Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

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1 Edge-and vertex-reinforced random walks with super-linear reinforcement on infinite graphs Codina Cotar University College London December 2016, Cambridge

2 Edge-reinforced random walks (ERRW): random walks on a graph G for which the probability of jump from a vertex to a neighbour is weight-proportional to the number of times (+initial edge weight) the edge between the two vertices has been visited in the past. Vertex-reinforced random walks (VRRW): random walks on a graph G for which the probability of jump from a vertex to a neighbour is weight-proportional to the number of times (+initial vertex weight) the respective vertex has been visited in the past They depend on the past behaviour of the process (which is then a non-markov process).

3 Types of reinforcement Positive reinforcement: edge/vertex reinforcement; discrete/continuous Super-linear Linear Sub-linear Negative reinforcement Self-avoiding

4 Reinforced random walks are connected to: Reinforced learning, Bayesian statistics, genetics, biology. Preferential attachment/random graphs models (Bhamidi, Moerters,van der Hofstad) Supersymmetric quantum mechanics/random matrices (linearly edge-reinforced random walks: Sabot/Tarres, Disertori/Spencer/Zirnbauer). Important models in statistical mechanics, such as the gradient model to which the Gaussian Free Field belongs (negative reinforcement: Funaki/Spohn, Toth, Werner). Models of polymers in chemistry and physics (self-avoiding walks: Bolthausen, Brydges, Slade, Smirnov).

5 Polya s urn Polya urn Suppose we have an urn which initially has A red balls and B green balls. We pick a ball at random and put it back together with a ball of the same color. The urn now has A + B + 1 balls. We repeat this procedure infinitely many times. Each color is picked infinitely many times. Moreover the proportion of balls of one color converge to a Uniform. This is the linear case.

6 Polya s urn Suppose that instead of reinforcing linearly, we use any positive function w. P(red = A + 1, green = B) = 1 P(red = A, green = B + 1) = w(a) w(a) + w(b). We would like to know if the numbers of balls of both colors tend to infinity.

7 Polya s urn Theorem (H. Rubin, see B. Davis (1990)) If n=1 1/w(n) < then only finitely balls of one of the two colors are picked. If n=1 1/w(n) = then infinitely many balls of both colors are picked.

8 Edge-reinforced random walks Edge-reinforced random walks (ERRW) Linearly edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. l e 0 0, e E(G), are initial edge weights. Reinforcement weight function w : [0, ) (0, ) We have for ERRW (i) if currently at vertex v V(G), in the next step the particle jumps to a vertex u V(G) adjacent to v. (ii) the probability of a jump to u is w-proportional to the number of previous traversals of the edge {v, u}.

9 Edge-reinforced random walks Let w be a super-linear weight, i.e. and let sup e E(G) i=1 1 w(i + l e 0 ) < G = {the (random) graph of all edges in G traversed by the walk infinitely often} (Sellke 1994) If G is an infinite bipartite graph of bounded degree, then P(G has only one edge) = 1.

10 Edge-reinforced random walks (Limic AOP 2001) Extension to infinite graphs G of bounded degree and weight function w(k) = k ρ, ρ > 1. (Limic and Tarres AOP 2007) Extension to infinite graphs G of bounded degree and increasing super-linear weight functions. Their method does not work for example on finite/infinite graphs with at least an odd cycle and weight: w(k) = k ρ /(2 + ( 1) k ), ρ > 1, or w(k) = exp{k(2 + ( 1) k }. Tools: Rubin s construction, martingale methods

11 Edge-reinforced random walks (Cotar-Thacker AOP-to appear) Let G be an infinite graph of bounded degree. For any super-linear weight function w and any initial weights l e 0 0, e E(G), we have P(G has only one edge) = 1. Extension to the case where each edge e has is own weight function w e satisfying sup e E(G) i=1 1 w e (i + l e 0 ) <. Extension also to a large class of cases in which an initial weight function w 0 is replaced a countable number of times by other super-linear weight functions w 1,..., w n,.... New technique based on the use of order statistics

12 Edge-reinforced random walks What about attraction time What can we say about the tail behavior of the time of attraction, that is, the first time after which only the attracting edge is traversed? Denote by T the attraction time.

