Statistical mechanics via asymptotics of symmetric functions

Transcription

1 symmetric Statistical mechanics via of symmetric (University of Pennsylvania) based on: V.Gorin, G.Panova, Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory, Annals of Probability arxiv:0.064 G. Panova, with free, arxiv: April 05

2 symmetric Overview Characters of U( ), boundary of the Gelfand-Tsetlin graph Normalized : S λ (x,..., x k ; N) = s λ(x,..., x k, N k ) s λ ( N ) Alternating Sign Matrices (ASM)/ 6-Vertex model: : : ζ y ζ x L

3 symmetric Tilings of a domain Ω (on a triangular lattice) with elementary rhombi of types ( lozenges ).

4 symmetric The many faces of lozenge tilings

5 symmetric The many faces of lozenge tilings

6 symmetric The many faces of lozenge tilings

7 symmetric The many faces of lozenge tilings

8 symmetric The many faces of lozenge tilings

9 symmetric The many faces of lozenge tilings

10 symmetric The many faces of lozenge tilings

11 symmetric The many faces of lozenge tilings

12 symmetric The many faces of lozenge tilings

13 symmetric The many faces of lozenge tilings

14 symmetric The many faces of lozenge tilings

15 symmetric The many faces of lozenge tilings

16 symmetric The many faces of lozenge tilings

17 symmetric Limit behavior Question: Fix Ω in the plane and let grid size 0, what are the properties of uniformly random tilings of Ω? [Cohn Larsen Propp, 998] Hexagonal domain: Tiling is asymptotically frozen outside inscribed ellipse. [Kenyon Okounkov, 005] Polygonal domain: Tiling is asymptotically frozen outside inscribed algebraic curve..jpg [Cohn Kenyon Propp, 00; Kenyon-Okounkov-Sheffield, 006] There exists a limit shape for the surface of the height function (plane partition). 6

18 symmetric Behavior near the boundary, interlacing particles x x x x x x x N Horizontal lozenges near a flat boundary: interlacing particle configuration Gelfand-Tsetlin patterns. Question: What is the joint distribution of {x i j }k i= as N (scale = N )? x x x x x x with an explanation what the answer should be. Subsequent results: [Gorin-P,0], [Novak, 04],[Mkrtchyan, 0, periodic weights, unbounded 7

19 symmetric Behavior near the boundary, interlacing particles x x x x x x x N Horizontal lozenges near a flat boundary: interlacing particle configuration Gelfand-Tsetlin patterns. x x x x x x Question: What is the joint distribution of {x i j }k i= as N (scale = N )? Conjecture [Okounkov Reshetikhin, 006 ]: The joint distribution converges to a GUE-corners (aka GUE-minors) process: eigenvalues of GUE matrices. Proven for the hexagon by Johansson- Nordenstam (006). with an explanation what the answer should be. Subsequent results: [Gorin-P,0], [Novak, 04],[Mkrtchyan, 0, periodic weights, unbounded 7

20 symmetric GUE in tilings: our setup Domain Ω λ(n) : width N and the positions λ(n) + N > λ(n) + N > > λ(n) N of its N horizontal lozenges at the right boundary λ(5) = (4,,, 0, 0) Note: N Ω λ(n) is not necessarily a finite polygon as N, e.g. λ(n) = (N, N,...,, ) E.g. λ = (a,..., a, 0,..., 0) }{{}}{{} c b the hexagon with side lengths (a, b, c, a, b, c). 8

21 symmetric Plane partitions/gelfand-tsetlin patterns λ + N x x x Line j = λ + N λn N λ = (5, 4,,, 0) x = (4,, 0) λ = (4,,, 0, 0) Question: What is the joint distribution of the positions rescaling) near the boundary as N (i.e. lattice scale = N )? { } xj i (under correct 9

