2. cover times, blanket times, and majorizing measures Jian Ding U. C. Berkeley James R. Lee University of Washington Yuval Peres Microsoft Research TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A A

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

4. 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

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 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]

7. 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

8. 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 (1-o(1)) n ln n · t cov ( G ) · min (4n 3 /27, 2mn) general bounds (n = vertices, m = edges) [Feige’95, Alelinuas-Karp-Lipton-Lovasz-Rackoff’79] [Feige’95, Matthews’88] (conjecture: the complete graph is extremal)

9. electrical resistance 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 + 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) = 2m R eff (u,v) (endows κ with special geometric properties)

10. computation Hitting time: easy to compute in deterministic poly time by solving system of linear equations H(u,u) = 0 H(u,v) = 1 + E w » u H(w,v) Cover time: easy to compute in deterministic exponential time Approximations (deterministic, poly-time): [Matthews’88, CRRST’89] Augmented Matthews bound yields an O(log log n) 2 approximation [Kahn-Kim-Lovasz-Vu’99] For trees, there is an 1 + ² approximation for every ² > 0 [Feige-Zeitouni’09] max u,v κ (u,v) yields an O(log n) approximation Open question: Does there exist an O( 1 )-approximation for general graphs? [Aldous-Fill’94]

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

12. blanket times, comparisons Conjecture [Winkler-Zuckerman’96]: For every graph G and 0 < ¯ < 1, t blanket (G, ¯ ) ³ t cov (G). Proved for many special cases. True up to (log log n) 2 by [KKLV’99] Comparison of cover times: If G and G’ are two graphs on the same set of nodes and κ G (u,v) · κ G ’ (u,v) for all u,v 2 V, does it follow that ? ³³

13. main theorem Talagrand introduced a functional on any metric space (X, d). T HEOREM: For any graph G, where ³ denotes equivalence up to a universal constant. Some consequences: - There is a deterministic O( 1 )-approximation to for any metric space, hence the same holds for t cov ( G ). - Postively resolves the Winkler-Zuckerman blanket time conjectures. - Bi-lipschitz stability. For instance, t cov ( G ) ³ t cov ( G’ ) where G’ is a spectral sparsifier of G.

14. main theorem Talagrand introduced a functional on any metric space (X, d). T HEOREM: For any graph G, for any 0 < ¯ < 1, where A. B denotes A · O(B). Some consequences: - There is a deterministic O( 1 )-approximation to for any metric space, hence the same holds for t cov ( G ). - Postively resolves the Winkler-Zuckerman blanket time conjectures. - Bi-lipschitz stability. For instance, t cov ( G ) ³ t cov ( G’ ) where G’ is a spectral sparsifier of G.

15. a fast randomized algorithm T HEOREM: If g is an n-dimensional Gaussian, then D = diagonal degree matrix A = adjacency matrix of G T HEOREM: For m-edge graphs, there is an O(m polylog(m))-time randomized algorithm to compute an O( 1 )-approximation to the cover time. Uses [Spielman-Teng] and [Spielman-Srivistava]

16. main theorem T HEOREM: For any graph G and δ 2 (0,1), where ³ denotes equivalence up to a universal constant.

17. Gaussian processes Consider a Gaussian process { X u : u 2 S } with E ( X u )=0 8 u 2 S (i.e. every linear combination ® 1 X 1 +  + ® k X k is normal) Such a process comes with a natural metric transforming (S,d) into a metric space. Equivalently, for S finite, consider S µ R n, and the process X u = h g, u i for u 2 S where g =( g 1, …, g n ) is an i.i.d. N( 0,1 ) vector. P ROBLEM: What is E max { X u : u 2 S } ?

18. 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

19. Gaussian processes Gaussian concentration: Sudakov minoration:

20. Gaussian processes

21. covering trees Recursively partition into pieces of diameter j=0, 1, 2, … Value of this path is where d j is the sequence of degrees down the path

22. covering trees

23. packing trees Main technical theorem:

24. majorizing measure theorem Majorizing measures theorem (Talagrand):

25. main theorem [Ding, L, Peres] T HEOREM: For any graph G and δ 2 (0,1), where ³ denotes equivalence up to a universal constant.

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

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

28. hints of a connection Gaussian concentration: KKLV concentration: Here, N t (w ) denotes the number of visits to w when the random walk started at u has returned to u for the (t deg(u)) th time.

29. hints of a connection - Trees + KKLV concentration suffice for upper bound - [Barlow-Ding-Nachmias-Peres] prove the “Dudley version”

30. an isomorphism theorem

32. a problem on Gaussian processes Gaussian process:

33. a problem on Gaussian processes Gaussian process: We need strong estimates on the size of this window as ε  0. (want to get a point there with probability at least 0.1 ) Problem: Majorizing measures handles first moments, but we need second moment bounds.

34. percolation on trees and the DGFF 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 “discrete Gaussian free field.”

35. percolation on trees and the DGFF 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.

36. open questions Holds for - complete graph - complete d-ary tree - discrete torus 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?

37. open questions Holds for - complete graph - complete d-ary tree - discrete torus 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?