf (0) 0 and lim σ σ2 Ef (h + σz) <, then σ 2 Eg(h + σz) ĝ(0) C(f, g) ˆf (0) σ2 Ef (h + σz) σ Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

59 2/3 Arugment A 1-D dispersive estimate A 1-D dispersive estimate Theorem (1-D Estimate) If f L 2 ((x 2 1)dx) L 1 even, then σ 2 Ef (h + σz) σe 1 2( h σ ) 2 ˆf (0) C σ f 1 Remark Suggests that curves are identified σ e 1 2( h σ ) 2 = const. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

60 2/3 Arugment A 1-D dispersive estimate Example of 1-D estimate: Asymptotics of q Idea: known integrals give sharp estimates on others Example (Estimating q on α = const. curves) 1-D estimate gives way to estimate q given α. α = ξ 0 (q ) ξ 0 (q ) σ2 Esech 4 (h + σz) β 2 ξ 0(q )(1 q ) = ξ 0 (q ) ξ 0 (q ) σ2 Esech 2 (h + σz) So for f = sech 4, g = sech 2 σ 2 Eg(h + σz) ĝ(0) C(f, g) ˆf (0) σ2 Ef (h + σz) σ 2 Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

61 2/3 Arugment A 1-D dispersive estimate Example of 1-D estimate: Asymptotics of q Idea: known integrals give sharp estimates on others Example (Estimating q on α = const. curves) 1-D estimate gives way to estimate q given α. α = ξ 0 (q ) ξ 0 (q ) σ2 Esech 4 (h + σz) β 2 ξ 0(q )(1 q ) = ξ 0 (q ) ξ 0 (q ) σ2 Esech 2 (h + σz) So for f = sech 4, g = sech 2 β 2 ξ 0(q )(1 q ) 3 Cξ 0 α (q ) 2 (ξ 0 (q )) 3/2 β Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

62 2/3 Arugment Proof of Thm 3 2/3 Argument: An a priori stability estimate Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

63 2/3 Arugment Proof of Thm 3 2/3 Argument: An a priori stability estimate Theorem If α 2 3 ξ ξ 0 (q ) 0 (1) then the minimizer is a dirac mass. Proof. Want to show G is convex. (1 ℵξ 0 (1) ) β(ξ 0 (q )) 3/2 Manipulations of derivatives = G is convex provided ξ (1)(1 q ) 1 Rearrange previous estimate on q from before and plug in bound to get above inequality. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

64 The Long Time argument Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

65 Theorem 2 Goal is to show: The Long Time argument Low Temperature Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

66 Theorem 2 The Long Time argument Low Temperature Goal is to show: Theorem For all models ξ 0 and nonzero external field h 0, there is an β L (ξ 0, h 0 ) such that in the region β β L, h h 0, the minimizer is 1 atomic provided α 1. Difficult to understand near α = 1 Above estimate allows assumption that α 2/3 (1 ɛ) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

67 The Long Time argument Low Temperature Strategy Low Temperature Want to study RS at low temperature up to α = 1 G (t) 0 provided Idea: I (t) = E h ( u 3 xxx (X t ) u 2 xx(x t ) ) C β 2 + o( 1 β 2 ) t [q, 1] 1 Fix t [q, 1] 2 Flow along α = const. to infinity, 3 Bound error between problem at infinity and problem at finite β. 4 Obtain uniform bound in t Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

68 The Long Time argument Low Temperature Strategy 2 Low Temperature Try to translate in to similar language as before Girsanov = I (t) EΨ(Y t, Y q ) Gaussian integral with Σ = β 2 ξ 0(q ) h = h (1, 1) ( ) ξ 0 (t)/ξ 0 (q ) Becoming rank 1 with exploding eigenvalues Needs 2D dispersive estimate Need to take limit in β for each t Interval [q, 1] shrinking = rescale: τ = ξ 0 (t) ξ 0 (q ) ξ 0 (1) ξ 0 (q ) Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

69 The Long Time argument A 2D Estimate: the Set-Up 2D Dispersive Estimates Gaussian vector in R 2. Covariance structure Σ(t) = λ 1 (t)v 1 v 1 + λ 2 (t)v 2 v 2 Have set up: {v 1, v 2 } {w, v} λ 1 ν = 3 4 ατ [0, 1], λ 2 σ 2 + m, v 1 0 Need to estimate Ef (Y t ) uniformly in t Uniform in t = uniform in ν [0, 3 4 α]. Naive approach: Bound errors in terms of conv. rate for operator Σ 1 : Ef (Y t ) Ef (Y ) C Σ 1 t Σ 1 Problem: Σ 1 becomes singular as ν 0 = RHS blows up Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