22 symmetric Answers: the Gaussian Unitary Ensemble (GUE) Gaussian Unitary Ensemble: matrices [X ij ] i,j : X = X T ReX ij, ImX ij i.i.d. N (0, /) for i j and X ii i.i.d. N (0, ) a a a a 4 a a a a 4 a a a a 4 a 4 a 4 a 4 a 44 (x k xk xk k ) eigenvalues of [X i,j ] k i,j= (top k k). Interlacing condition: x j i xj i xj i x 4 x 4 x 4 x4 4 x x x x x The joint distribution of {x j i } i j k is known as GUE corners (also, GUE-minors) process, =: GUE k. x 0

23 symmetric GUE in tilings: our results Limit profile f (t) of λ(n) as N ( : ) λ(n) i i N f N N λ(n) f (t) Ω λ(n) domain: Theorem (Gorin-P (0), Novak (04)) Let λ(n) = (λ (N)... λ N (N)), N =,,... be a partition. If a piecewise-differentiable weakly decreasing function f (t) (limit profile of λ(n)) s.t. N ( ) λ i (N) i N f = o( N) as N N i= and also sup i,n λ i (N)/N <. Let Υ k λ(n) = {xj i } be the collection of the positions of the horizontal lozenges on lines j =,..., k. Then for every k as N x λ + N λ + N Υ k λ(n) NE(f ) NS(f ) GUE k (GUE-corners proc. rank k) x in the sense of weak convergence, where x E(f ) = f (t)dt, 0 Line j = λn S(f ) = f (t) dt E(f ) + f (t)( t)dt. 0 0

24 symmetric Towards the proof:...or more generally symmetric, Lie group characters. Irreducible (rational) representations V λ of GL(N) (or U(N)) are indexed by dominant weights (signatures/young diagrams/integer partitions) λ: where λ i Z, e.g. λ = (4,, ), λ λ λ N, : s λ (x,..., x N ) characters of V λ. Weyl s determinantal formula: s λ (x,..., x N ) = [ det x λ j +N j i ] N ij= i<j (x i x j ) Semi-Standard Young tableaux( Gelfand-Tsetlin patterns) of shape λ : s (,) (x, x, x ) = s (x, x, x ) = x x +x x +x x +x x x +x x x +x x x.

25 symmetric Main object: Normalized N k {}}{ S λ(n) (x,..., x k ) := s λ(n)(x,..., x k,,..., ) s λ(n) (,..., ) }{{} N or more generally normalized Lie group characters: X γ(n) (x,..., x k ) := χ γ(n)(x,..., x k, N k ) χ γ(n) ( N ) Meaning also via the Harish-Chandra/Itzykson Zuber integral: s λ (e a,..., e an ) s λ (,..., ) = a i a j bj =λ j +N j e a i e a j i<j exp(trace(aubu ))du U(N)

26 symmetric Integral formula, k = Theorem (Gorin-P) For any partition λ and any x C \ {0, } we have (N )! x z S λ (x; N, ) = (x ) N πi N C i= (z (λ i + N i)) dz, where the contour C includes all the poles of the integrand. Similar formulas hold for the other normalized Lie group characters. Theorem (Gorin-P) ( ) If λ(n) N f i [under certain convergence conditions], for all fixed y 0: N lim N N ln S λ(n)(e y ; N, ) = yw 0 F(w 0) ln(e y ), where F(w; f = 0 ln(w f (t) + t)dt, w0 root of F(w; f ) = y. If ( ) w λ(n) N f i [ other conv. cond.], for any fixed h R: N ( ) NE(f S λ(n) (e h/ N ; N, ) = exp )h + S(f )h + o(), where E(f ) = f (t)dt, S(f ) = f (t) dt E(f ) + f (t)( t)dt Remark. Similar statements hold for a larger class of, e.g symplectic characters, Jacobi...+ q analogues. Remark. Integral formula appears also in [Colomo,Pronko,Zinn-Justin], random matrix interpretation in [Guionnet Maida], new analysis in [Novak]... 4

