cover times, blanket times, and the GFF

1. Jian Ding Berkeley-Stanford-Chicago cover times, blanket times, and the GFF James R. Lee University of Washington Yuval Peres Microsoft Research

2. random walks on graphs • By putting conductances{cuv} on the edges of the graph, • we can get any reversible Markov chain.

3. Hitting time: H(u,v)= expected # of steps to hitvstarting at u • Commute time: κ(u,v) = H(u,v) + H(v,u) • (metric) hitting and covering • Cover time: tcov(G)=expected time to hit all vertices of G, • starting from the worst vertex

4. Hitting time: H(u,v)= expected # of steps to hitvstarting at u • Commute time: κ(u,v) = H(u,v) + H(v,u) • (metric) hitting and covering • Cover time: tcov(G)=expected time to hit all vertices of G, • starting from the worst vertex orders of magnitude of some cover times • path • complete graph • expander • 2-dimensional grid • 3-dimensional grid • complete d-ary tree • n2 • n log n • n log n • n (log n)2 • n log n • n (log n)2/log d • [coupon collecting] • [Broder-Karlin 88] • [Aldous 89, Zuckerman 90] • [Aldous 89, Zuckerman 90] • [Zuckerman 90]

5. Hitting time: H(u,v)= expected # of steps to hitvstarting at u • Commute time: κ(u,v) = H(u,v) + H(v,u) • (metric) hitting and covering • Cover time: tcov(G)=expected time to hit all vertices of G, • starting from the worst vertex general bounds (n= # vertices, m = # edges) • (1-o(1)) n logn·tcov(G)·2nm • [Feige’95, Matthews’88] • [Alelinuas-Karp-Lipton-Lovasz-Rackoff’79]

6. Hitting time: H(u,v)= expected # of steps to hitvstarting at u • Commute time: κ(u,v) = H(u,v) + H(v,u) • (metric) electrical resistance • Cover time: tcov(G)=expected time to hit all vertices of G, • starting from the worst vertex • [Chandra-Raghavan-Ruzzo-Smolensky-Tiwari’89]: • If G has m edges, then for every pair u,v u v • κ(u,v) = 2mReff (u,v) + • (endows κ with certain • geometric properties) • Reff (u,v) = inverse of electrical current • flowing from u to v

7. Hitting time: Easy to compute in deterministic poly time by solving • system of linear equations computation H(u,u) = 0 H(u,v) = 1+ w»uH(w,v) • Cover time: Easy to compute in exponential time by a determinsitic algorithm • and polynomial time by a randomized algorithm • Natural question: Does there exist a poly-time deterministically computable • O(1)-approximation for general graphs? • [Aldous-Fill’94]

8. Clearly: approximation in derministic poly-time • [Matthews’88] proved: • and • [Kahn-Kim-Lovasz-Vu’99] show that Matthews’ lower bound gives an • O(log logn)2approximation to tcov • [Feige-Zeitouni’09] give a (1+²)-approximation for trees • for every ² > 0, using recursion.

9. blanket times 5 100 3 • Blanket times[Winkler-Zuckerman’96]: • The ¯-blanket time tblanket(G,¯)is the expected first time T at which all the local times, • are within a factor of ¯.

10. blanket times 0.01 0.2 0.333 • Blanket times[Winkler-Zuckerman’96]: 0.99 • The ¯-blanket time tblanket(G,¯)is the expected first time T at which all the local times, • are within a factor of ¯.

11. Conjecture [Winkler-Zuckerman’96]: • For every graph G and 0 < ¯ < 1, tblanket(G,¯)³tcov(G). blanket time conjecture ³ ³ ³ • Proved for some special cases. • True up to (log log n)2by [Kahn-Kim-Lovasz-Vu’99]

12. A centered Gaussian process {gv}v2V satisfying for all u,v2V: Gaussian free field on a graph • and for some fixed v02V.