70 The Long Time argument A 2D Estimate: the Set-Up 2D Dispersive Estimates Gaussian vector in R 2. Covariance structure Σ(t) = λ 1 (t)v 1 v 1 + λ 2 (t)v 2 v 2 Have set up: {v 1, v 2 } {w, v} λ 1 ν = 3 4 ατ [0, 1], λ 2 σ 2 + m, v 1 0 Need to estimate Ef (Y t ) uniformly in t Uniform in t = uniform in ν [0, 3 4 α]. Naive approach: Bound errors in terms of conv. rate for operator Σ 1 : Ef (Y t ) Ef (Y ) C Σ 1 t Σ 1 Problem: Σ 1 becomes singular as ν 0 = RHS blows up Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

71 The Long Time argument Don t Invert in the Bad Directions 2D Dispersive Estimates Problem: Σ becomes increasingly singular in ν 0 Solution: Don t invert in bad directions, Play off regularity of f. let f (x) = f (xw + yv)dx Get estimate I (t) λ 1/2 2 E f (ν 1/2 1 z) + O( 2π β 5/2 δ ) e h2 λ 2 C β 2 E f (ν1/2 z) + o( 1 β 2 ) Leave non-exploding directions alone = Gaussian integral Do standard estimate in exploding direction = Lebesgue average Just need E f (ν 1/2 z) C for some C > 0. For us f (x) < 0 always! Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

72 The Long Time argument Don t Invert in the Bad Directions 2D Dispersive Estimates Problem: Σ becomes increasingly singular in ν 0 Solution: Don t invert in bad directions, Play off regularity of f. let f (x) = f (xw + yv)dx Get estimate I (t) λ 1/2 2 E f (ν 1/2 1 z) + O( 2π β 5/2 δ ) e h2 λ 2 C β 2 E f (ν1/2 z) + o( 1 β 2 ) Leave non-exploding directions alone = Gaussian integral Do standard estimate in exploding direction = Lebesgue average Just need E f (ν 1/2 z) C for some C > 0. For us f (x) < 0 always! Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

73 The Long Time argument Don t Invert in the Bad Directions 2D Dispersive Estimates Problem: Σ becomes increasingly singular in ν 0 Solution: Don t invert in bad directions, Play off regularity of f. let f (x) = f (xw + yv)dx Get estimate I (t) λ 1/2 2 E f (ν 1/2 1 z) + O( 2π β 5/2 δ ) e h2 λ 2 C β 2 E f (ν1/2 z) + o( 1 β 2 ) Leave non-exploding directions alone = Gaussian integral Do standard estimate in exploding direction = Lebesgue average Just need E f (ν 1/2 z) C for some C > 0. For us f (x) < 0 always! Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

74 The Long Time argument Don t Invert in the Bad Directions 2D Dispersive Estimates Problem: Σ becomes increasingly singular in ν 0 Solution: Don t invert in bad directions, Play off regularity of f. let f (x) = f (xw + yv)dx Get estimate I (t) λ 1/2 2 E f (ν 1/2 1 z) + O( 2π β 5/2 δ ) e h2 λ 2 C β 2 E f (ν1/2 z) + o( 1 β 2 ) Leave non-exploding directions alone = Gaussian integral Do standard estimate in exploding direction = Lebesgue average Just need E f (ν 1/2 z) C for some C > 0. For us f (x) < 0 always! Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

75 Table of Contents 1 Introduction Mean Field Spin Glasses The Parisi Variational Problem Main Questions 2 Main Results Optimality conditions AT line 3 Theorem 1 Formal Sketch Thm 1 Part 1 Part 2 4 Plan of Attack for Main Thm 2 Main Idea 5 A Shift of Viewpoints Change of coordinates 6 2/3 Arugment A 1-D dispersive estimate Proof of Thm 3 7 The Long Time argument Low Temperature 2D Dispersive Estimates 8 Conclusion Conclusion Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

76 Conclusion Conclusions and Open questions PDE and Stoch. Analysis techniques simplify analysis New charactierzation of Parisi measures AT line as local-to-global theorem AT criterion correct in (T, h)-plane less compact set for all models Behavior in compact set has yet to be studied in full generality even in the physics literature New class of gaussian estimates in mixed large noise-small noise regime. Open Question: Explain bifurcation of fixed points in general models. Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

77 Thanks Conclusion Thanks for listening! Thanks to: G. Ben Arous, R.V. Kohn, NYU Paris GRI Institute This research was supported by an NSF Graduate Research Fellowship DGE , NSF Grants DMS and OISE Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 42

78 What is the minimizer? The interpretation and Positivity problems Question What is the minimizer in terms of the problem at finite N? Conjectured to be order parameter/sufficient statistic Conjectured to be related to limiting distribution of overlap: µ ( ) = lim EG 2 N (R 12 ) Known for some models (β p 0 for p : t p is total) Positivity Problem: µ Pr[0, 1]. A priori limit overlap dist n in Pr[ 1, 1]. For models with Z 2 symmetry, e.g. SK with h = 0, how does this happen? Geometrically must be the case at N =, how does the symmetry get broken at large N? Aukosh S. Jagannath (NYU) The Parisi Variational Problem September 28, / 1