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

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

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

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

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

S 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

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

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

Commute time and resistance October effective resistence between u and v

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

11 1{ 7 5 } Hitting time and rubber bands October

Random maze October

Random maze October

October We obtain every maze with the same probability! Random maze

Random spanning tree October

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

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

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

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 Sampling by random walk

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

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

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

Volume computation by multiphase Monte-Carlo October

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

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

Sampling by ball walk October

Sampling by hit-and-run walk October

steplength can be large! Sampling by reflecting walk October

- 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

bottleneck isoperimetric quantity Conductance October

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

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

- The Slicing Conjecture - Reflecting walk Possibilities for further improvement October

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

Smallest bisecting surface F H Smallest bisecting hyperplane ? ? The Slicing Conjecture October