Random tilings and representations of classical Lie groups Alexey - PowerPoint PPT Presentation

Random tilings and representations of classical Lie groups Alexey Bufetov University of Bonn 21 November, 2019

Domino tilings of Aztec diamond The Aztec diamond of size n is the set of all lattice squares which are (fully) contained in { ( x , y ) : | x | + | y | ≤ n + 1 } .

Domino tilings of Aztec diamond Let us consider a chessboard coloring of the Aztec diamond. It is useful to distinguish not two, but four different types of dominoes.

Domino tilings of Aztec diamond The Aztec diamond of size n is the set of all lattice squares which are (fully) contained in { ( x , y ) : | x | + | y | ≤ n + 1 } . Domino tilings of the Aztec diamond were introduced by Elkies-Kuperberg-Larsen-Propp’92. They proved that the number of tilings is equal to 2 n ( n +1) / 2 . Question: What happens when we consider a uniformly random domino tiling of the Aztec diamond of large size ? We will color four types of dominoes by different colors in the next picture.

We see that a uniformly random domino tiling has some structure ! Theorem (Jockusch-Propp-Shor’95): Asymptotically a uniformly random tiling becomes frozen outside of a certain circle. There are many more interesting properties of these tilings.

More general domains

Bufetov-Knizel’16

Domino tilings General domains: Concentration phenomenon (existence of limit curve and shape): Cohn-Kenyon-Propp’01, Kenyon-Okounkov-Sheffield’06, Kenyon-Okounkov’07. The most studied example: Aztec diamond; Jockusch-Propp-Shor (1995), Johansson (2003), Chhita-Johansson-Young (2012). Limit shape, global fluctuations. Bufetov-Knizel’16: rectangular Aztec diamonds: arbitrary boundary conditions on one of the sides. Limit shapes, explicit formulae for frozen boundary curves, central limit theorem for global fluctuations.

Lozenge tilings N = 6

Bufetov-Gorin’16: LLN+CLT, extending results of Petrov’12.

Lozenge tilings Kenyon-Okounkov conjecture: Kenyon (2004) : a class of domains with no frozen regions. Borodin-Ferrari (2008): Some infinite domains with frozen regions. Petrov (2012), Bufetov-Gorin (2016): A class of simply-connected domains with arbitrary boundary conditions on one side. (Boutillier-de Tili` ere (2009), Dubedat (2011)),Berestycki-Laslier-Ray (2016) ) : Some non-planar domains. Bufetov-Gorin (2017): Some domains with holes.

Bufetov-Gorin’17: LLN+CLT.

Signatures and Schur functions A signature of length N is an N -tuple of integers λ = l 1 > l 2 > · · · > l N Sign( N ) — the set of all signatures of length N . The Schur function is defined by � � l j det i , j =1 ,..., N x i s λ ( x 1 , . . . , x N ) := 1 ≤ i < j ≤ N ( x i − x j ) , � where λ is a signature of length N . The Schur function is a Laurent polynomial in x 1 , . . . , x N .

Representations of U ( N ) Let U ( N ) denote the group of all N × N unitary matrices. It is known that all irreducible finite-dimensional representations of U ( N ) are parameterized by signatures (highest weights). Let π λ be an irreducible representation of U ( N ) corresponding to λ . The character of π λ is a function on U ( N ). Its value on all matrices with the same eigenvalues is the same. The values of the character of π λ is the Schur function s λ ( x 1 , . . . , x N ), where x i are eigenvalues of an element from U ( N ).

Application to tilings Let λ be a signature of length N . We have branching rule: � s λ ( x 1 , . . . , x N − 1 , 1) = s µ ( x 1 , . . . , x N − 1 ) , µ ≺ λ where µ ≺ λ means that they interlace : λ 1 > µ 1 ≥ λ 2 > µ 2 ≥ · · · ≥ λ N − 1 > µ N − 1 ≥ λ N . λ 5 λ 4 λ 3 λ 2 λ 1 ❤ ❤ ❤ ❤ ❤ µ 4 µ 3 µ 2 µ 1 ❤ ❤ ❤ ❤

Gelfand-Tsetlin arrays Considering s λ ( x 1 , . . . , x M , 1 , 1 , . . . , 1), for M < N , one obtains the following picture: λ 4 λ 3 λ 2 λ 1 ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ For large N we consider uniformly random Gelfand-Tsetlin arrays with fixed upper row λ . What is the behavior of the signature on level [ α N ], 0 < α < 1. ?

Lozenge tilings N = 6

Bufetov-Gorin’16: LLN+CLT, extending results of Petrov’12.

Asymptotic representation theory Tensor products of representations: Let λ and µ be signatures of length N . We consider the decomposition of the (Kronecker) tensor product π λ ⊗ π µ into irreducible components π λ ⊗ π µ = � c λ,µ π η , η η where η runs over signatures of length N . The decomposition is given by the classical Littlewood-Richardson rule. However, for large N it is hard to “extract information” from this rule.

