1 / 23

Critical resonance in the non-intersecting lattice path model

Critical resonance in the non-intersecting lattice path model. Richard W. Kenyon Université Paris-Sud. David B. Wilson Microsoft Research. Non-intersecting lattice paths. Partition Function Z. Weights a,b,c and the Phases. a. Solid `a’ type edges. Solid `b’ type edges.

neal
Download Presentation

Critical resonance in the non-intersecting lattice path model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Critical resonance in the non-intersecting lattice path model Richard W. Kenyon Université Paris-Sud David B. Wilson Microsoft Research

  2. Non-intersecting lattice paths

  3. Partition Function Z

  4. Weights a,b,c and the Phases a Solid `a’ type edges Solid `b’ type edges Solid `c’ type edges c b

  5. Solid and liquid phases and the melting transition between them

  6. The transition: a=1, b=c=1/2

  7. Partition function and edge density

  8. If m, n even, each loop has even length, (-1)=(-1)#loops Kasteleyn Matrices Topology: If a loop windsptimes horizontally,qtimes vertically, then all loops wind (p,q) times, where gcd(p,q)=1 Weight horizontal cut-edges by (-1) vertical cut-edges by (-1)

  9. Kasteleyn Matrices

  10. Simplifying Z

  11. Multiplicands for Z

  12. Multiplicands for three regions with different aspect ratios

  13. Integral approximation for log Z

  14. Anatomy of the resonant spikes

  15. Anatomy of the resonant spikes

  16. Ratcheting(?) vs Many loops, each winding around once One loop, winding around many times

  17. Free fermions in 1 dimension Fermions are particles that repel one another. In 1 dimension, log(Z)/ -Li3/2(-) density / -Li1/2(-). Same equations for lattice paths when (presumed) ratchet number is 0. When (presumed) ratchet number is nonzero, behavior differs from free fermions.

  18. What happens away fromthe simple rationals?

  19. Resonance Many physical systems exhibit resonance when ratios of frequencies nearly rational. Does resonance for lattice paths resemble resonance in other systems?

  20. Branch cuts and the integral

  21. Portion of domino tiling of the plane

  22. Matchings, tilings, & lattice paths

  23. Three solid phases and liquid phase

More Related