27 symmetric From k = to general k, multiplicativity Theorem (Gorin-P ) Let D i, = x i, Vandermonde det. Then λ, k N, we have x i N k {}}{ S λ (x,..., x k ; N) = s λ(x,..., x k,,..., ) s λ (,..., ) }{{} N [ ] k det D j k (N i)! = (N )!(x i= i ) N k i, k i,j= S λ (x j ; N, )(x j ) N. (x,..., x k ) j= Corollary (Gorin-P) Suppose that the sequence λ(n) is such that, as N, ln ( S λ(n) (x; N, ) ) Ψ(x) uniformly on a compact M C. Then for any k N ln ( S λ(n) (x,..., x k ; N, ) ) lim = Ψ(x ) + + Ψ(x k ) N N uniformly on M k. More informally, under various regimes of convergence for λ(n) we have S λ(n) (x,..., x k ) S λ(n) (x ) S λ(n) (x k ). Similar for symplectic characters, Jacobi; and q-analogues. Note: appears in [de Gier, Nienhuis, 5

28 symmetric GUE in tilings I: combinatorics x 0 x x x Tilings of domain Ω λ(n) Gelfand-Tsetlin schemes with bottom row λ(n) Semi-Standard Young Tableaux of shape λ(n) T = Positions of the horizontal lozenges on line j: x j shape of the subtableaux of T of the entries,..., j. Note: actual positions on the d projection are at (x k + (k/, k/,..., k/, k/)) 6

29 symmetric GUE in tilings I: combinatorics x 0 x x x Tilings of domain Ω λ(n) Gelfand-Tsetlin schemes with bottom row λ(n) Semi-Standard Young Tableaux of shape λ(n) T = x = (,, 0). Positions of the horizontal lozenges on line j: x j shape of the subtableaux of T of the entries,..., j. Note: actual positions on the d projection are at (x k + (k/, k/,..., k/, k/)) 6

30 symmetric GUE in tilings II: moment generating Proposition In a uniformly random tiling of Ω λ the distribution of the positions of the horizontal lozenges on the kth line x k (λ) is given by: Prob{x k (λ) = η} = sη(k )s λ/η ( N k ) s λ ( N, ) where s λ/η is the skew Schur polynomial. Proof: combinatorial definition of as sums over SSYTs. Proposition For any variables y,..., y k, the following moment generating function of x k (as above) is given by N k {}}{ s E x k (y,..., y k ) s x k (,..., ) = s λ(y,..., y k,,..., ) = S λ (y,..., y k ). s λ (,..., ) }{{}}{{} k N 7

31 symmetric GUE in tilings III: MGF Proposition ( ) EB k (y; GUE k ) = exp (y + + y k ), where B k (y; ν) = s ν δ k (y,..., y k ) when ν strict partition. s ν δk (,..., ) }{{} k 8

32 symmetric GUE in tilings III: MGF Proposition ( ) EB k (y; GUE k ) = exp (y + + y k ), where B k (y; ν) = s ν δ k (y,..., y k ) when ν strict partition. s ν δk (,..., ) }{{} k Compare: s S λ (y,..., y k ) = E tiling x k (y,..., y k ) s x k (,..., ) = E tiling B k (y; x k + δ k ) }{{} k Proposition (Gorin-P) For any k real numbers h,..., h k and λ(n)/n f (as earlier) we have: ( h lim S NS(f λ(n) e ),..., e N ) h NS(f k ) e ( ) E(f ) ( ki= h NS(f ) i = exp ) k hi. i= 8

33 symmetric GUE in tilings III: MGF Proposition ( ) EB k (y; GUE k ) = exp (y + + y k ), where B k (y; ν) = s ν δ k (y,..., y k ) when ν strict partition. s ν δk (,..., ) }{{} k Compare: s S λ (y,..., y k ) = E tiling x k (y,..., y k ) s x k (,..., ) = E tiling B k (y; x k + δ k ) }{{} k Proposition (Gorin-P) For any k real numbers h,..., h k and λ(n)/n f (as earlier) we have: ( h lim S NS(f λ(n) e ),..., e N ) h NS(f k ) e ( ) E(f ) ( ki= h NS(f ) i = exp Theorem. Let Υ k λ(n) = {xk, x k,...} collection of positions of the horizontal lozenges on lines k, k,..., of tiling from Ω λ(n), then Υ k λ(n) NE(f ) GUE k (GUE-corners process of rank k). NS(f ) ) k hi. i= 8

