limit shapes beyond dimers

Transcription

1 limit shapes beyond dimers Richard Kenyon (Brown University) joint work with Jan de Gier (Melbourne) and Sam Watson (Brown)

2 What is a limit shape?

3 Lozenge tiling limit shape Thm[Cohn,K,Propp (2000)] The function h : R! R describing the limit shape is the unique minimizer of the surface tension integral ZZ min (rh) dx dy. h R (s, t) is the entropy for tilings with slope (s, t).

4 The six-vertex model a 1 a 2 a 3 a 4 a 5 a 6 Lieb ( 69): a 1 = a 2,a 3 = a 3,a 5 = a 6 free fermions a 1 a 2 + a 3 a 4 a 5 a 6 =0 a 1 a 2 a 3 a 4 a 5 a 6

5 The five vertex model: a generalization of the lozenge tiling model a special case of the six-vertex model The five-vertex model 1 0 e X e Y re X+Y 2 re X+Y 2 A configuration has probability 1 Z evx+hy r c where r is the number of corners, v is the number of vertical edges, h is the number of horizontal edges. X =0,Y =0,r = 1:

6 lozenge tilings and the 5-vertex model The 5 vertex model with r = 1 is the lozenge tiling model. r 6= 1 means blue and green lozenges interact.

7 X, Y play the role of a magnetic field ; 1 e X e Y re X+Y 2 re X+Y 2 For fixed r, varying (X, Y ) corresponds to varying density (s, t) of lines s = horizontal density t = vertical density However the relationship between (X, Y ) and (s, t) is far from trivial. In fact knowing this relationship is equivalent to knowing the free energy: rf (X, Y )=(s, t).

8 In the r = 1 (lozenge tiling) case, the relationship between (s, t) and (X, Y )is both simple and mysterious: z e X e Y s t 0 1 (0, 1) Y 2 t r rf 1 0 s (0, 0) (1, 0) X X

9 The surface tension (s, t) is the Legendre dual of the free energy F (X, Y ) In case r = 1, (s, t) = D(z), the Bloch-Wigner dilogarithm

10 General r case: how to find the free energy F (X, Y )? No determinant formulas... need to use Bethe Ansatz that is, find an explicit diagonalization of the 2 N 2 N transfer matrix T T ({1, 5, 6}, {3, 5, 7}) =r 4 T has a partial diagonalization into blocks T k : 0 1 T 0 T 1 T = C A TN T k is the n k n k transfer matrix for k particles

11 For T 1, eigenvectors have the form f (x) = x where N = 1. For T 2, eigenvectors have the form f 1, 2 (x 1,x 2 )=A 12 x 1 1 x A 21 x 2 1 x 1 2 For T k, eigenvectors have the form f 1,..., k (x 1,...,x k )= X (for x i 2 [N]) 2S k A x1 (1)... x k (k) =det A for some constants i and A = A ( 1,..., N ) 0 x x 1 k.. x k 1... x k k 1 C A

12 For T n, the Bethe roots satisfy a system of polynomial equations N i =( 1) n 1 n Y j=1 1 (1 r 2 )e Y 1 j 1 (1 r 2 )e Y 1 i, Let (1 r 2 )e Y w j = j. w N n i (1 w i ) n = C ny j=1 w j 1 w j { symmetric in all w j s

13 w 12 (1 w) 4 = const ovals with the same / The rescaled Bethe roots w i lie on Cassini ovals C, = {w : log w + log 1 w =1}

14 The leading eigenvalue is = e Xn e Y (N n) 1 ( 1) n Ar 2n (1 r 2 ) N n Y j=1 r 2 1 (1 r 2 )w j where w N n i (1 w i ) n = A

15 There is an explicit formula for r(s, t) when r<1, (s, t) is piecewise analytic: neutral 1 s t +(1 r 2 )st =0 0.2 repulsive

16 r (s, t) =(X, Y ), where X, Y are defined as follows. (Case r<1) z tθ w = (r, s, t) 0 sθ (1-s)θ (1-s-t)θ 1 1-r 2 Note: (z 1)(w 1) = r 2 X = log(1 r 2 ) z B( 1 r 2 ) Y = log(1 r 2 ) w B( 1 r 2 ) B(u) = 1 (arg(u) log 1 u + Im Li(u))

17 r rr

18 r>1 case F

19 For r = 1, the PDE describing the limit shape is reducible to the complex Burgers equation z x + zz y =0 [K, Okounkov 07]

20 Thm: (...analogous limit shape theorem for 5-vertex model) Cor: The PDE for the limit shape can be reduced to the PDE for z = z(x, y): z x + f(z)z y =0 where f(z) is explicit (but only real analytic, not complex analytic.) [z arg z +(1 z) arg(1 z)]z x +[w arg w +(1 w) arg(1 w)]w y =0 where (z 1)(w 1) = r 2.

21 Simulations r = 10 r =1 r =.1

22 What are the fluctuations in the neutral phase?

23 The leading eigenvector of T k as the form f 1,..., k (x 1,...,x k )= X 2S k A x1 (1)... x k (k) =det A 0 x x 1 k.. x k 1... x k k 1 C A and Pr(particles at x 1,x 2,...,x k ) / f(x 1,...,x k )f(n x 1,...,N x k ) in our case A =( 1) Y 1applei<japplen (1 1 (j) ) leading to f = Q x 1+ +x n det with Q =( Q n j=1 j) 1/n. 0 x x 1 n (1 1 1 ) x (1 n 1 ) x 2 n.. ( x n 1... (1 n 1 ) n 1 x n n 1 C A

24 1)N/k); in the at- We can compute f(1, 2, 3,...,k) and f(n/k, 2N/k,..., (k tracting phase their ratio tends to 1 as N,k!1. These 10 particles are distributed like electrons in a conductor Thm: In the attracting phase particles on a given row or column are uniformly located (conditionally on being disjoint). Height fluctuations are O( p N).

25 In the r! 0 limit, (S, T ) is defined on ST < 1 and is a function of ST.

26 Thank you