1. Why is it useful to walk randomly? László Lovász Mathematical Institute Eötvös Loránd University October 20121

2. Random walk on a graph October 20122 Graph G=(V,E)

3. Random walk on a graph October 20123  t (v): probability of being at node v after t steps Stationary distribution

4. Hitting time H(s,t) = hitting time from s to t = expected # of steps, starting at s, before hitting t  (s,t) = commute time between s and t = H(s,t) + H(t,s) October 20124

5. Every nonconstant function has at least 2 poles. Harmonic functions Every function defined on S  V (S  ) has a unique extension harmonic on V \ S. G=(V,E) graph, f: V   October 20125

6. S 2 3 1 f(v)= E(f(Z v )) Z v : (random) point where random walk from v hits S v 0 v 1 f(v)= P( random walk from v hits t before s) s t Harmonic functions and random walks October 20126

7. 0 v 1 f(v)= electrical potential s t Harmonic functions and electrical networks October 20127

8. f(v) = position of nodes 0 1 Harmonic functions and rubber bands October 20128

9. Commute time and resistance October 20129 effective resistence between u and v

10. Distance from s to t = H(s,t). t weight=degree strength=1 Hitting time and rubber bands October 201210

11. 11 1{ 7 5 }1 2 7 7 9 3 5 Hitting time and rubber bands October 201211

12. Random maze October 201212

13. Random maze October 201213

14. October 201214 We obtain every maze with the same probability! Random maze

15. Random spanning tree October 201215

16. - card shuffling - statistics - simulation - counting - numerical integration - optimization - … Sampling: a general algorithmic task October 201216

17. polynomial time algorithm certificate October 201217 L: a „language” (a family of graphs, numbers,...) Sampling: a general algorithmic task

18. Find: - a certificate Given: x - an optimal certificate - the number of certificates - a random certificate (uniform, or given distribution) October 201218 L: a „language” (a family of graphs, numbers,...) Sampling: a general algorithmic task

19. One general method for sampling: Random walks (+rejection sampling, lifting,…) Construct regular graph with node set V Want: sample uniformly from V Simulate (run) random walk for T steps Output the final node ???????????? mixing time October 201219 Sampling by random walk

20. Given: convex body K   n Want: volume of K Not possible in polynomial time, even if an error of n n/10 is allowed. Elekes, Bárány, Füredi Volume computation October 201220

21. Dyer-Frieze-Kannan 1989 But if we allow randomization: There is a polynomial time randomized algorithm that computes the volume of a convex body with high probability with arbitrarily small relative error Volume computation October 201221

22. Why not just.... * * * * * * * * * * ** * * * * * * S Need exponential size S to get nonzero! Volume computation by plain Monte-Carlo October 201222

23. Volume computation by multiphase Monte-Carlo October 201223

24. Can use Monte-Carlo! But... Now we have to generate random points from K i+1. Need sampling to compute the volume Volume computation by multiphase Monte-Carlo October 201224

25. Do sufficiently long random walk on centers of cubes in K Construct sufficiently dense lattice Pick random point p from little cube If p is outside K, abort; else return p Dyer-Frieze-Kannan 1989 Sampling by random walk on lattice October 201225

26. Sampling by ball walk October 201226

27. Sampling by hit-and-run walk October 201227

28. steplength can be large! Sampling by reflecting walk October 201228

29. - Stepsize - Where to start the walk? - How long to walk? - How close will be the returned point to random? Issues with all these walks October 201229

30. bottleneck isoperimetric quantity Conductance October 201230

31. Dyer-Frieze-Kannan 1989 Polynomial time! Cost of volume computation (number of oracle calls) Amortized cost of sample point Cost of sample point Time bounds October 201231

32. Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 Kannan-Lovász-Simonovits 1997 Lovász 1999 Kannan-Lovász 1999 Lovász-Vempala 2002 Lovász-Vempala 2003 Time bounds October 201232

33. - The Slicing Conjecture - Reflecting walk Possibilities for further improvement October 201233

34. Reflecting random walk in K steplength h large How fast does this mix? Stationary distribution: uniform Chain is time-reversible (e.g. exponentially distributed with expectation = diam( K )) October 201234

35. Smallest bisecting surface F H Smallest bisecting hyperplane ? ? The Slicing Conjecture October 201235