34 symmetric Free boundary domains N M N T f (N, M) := λ : l(λ)=n,λ M tilings of Ω λ, set of all tilings in an N M N trapezoid with N free (unrestricted) horizontal rhombi on the right border. Symmetric tilings of the N M N N M N hexagon/ symmetric boxed Plane Partitions. M N Questions:. Existence of a limit shape (surface)? Equation derived in [Di Francesco Reshetikhin, 009].. Behavior near boundary (GUE?). 9

35 symmetric with free : GUE m n y y y Line k = λ + N λ + N λ N Theorem (P) Let Yn,m k = (y k,..., y k k ) denote the positions of the horizontal lozenges on the kth vertical line. As n, m with m/n a for 0 < a <, we have Y k n,m m/ n(a + a)/8 GUE k weakly as RVs. Moreover, the collection { } Y j k n,m m/ weakly converge as n(a +a)/8 j= RVs to the GUE corners process Proof method: Generating function for free boundary tilings is an SO n+ character: λ (m n ) s λ (x,..., x n) = det[xm+n i j det[x n i j x i j ] i,j n x i j ] i,j n = γ (m n )(x), Apply the same asymptotic techniques to this MGF as for GUE eigenvalues MGF as N. 0

36 symmetric with free : limit shape (limit surface) Theorem (P) Let n, m Z, such that m/n a as n, where a (0, + ). Let H(u, v) be the height function (plane partition) of the uniformly random lozenge tiling in T f (n, m),i.e. H(u, v) = n y nu nv v. For all u (0, ), u v 0, as n we have that H(u, v) converges uniformly in probability to a deterministic function L(u, v), referred to as the limit shape. Moreover, for any fixed u (0, ), the function L(u, v) is the distribution function of the limit measure m whose moments are given by r ( r x r m(dx) = R l) l (l + )! t l Φa(t), l=0 t= where Φ a(e y ) = y a + φ(y; a) and... h(y) = 4 ( (e y ) + ) + (e y + ) + 4(a + a) (e y ) φ(y; a) = ( a ( + ) ln h(y) ( a + )(e y ) ) ( a + ( ) ln h(y) ( a + )(e y ) ) + a ( ln h(y) + a (e y ) ) ( a ( ) ln h(y) + ( a )(e y ) )

37 symmetric with free : limit shape (limit surface) Theorem (P) Let n, m Z, such that m/n a as n, where a (0, + ). Let H(u, v) be the height function (plane partition) of the uniformly random lozenge tiling in T f (n, m),i.e. H(u, v) = n y nu nv v. For all u (0, ), u v 0, as n we have that H(u, v) converges uniformly in probability to a deterministic function L(u, v), referred to as the limit shape. Proof, idea: 4 Horizontal lozenges at line x = un at positions µ = y un distributed ρ (unif. on all tilings), giving a sequence of random measures m[µ] = ( µi ) δ. n n i S ρ(u,..., u N ) := µ:l(µ)=n ρ(µ) sµ(u,..., u N ) s µ( N. ) (and observe S ρ = γ m n (x,...,x N ) our MGF) γ (m n ) ( N ) Using this MGF, its and operators, obtain a concentration phenomenon: m[µ] m a deterministic measure, which is the limit shape L. 4 after [Borodin-Bufetov-Olshanski, Bufetov-Gorin]

