A third-order phase transition in random domino tilings

Transcription

1 Statistical Field Theory, Low-Dimensional Systems, Integrable Models and Applications Firenze, 6-7 Giugno 2013 A third-order phase transition in random domino tilings Filippo Colomo INFN, Firenze Joint work with: Andrei Pronko (PDMI-Steklov, Saint-Petersbourg) [arxiv:0613.xxxx]

2 A well-known fact: Local defect: Extended defect line: Geometric constraints can induce effective long-range interactions.

3 As a result, thermodynamic limit may depend on boundary conditions. In particular, we can have: spatial dependence of order parameter free-energy density entropy density spatial phase separation i.e.,emergence of regions of order and disorder sharply separated by some smooth curves.

4 Domino tiling of a square domino:= tile

5 Square Aztec Diamond of Order ( )

6 Domino tiling of an Aztec diamond [Jockush-Propp-Shor '95]

7 Domino tiling of an Aztec diamond [Jockush-Propp-Shor '95] The Arctic Circle Theorem [Jockush-Propp-Shor '95] such that almost all (i.e. with probability ) randomly picked domino tilings of have a temperate region whose boundary stays uniformly within distance from the circle of radius.

8 Boxed plane partitions [Cohn-Larsen-Propp'98]

9 Corner melting of a crystal [Ferrari-Spohn '02] Plane partitions [Cerf-Kenyon'01][Okounkov-Reshetikhin'01]

10 Skewed plane partitions [Okounkov-Reshetikhin '05-'07] [Boutillier-Mkrtchyan-Reshetikhin-Tingley '12]

11 So we have seen: domino tilings; rhombi tilings; partitions; plane partitions; skewed plane partitions... Actually they are all avatars of the same model, `dimer covering of regular planar bipartite lattices', exhibiting emergence of phase separation, limit shapes, frozen boundaries /arctic curves, and fluctuations governed by Random Matrix models. The model has been solved in full generality [Kenyon, Sheffield, Okounkov, '03-'05] with deep implications in algebraic geometry and algebraic combinatorics.

12 End of introductory part.

13 Domino tiling of the Aztec diamond [Jockush-Propp-Shor'95]

14 [FC-Sportiello]

15

16 s+q r s N r+s+q=n

17 Partition function [FC-Pronko'08],[A. Pronko'13] (Hankel Determinant) Introducing `time' via i.e.,, Sylvester identity for determinants implies: is the tau-function of a semi-infinite Toda chain. Extracting the sum from the determinant, one can rewrite: Hermitean random matrix model (with discrete measure) [Douglas-Kazakov'93]

18 Free energy: Random Matrix Model approach For simplicity, we restrict to the symmetric situation NB: we are interested in the scaling limit, with : fixed. Free energy: To investigate the large In the large by integrals: behaviour of, one need to rescale: limit, sums can now be reinterpreted as Riemann sums, and replaced

19 Saddle-point approximation Write the integrand as: Saddle-point eqs. read: The solution of the saddle-point eqs. is given by the equilibrium configuration of a set of mutually repelling charged particles, in the linear potential, confined to the real interval :

20 Saddle-point approximation Introduce a normalized density of solutions of saddle-point eqs.: Discreteness of eigenvalues implies [Douglas-Kazakov'93] Standard methods (e.g. using the resolvent) can be exploited to determine and solve the model. The only caveat is the implementation of the constraint: (in fact, just a minor technical complication) [Douglas-Kazakov'93],[Brézin-Kazakov'99],[Zinn-Justin'00]

21 NB: We have two `hard walls' at and Near the edges of its support the density as a universal behaviour: If, then, e.g. in the vicinity of The discreteness constraint : thus implies: In the vicinity of an hard wall:

22 Two scenarios i) large (or small ): potential well is deep and narrow The eigenvalues accumulate to the left:

23 Two scenarios i) large (or small ): potential well is deep and narrow The eigenvalues accumulate to the left: ii) small (or large ): potential well is wide and not so deep The eigenvalues expand till the right wall:

24 Scenario i) We have: and, thus: 1 a where and b c are determined from the solution of the saddle-point eqs: NB1: Exactly the same Random Matrix Model appears: (but with ): in [Brezin-Kazakov'00] (statistics of partitions for the permutation group) in [Zinn-Justin'00], (ferroelectric phase ( ) of the DW 6VM partition function. NB2: This scenario holds as long as

25 Scenario ii) We have: and, thus: 1 a where and b are determined by the solution of the saddle-point equation: c

26 c=2 c=1.9 c=1.6 c=1.1

27 In all we have the following result for the free energy density of the somino tilings of the `modified' Aztex diamond: where. NB1: NB2: is the value corresponding to the Arctic Ellipse 3rd order phase transition at

28 Generic Potential case: is not linear any more: bulky calculations, but everything remains qualitatively the same: two scenarios separated by critical curve which is the Arctic Circle 3rd order phase transition

29 3rd-order phase transition in Matrix Models [Gross-Witten'80,Wadia-80]: standard phase invasive phase (compact interval) somewhere everywhere [Douglas-Kazakov'93] : (discrete eigenvalues) standard phase saturated phase everywhere somewhere Our model: (discrete eigenvalues & compact interval) saturated phase invasive saturared phase somewhere but also somewhere everywhere, with two saturated regions NB1: the appearance of 3rd-order phase in all three cases is obvious NB2: however the simultaneous occurrence of discreteness & hard walls is unusual, and a novel edge behaviour ( Airy ) might appear at the transition.