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Boundary Partitions in Trees and Dimers

Boundary Partitions in Trees and Dimers. (Connection probabilities in multichordal SLE 2 , SLE 4 , and SLE 8 ). arXiv:math.PR/0608422. Richard W. Kenyon and David B. Wilson. University of British Columbia Brown University. Microsoft Research.

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Boundary Partitions in Trees and Dimers

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  1. Boundary Partitions in Trees and Dimers (Connection probabilities in multichordal SLE2, SLE4, and SLE8) arXiv:math.PR/0608422 Richard W. Kenyon and David B. Wilson University of British Columbia Brown University Microsoft Research

  2. Boundary connections(Razumov & Stroganov)

  3. Exponents from networks(Duplantier & Saleur)

  4. Kirchoff’s formula for resistance 1 3 Arbitrary finite graph with two special nodes 5 4 2 1 3 1 3 1 3 1 3 1 3 5 4 5 4 5 4 5 4 5 4 2 2 2 2 2 5 2-tree forests with nodes 1 and 2 separated 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2 3 spanning trees

  5. 1 3 5 4 2 1 3 5 4 2 Matrix-tree theorem (Kirchoff) 1 3 5 4 2 Spanning forest rooted at {1,2,3} Spanning tree Kirchoff matrix (negative Laplacian)

  6. 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2 1 3 1 3 1 3 5 4 5 4 5 4 2 2 2

  7. 1 3 3 Arbitrary finite graph with two special nodes three 5 4 (Kirchoff) 2

  8. Arbitrary finite graph with four special nodes? 1 4 All pairwise resistances are equal 5 2 3 1 4 All pairwise resistances are equal 3 2 Need more than boundary measurements (pairwise resistances) Need information about internal structure of graph

  9. Circular planar graphs 1 1 4 1 3 4 3 5 5 4 2 2 3 planar, not circular planar 2 circular planar circular planar Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face

  10. Noncrossing (planar) partitions 4 4 1 3 1 3 2 2 4 1 3 2

  11. 1 3 5 4 2 Goal: compute the probability distribution of partition from random grove

  12. Carroll-Speyer groves Carroll-Speyer ’04 Petersen-Speyer ’05

  13. Multichordal SLE Crossing probabilities: Percolation -- Cardy ’92 Smirnov ’01 Critical Ising – Arguin & Saint-Aubin ’02 Smirnov ’06 Bichordal SLE – Bauer, Bernard, Kytölä ’05 Trichordal percolation, multichordal SLE – Dubédat ’05 Covariant measure for parallel crossing – Kozdron & Lawler ’06 Multichordal SLE2, SLE4, SLE8, double-dimer paths – Kenyon & W ’06 SLE4 characterization of discrete Guassian free field – Schramm & Sheffield ’06 SLE and ADE (from CFT) – Cardy ’06 Surprising connection between =4 and =8,2

  14. Uniformly random grove

  15. Peano curves surrounding trees

  16. Multichordal loop-erased random walk

  17. Double-dimer configuration

  18. Noncrossing (planar) pairings 4 4 1 3 1 3 2 2 4 1 3 2

  19. Double-dimer model in upper half plane with nodes at integers

  20. Contours in discrete Gaussian free field(Schramm & Sheffield)

  21. DGFF vs double-dimer model • DGFF has SLE4 contours (Schramm-Sheffield) • Double-dimer believed to have SLE4 contours, no proof • Connection probabilities are the same in the scaling limit (Kenyon-W ’06)

  22. (negative of) Dirichlet-to-Neumann matrix Electric network

  23. 1 3 5 4 2

  24. 1 3 5 4 2 0

  25. 4 1 3 2

  26. 4 1 3 2

  27. Grove partition probabilities

  28. Bilinear form onplanar partitions / planar pairings

  29. Ko & Smolinsky determine when matrix is singular Gram Matrix of Temperley-Lieb Algebra Meander Matrix Di Francesco, Golinelli, Guitter diagonalize matrix

  30. Bilinear form onplanar partitions / planar pairings

  31. (extra term in recent work by Caraciollo-Sokal-Sportiello on hyperforests) These equivalences are enough to compute any column!

  32. Computing column  By induction find equivalent linear combination when item n deleted from . If {n} is a part of , use rule for adjoining new part. Otherwise, n is in same part as some other item j, use splitting rule. n n Now induct on # parts that cross part containing j & n Use crossing rule with part closest to j j

  33. Grove partition probabilities

  34. 3 1 3 5 4 3 1 2 4 2 1 2 4 1 3 5 4 2 Dual electric network & dual partition Planar graph Dual graph Grove Dual grove

  35. Curtis-Ingerman-Morrow formula 1 8 2 7 3 6 4 5 Fomin gives another version of this formula, with combinatorial proof

  36. Pfaffian formula 6 5 1 4 2 3

  37. Double-dimer pairing probabilities

  38. Planar partitions & planar pairings

  39. Planar partitions & planar pairings

  40. Assume nodes alternate black/white

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