38 symmetric with free : limit shape (limit surface) Theorem (P) Let n, m Z, such that m/n a as n, where a (0, + ). Let H(u, v) be the height function (plane partition) of the uniformly random lozenge tiling in T f (n, m),i.e. H(u, v) = n y nu nv v. For all u (0, ), u v 0, as n we have that H(u, v) converges uniformly in probability to a deterministic function L(u, v), referred to as the limit shape. Corollary (P) The height function of the uniformly random lozenge tilings of a half-hexagon with free right boundary converges (in probability) to a unique limit shape (surface) H(x, y), which coincides over the half-hexagon with the limit shape for the tilings of the full hexagon (fixed boundary). The shifted by m/ and rescaled by n(a + a)/8 positions of the horizontal lozenges on the k-th vertical line have the same joint distributions as n, m, which is GUE k.

39 symmetric 6 Vertex model / ASM Six vertex types: H H O O H O H H O H O H O H H a a b b c c Alternating Sign Matrix: H H H A 6 vertex model configuration: H H H O H O H O H O H O H H H H H O H O H O H O H O H H H H H O H O H O H O H O H H H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H

40 symmetric Definitions and background on ASMs Definition: A is an Alternating Sign Matrix ASM of size n if: n n A {0, +, } n n, A i,j =, A i,j = i= j= and (A i,k, i =... n s.t. A i,k 0) = (,,,,...,, ) A monotone triangle is a Gelfand-Tsetlin pattern with strictly increasing rows. 6 Vertex model ASM monotone triangles. Uniform measure on ASMs vertices in 6V model have same weights ( ice ) ASM: positions of s Monotone triangle: 4 5 in sum of first k rows < Question: As n : Uniformly random ASM. What is the distribution of the positions of the s and s near the boundary Distribution of the top k rows of the monotone triangle? Known results: Limit behavior(conj): Behrend, Colomo, Pronko, Zinn-Justin, Di Francesco. Free fermions point (weight(,-)=) domino tilings, Aztec diamond. Exact gen. fs. for certain statistics (e.g. positions of s on boundary).

41 symmetric ASMs/6Vertex: new results k : ASM A: Ψ k (A) := j=:n, A kj = j Monotone triangle M = [m i j ] j i : Ψ k (M) = j=:n, A kj = k j= j k mj k m k j j= Ψ = = Ψ = ( + 5) (4) = Ψ = Ψ = ( + + 5) ( + 5) 4

42 symmetric ASMs/6Vertex: new results k : ASM A: Ψ k (A) := j=:n, A kj = j Monotone triangle M = [m i j ] j i : Ψ k (M) = j=:n, A kj = k j= j k mj k m k j j= Ψ = = Ψ = ( + 5) (4) = Ψ = Ψ = ( + + 5) ( + 5) Theorem (Gorin-P) If A unif. rand. n n ASM, then Ψ k (A) n/ n, k =,,... converge as n to the collection of i.i.d. Gaussian random variables, N(0, /8). 4

43 symmetric ASMs/6Vertex: new results k : ASM A: Ψ k (A) := j=:n, A kj = j Monotone triangle M = [m i j ] j i : Ψ k (M) = j=:n, A kj = k j= j k mj k m k j j= Ψ = = Ψ = ( + 5) (4) = Ψ = Ψ = ( + + 5) ( + 5) Theorem (Gorin-P) If A unif. rand. n n ASM, then Ψ k (A) n/ n, k =,,... converge as n to the collection of i.i.d. Gaussian random variables, N(0, /8). Using this Theorem on Ψ k (n) and the Gibbs property: Theorem (G, 0; Conjecture in [Gorin-P] ) Fix any k. The [centered,rescaled] positions of s on the first k rows (top k rows of the monotone triangle M) tend to the GUE-corners process: 8 ( [M] i=:k n ) GUE k. n 4

