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

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

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

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

5. hitting and covering Hitting time: H(u,v) = expected # of steps to hit v starting at u Commute time: κ (u,v) = H(u,v) + H(v,u) (metric) Cover time: t cov ( G ) = expected time to hit all vertices of G, starting from the worst vertex regular trees random graphs discrete torus, lattices [Aldous 91] [Cooper-Frieze 08] [Dembo-Peres-Rosen-Zeitouni 04] asymptotically optimal bounds

6. hitting and covering (1- o (1)) n log n · t cov ( G ) · 2 nm general bounds ( n = # vertices, m = # edges) [Alelinuas-Karp-Lipton-Lovasz-Rackoff’79] [Feige’95, Matthews’88] Hitting time: H ( u, v ) = expected # of steps to hit v starting at u Commute time: κ ( u, v ) = H ( u, v ) + H ( v, u ) (metric) Cover time: t cov ( G ) = expected time to hit all vertices of G, starting from the worst vertex

7. electrical resistance + u v R eff ( u, v ) = inverse of electrical current flowing from u to v [Chandra-Raghavan-Ruzzo-Smolensky-Tiwari’89]: If G has m edges, then for every pair u, v κ ( u, v ) = 2 m R eff ( u, v ) (endows κ with certain geometric properties) Hitting time: H ( u, v ) = expected # of steps to hit v starting at u Commute time: κ ( u, v ) = H ( u, v ) + H ( v, u ) (metric) Cover time: t cov ( G ) = expected time to hit all vertices of G, starting from the worst vertex

8. computation Hitting time: Easy to compute in deterministic poly time by solving system of linear equations 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]

9. approximation in derministic poly-time Clearly: [Kahn-Kim-Lovasz-Vu’99] show that Matthews’ lower bound gives an O (log log n ) 2 approximation to t cov [Matthews’88] proved: and [Feige-Zeitouni’09] give a ( 1 + ² )-approximation for trees for every ² > 0, using recursion.

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

11. blanket times Blanket times [Winkler-Zuckerman’96] : The ¯ -blanket time t blanket ( G, ¯ ) is the expected first time T at which all the local times, are within a factor of ¯. 0.010.20.333 0.99

12. blanket time conjecture Conjecture [Winkler-Zuckerman’96]: For every graph G and 0 < ¯ < 1, t blanket ( G, ¯ ) ³ t cov ( G ). Proved for some special cases. ³³ True up to (log log n) 2 by [Kahn-Kim-Lovasz-Vu’99] ³

13. Gaussian free field on a graph A centered Gaussian process { g v } v 2 V satisfying for all u, v 2 V : and for some fixed v 0 2 V.

14. Gaussian free field on a graph A centered Gaussian process { g v } v 2 V satisfying for all u, v 2 V : and for some fixed v 0 2 V. Density proportional to

15. Gaussian free field on a graph A centered Gaussian process { g v } v 2 V satisfying for all u, v 2 V : and for some fixed v 0 2 V. Covariance given by the Green function of SRW killed at v 0 : expected # visits to v from RW start at u before hitting v 0

16. main result For every graph G =( V, E ), for every 0 < ¯ < 1, where { g v } v 2 V is the GFF on G. Positively resolves the Winkler-Zuckerman blanket time conjectures. ³³ ³

17. geometrical interpretation where D is the diameter of the projection of this embedding onto a random line:

18. Gaussian processes

19. Consider a Gaussian process { X u : u 2 S } with E ( X u )=0 8 u 2 S Such a process comes with a natural metric transforming ( S, d ) into a metric space. PROBLEM: What is E max { X u : u 2 S } ?

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

21. main theorem, restated For every graph G =( V, E ), where · = commute time. We also construct a deterministic poly-time algorithm to approximate within a constant factor. T HEOREM: C OROLLARY: There is a deterministic poly-time O( 1 )-approximation for t cov Positively answers the question of Aldous and Fill.

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

23. Gaussian processes P ROBLEM: What is E max { X u : u 2 S } ? ® If random variables are “independent,” expect the union bound to be tight. Gaussian concentration: Expect max for k points is about

24. Gaussian processes Gaussian concentration: Sudakov minoration:

25. chaining

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

28. hints of a connection Gaussian concentration: Sudakov minoration:

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

30. hints of a connection Gaussian concentration: KKLV’99 concentration:

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

32. hints of a connection

33. Dynkin isomorphism theory

34. 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)

35. Generalized Ray-Knight theorem law

36. blanket time vs. GFF

37. the lower bound law

38. only one Everest, but...

39. a problem on Gaussian processes Gaussian free field:

40. a problem on Gaussian processes Gaussian free field: 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.

41. percolation on trees and the GFF

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

44. percolation on trees and the GFF First and second moments agree for percolation on balanced trees 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.

45. example: precise asymptotics in 2D Bolthausen, Deuschel and Giacomin (2001): Dembo, Peres, Rosen, and Zeitouni (2004):

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