Finite level Let A and B be two Hermitian matrices with known eigenvalues. What can we say about the eigenvalues of A + B ? For which triples of signatures ( λ, µ, η ) the Littlewood-Richardson coefficient c λ,µ is strictly positive ? η The two questions above are closely related. The final answer to both of them was found by Knutson-Tao’98. One can say that we will deal with randomized, asymptotic versions of these questions. What happens in a typical situation ?

Random matrices Let A be a N × N Hermitian matrix with eigenvalues { a i } N i =1 . Let N m [ A ] := 1 � δ ( a i ) N i =1 be the empirical measure of A . For each N = 1 , 2 , . . . take two sets of real numbers a ( N ) = { a i ( N ) } N i =1 and b ( N ) = { b i ( N ) } N i =1 . Let A ( N ) be the uniformly (“Haar distributed”) random N × N Hermitian matrix with fixed eigenvalues a ( N ) and let B ( N ) be the uniformly (“Haar distributed”) random N × N Hermitian matrix with fixed eigenvalues b ( N ) such that A ( N ) and B ( N ) are independent.

Free convolution Suppose that as N → ∞ the empirical measures of A ( N ) and B ( N ) weakly converge to probability measures m 1 and m 2 , respectively. Theorem (Voiculescu, 1991) The random empirical measure of the sum A ( N ) + B ( N ) converges (weak convergence; in probability) to a deterministic measure m 1 ⊞ m 2 which is the free convolution of m 1 and m 2 . Central Limit Theorem: explicit formulae for covariance (Pastur-Vasilchuk’06); second order freeness (Mingo-Speicher’04, Mingo-Sniady-Speicher’04), ...

Representation theory → Probability Vershik-Kerov, 80’s. Given a finite-dimensional representation π of U ( N ) we can decompose it into irreducible components: � c λ π λ , π = λ where non-negative integers c λ are multiplicities. This decomposition gives rise to a probability measure ρ π on signatures of length N such that ρ π ( λ ) := c λ dim( π λ ) . dim( π )

representation π λ ⊗ π µ → random signature ρ π λ ⊗ π µ → random probability measure m [ ρ π λ ⊗ π µ ] on R . � l i � N m [ λ ] := 1 � i =1 δ . N N Assume that two sequences of signatures λ = λ ( N ) and µ = µ ( N ) satisfy N N 1 � λ i � 1 � µ i � � � δ − N →∞ m 1 , − − → δ − N →∞ m 2 , − − → N N N N i =1 i =1 where m 1 and m 2 are probability measures. We are interested in the asymptotic behaviour of the decomposition of the tensor product into irreducibles, i.e., we are interested in the asymptotic behaviour of the random probability measure m [ ρ π λ ⊗ π µ ].

Limit results for tensor products Under assumptions above, we have (Bufetov-Gorin’13, Bufetov-Gorin’16): Law of Large Numbers: N →∞ m [ ρ π λ ⊗ π µ ] = m 1 ⊗ m 2 , lim where m 1 ⊗ m 2 is a deterministic measure on R . We call m 1 ⊗ m 2 the quantized free convolution of measures m 1 and m 2 . Central Limit Theorem: Fluctuations around the limit measure are Gaussian and given by an explicit double contour integral formula.

Related results In the case of the symmetric group related results were obtained by Biane (1998), Sniady (2005). Bufetov-Gorin’13, Bufetov-Gorin’16: Similar results for symplectic and orthogonal groups. LLN for Perelomov-Popov measures (Bufetov-Gorin’13); it is given by a conventional free convolution. Free independence: Collins-Novak-Sniady’16.

� � l j det i , j =1 ,..., N x i s λ ( x 1 , . . . , x N ) := 1 ≤ i < j ≤ N ( x i − x j ) , � An important role in all these applications is played by the following asymptotics. r is fixed, N → ∞ . The following two relations are equivalent (Guionnet-Maida’04, and also Gorin-Panova’13, Bufetov-Gorin’13). N log s λ ( N ) ( x 1 , . . . , x r , 1 N − r ) 1 → F 1 ( x 1 ) + · · · + F 1 ( x r ) s λ (1 N ) � � n l ( N ) ⇒ 1 � i ⇐ → µ 1 δ N N i =1 Notation: 1 N := (1 , 1 , . . . , 1) – N -tuple of 1’s.

Extreme characters of the infinite-dimensional unitary group Consider the tower of embedded unitary groups U (1) ⊂ U (2) ⊂ · · · ⊂ U ( N ) ⊂ U ( N + 1) ⊂ . . . . The infinite–dimensional unitary group U ( ∞ ) is the union of these groups. Character of U ( ∞ ) is a positive-definite class function χ : U ( ∞ ) → C , normalised at unity: χ ( e ) = 1. We consider characters instead of representations. Extreme characters serve as an analogue of irreducible representations.