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

2. Boundary connections (Razumov & Stroganov)

4. Exponents from networks (Duplantier & Saleur)

5. 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 Arbitrary finite graph with two special nodes Kirchoff’s formula for resistance 3 spanning trees 5 2-tree forests with nodes 1 and 2 separated

6. 54 2 13 54 2 13 Spanning tree Kirchoff matrix (negative Laplacian) 54 2 13 Spanning forest rooted at {1,2,3} Matrix-tree theorem (Kirchoff)

7. 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13 54 2 13

8. 54 2 13 Arbitrary finite graph with two special nodes (Kirchoff) 3 three

9. Arbitrary finite graph with four special nodes? 5 3 2 14 All pairwise resistances are equal 3 2 14 Need more than boundary measurements (pairwise resistances) Need information about internal structure of graph

10. 54 2 13 Planar graph Special vertices called nodes on outer face Nodes numbered in counterclockwise order along outer face Circular planar graphs 5 3 2 14 circular planar 3 2 1 4 planar, not circular planar

11. Noncrossing (planar) partitions 2 13 4 2 13 4 2 13 4

12. 54 2 13 Goal: compute the probability distribution of partition from random grove

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

15. Multichordal SLE 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 Crossing probabilities: Multichordal SLE 2, SLE 4, SLE 8, double-dimer paths – Kenyon & W ’06 SLE 4 characterization of discrete Guassian free field – Schramm & Sheffield ’06 SLE and ADE (from CFT) – Cardy ’06 Surprising connection between  =4 and  =8,2

16. Uniformly random grove

17. Peano curves surrounding trees

18. Multichordal loop-erased random walk

19. Double-dimer configuration

20. Noncrossing (planar) pairings 2 13 4 2 13 4 2 13 4

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

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

23. DGFF vs double-dimer model DGFF has SLE 4 contours (Schramm- Sheffield) Double-dimer believed to have SLE 4 contours, no proof Connection probabilities are the same in the scaling limit (Kenyon-W ’06)

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

25. 54 2 13

26. 54 2 13 0

27. 1 2 4 3

28. 1 2 4 3

29. Grove partition probabilities

31. Bilinear form on planar partitions / planar pairings

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

33. Bilinear form on planar partitions / planar pairings

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

38. 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. j n n Now induct on # parts that cross part containing j & n Use crossing rule with part closest to j

39. Grove partition probabilities

40. Dual electric network & dual partition 54 2 13 1 2 3 4 Planar graph Dual graph Grove Dual grove 1 2 3 4 54 2 13

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

43. Pfaffian formula 1 2 3 4 56

45. Double-dimer pairing probabilities

47. Planar partitions & planar pairings

49. Assume nodes alternate black/white

53. arXiv:math.PR/0608422

54. Caroll-Speyer groves