13. A centered Gaussian process {gv}v2V satisfying for all u,v2V: Gaussian free field on a graph • and for some fixed v02V. • Density proportional to

14. A centered Gaussian process {gv}v2V satisfying for all u,v2V: Gaussian free field on a graph • and for some fixed v02V. • Covariance given by the Green function of SRW killed at v0 : • expected # visits to v from RW start at u before hitting v0

15. For every graph G=(V, E), main result • for every 0 < ¯ < 1, where {gv}v2V is the GFF on G. • Positively resolves the Winkler-Zuckerman blanket time conjectures. ³ ³ ³

16. embedded in Euclidean space via GFF. geometrical interpretation where D is the diameter of the projection of this embedding onto a random line: An illustration for the complete graph

17. Gaussian processes

18. Consider a Gaussian process {Xu : u2S}with E (Xu)=0 8u2S PROBLEM: What is E max { Xu : u2S } ? Gaussian processes Such a process comes with a natural metric transforming (S, d) into a metric space.

19. Majorizing measures theorem [Fernique-Talagrand]: majorizing measures • is a functional on metric spaces, given by an explicit formula which • takes exponential time to calculate from the definition

20. THEOREM: • For every graph G=(V, E), main theorem, restated • where · = commute time. • We also construct a deterministic poly-time algorithm to approximate • within a constant factor. • COROLLARY: There is a deterministic poly-time O(1)-approximation for tcov • Positively answers the question of Aldous and Fill.

21. Majorizing measures theorem [Fernique-Talagrand]: majorizing measures

22. PROBLEM: What is E max { Xu : u2S } ? • If random variables are “independent,” • expect the union bound to be tight. Gaussian processes ® • Expect max for k points is about • Gaussian concentration:

23. Gaussian concentration: Gaussian processes • Sudakov minoration:

24. chaining

25. chaining • Best possible tree upper bound yields • [Dudley’67]: • where N(S,d,²) is the minimal number of ²-balls needed to cover S

26. Gaussian concentration: hints of a connection • Sudakov minoration:

27. Sudakov minoration: hints of a connection • Matthew’s bound (1988):

28. Gaussian concentration: hints of a connection • KKLV’99 concentration: For all nodes and • global time. • is local time at and

29. Dudley’67 entropy bound: hints of a connection • Barlow-Ding-Nachmias-Peres’09 analog for cover times

30. hints of a connection

31. Dynkin isomorphism theory

32. Dynkin isomorphism theory Ray-Knight(1960s): Characterize the local times of Brownian motion. Dynkin (1980): General connection of Markov process to Gaussian fields. The version we used is due to Eisenbaum, Kaspi, Marcus, Rosen, and Shi(2000)

33. For define. Letbe the GFF on with Then where Generalized Ray-Knight theorem law law

34. blanket time vs. GFF

35. ³|E| law the lower bound To lower bound cover time: Need to show that for with good chance small small w.h.p.

36. only one Everest, but...

37. Gaussian free field: a problem on Gaussian processes

38. Gaussian free field: a problem on Gaussian processes • We need strong estimates on the size of this window. • (want to get a point there with probability at least 0.1) • Problem: Majorizing measures handles first moments, but we • need second moment bounds.

39. percolation on trees and the GFF

40. percolation on trees and the GFF

41. First and second moments agree for • percolation on balanced trees percolation on trees and the GFF • Problem: General Gaussian processes behaves nothing like percolation! • Resolution: Processes coming from the Isomorphism Theorem all arise • from a GFF

42. First and second moments agree for • percolation on balanced trees percolation on trees and the GFF • For DGFFs, using electrical network theory, we show that it is possible • to select a subtree of the MM tree and a delicate filtration of the • probability space so that the Gaussian process can be coupled to a • percolation process.

43. Bolthausen, Deuschel and Giacomin (2001): For Gaussian free field onlattice of vertices example: precise asymptotics in 2D Dembo, Peres, Rosen, and Zeitouni (2004): For 2D torus

44. Upper bound is true. • Lower bound holds for: • - complete graph • - complete d-ary tree • - discrete 2D torus • - bounded degree graphs [Ding’11] open questions • QUESTION: Is there a deterministic, polynomial-time • (1+²)-approximation to the cover time for every ² > 0 ? • QUESTION: Is the standard deviation of the time-to-cover bounded • by the maximum hitting time?