44 symmetric 6Vertex/ASMs: proofs Vertex weights at (i, j): type a : q ui qvj, type b : q vj qui, type c : (q q)u i v j (v,..., v N, u,..., u N parameters, q = exp(πi/) ) Weight W (ϑ) of a configuration ϑ is = weight(vtx). vtx ϑ Set λ(n) := (N, N, N, N,...,,, 0, 0) GT N. Proposition (Okada;Stroganov) Let ג N be the set of all 6-Vertex configurations on an N N grid. N W (ϑ) = ( ) N(N )/ (q q) N (v i u i ) s λ(n) (u,..., u N, v,..., v N ). ג ϑ N i= 5

45 symmetric 6Vertex/ASMs: proofs Vertex weights at (i, j): type a : q ui qvj, type b : q vj qui, type c : (q q)u i v j (v,..., v N, u,..., u N parameters, q = exp(πi/) ) Weight W (ϑ) of a configuration ϑ is = weight(vtx). vtx ϑ Set λ(n) := (N, N, N, N,...,,, 0, 0) GT N. Proposition Let x i =number of vertices of type x on row i, then collection of rows i,..., i m m q E N ( qv )âil ( l q vl q ) bil q q q (v l q )ĉjl l= ( n ) ( = v sλ(n) (v,..., v m, N m ) n ) l s λ(n) ( N = v l S λ(n) (v,..., v m) ) l= l= 5

46 symmetric 6Vertex/ASMs: proofs Vertex weights at (i, j): type a : q ui qvj, type b : q vj qui, type c : (q q)u i v j (v,..., v N, u,..., u N parameters, q = exp(πi/) ) Weight W (ϑ) of a configuration ϑ is = weight(vtx). vtx ϑ Set λ(n) := (N, N, N, N,...,,, 0, 0) GT N. Proposition Let x i =number of vertices of type x on row i, then collection of rows i,..., i m m q E N ( qv )âil ( l q vl q ) bil q q q (v l q )ĉjl l= ( n ) ( = v sλ(n) (v,..., v m, N m ) n ) l s λ(n) ( N = v l S λ(n) (v,..., v m) ) l= l= Proof of Theorem: Proposition moment generating function for Ψ k = â k as a normalized Schur function. Using c k k (bounded), approximate the weights to get the MGF for â k. Asymptotics: S λ(n) (e y / n,..., e y k / n ) = k i= [ nyi exp + 5 ] y i + o() 5

47 symmetric The dense loop model Finite grid (here: vertical strip of width L and height ) tiled with squares, boundary triangales: y ζ ζ x The mean total current between points x and y: F x,y the average number of paths connecting the and passing between x and y. Similar observables in the critical percolation model [Smirnov, 009]. L 6

48 symmetric : the mean current Let λ L = ( L L,,...,, 0, 0) Define: [ u L (ζ, ζ ; z,..., z L ) = ( ) L χλ ı ln L+ (ζ, z,..., z L )χ λ L+ (ζ, z,..., z L ) ] χ λ L (z,..., z L )χ λ L+ (ζ, ζ, z,..., z L ) where χ ν is the character for the irreducible representation of highest weight ν of the symplectic group Sp(C). X (j) L = z j z j u L (ζ, ζ ; z,..., z L ) Y L = w w u L+(ζ, ζ ; z,..., z L, vq, w) v=w, Proposition (De Gier, Nienhuis, Ponsaing) Under certain assumptions the mean total current between two horizontally adjacent points is X (j) L = F (j,i),(j+,i), and Y is the mean total current between two vertically adjacent points in the strip of width L: Y (j) L = F (j,i),(j,i+). 7

49 symmetric : of the mean current Theorem As L we have and X (j) L zj =z; z i =, i j = i 4L (z z ) + o ( ) L Y L = i ( ) zi =, i=,...,l 4L (w w ) + o L Remark. When z =, F (j,i),(j+,i) is (trivially) identically zero. Remark. The fully homogeneous case when w = exp iπ/6, q = e πi/, then ( ) Y L = L + o. L Proof: same type of asymptotic methods and results hold for symplectic characters + some tricks with the multivariate formula. 8

50 symmetric Thank you 9