13 Edge-reinforced random walks Assume first that G is a finite graph with l edges. Let m(g) be the maximum number of pairs of edges that meet at a vertex. Cotar and Limic (AAP-2009) proved the following If w is increasing, then P G (T > k) m(g)c(l) P 1,1 (T > k/m(g)), where P 1,1 is the corresponding probability for a two-edge graph. Let G be an infinite tree of bounded degree. If w super-linear, then for any c > 1 ( ) 1 P(T > k) O + max P Gk,c (T > k c log k), G k,c k c where the above maximum is taken overall trees G k,c having fewer than c log k vertices and degree bounded by the degree of G.

14 Edge-reinforced random walks We can find optimal bounds for T for a finite graph. In particular, if w(k) = k ρ for some fixed ρ > 1, and if ρ = (ρ 1)/ρ P(T > k) 1 k ρ ρ. Moreover, E(T) is infinite if ρ and finite if ρ > What about infinite graphs of bounded degree? Weaker results than this in Cotar-Limic (2009).

15 Vertex-reinforced random walks Vertex-reinforced random walks (VRRW) l v 0 0, v V(G), are initial vertex weights. Reinforcement weight function w : [0, ) (0, ) We have for VRRW (i) if currently at vertex v V(G), in the next step the particle jumps to a vertex u V(G) adjacent to v. (ii) the probability of a jump to u is w-proportional to the number of previous traversals of the vertex u. Let w be a super-linear weight, i.e. sup v E(G) i=1 1 w(i + l v 0 ) <.

16 Vertex-reinforced random walks Let G = {the (random) graph of all vertices in G traversed by the walk infinitely often} (Volkov AOP 2006, Basdevant/Singh/Schapira AOP 2014) For VRRW on Z and increasing super-linear weight w, we have P(G has only two vertices) = 1.

17 Vertex-reinforced random walks (Benaim/Raimond/Schapira ALEA 2013) Let G be a complete graph with n vertices and assume w(k) = k ρ, ρ > 1. Let 3 m n. If ρ < (m 1)/(m 2), then for any 2 l m, we have P(G has exactly l vertices) > 0.

18 Vertex-reinforced random walks (Cotar-Thacker AOP-to appear) Let G be an infinite connected graph of bounded degree. Assume w satisfies sup v V(G) i 1 i w(i + l v 0 ) <. Then the walk traverses exactly two random neighbouring attracting vertices at all large times a.s., that is P(G has exactly two vertices) = 1. The assumptions above are satisfied by a large class of weight functions, among which are all the weight functions of order equal to or higher than w(l) = l 2 log 1+ɛ l, ɛ > 0.

19 Vertex-reinforced random walks (Cotar-Thacker AOP-to appear) Let G be a connected triangle-free graph, either finite or an infinite graph of bounded degree. If w satisfies either (a) (b) w super-linear and sup v V(G) sup i 1 sup v V(G) l 1 i 1/2 w(i + l v <, or 0 ) l w(l + l v 0 ) <, then the walk traverses exactly two random neighbouring attracting vertices at all large times a.s..

20 Main ideas of the proof Assume E(G) = n. We re-label the number of edge traversals at time k 0 in increasing order. More precisely, we define the order statistics at time k as a vector R k = (R 1 k,..., Rn k ). The components of this vector are defined as the values of the edge traversals at time k put in non-increasing order; this defines the vector R k uniquely. Then 0 R n k Rn 1 k... R 1 k k, R i k Ri k+1, i = 1, 2,..., n, and n R j k = k. j=1 We will show that R 2 <. Suffices to show E(g(R 2 )) <.

21 Main ideas of the proof For simplicity, assume l e 0 l 0. From Cotar-Limic (2009), we have for 0 l n k... l1 k with n i=1 li k = k P(R 1 k = l 1 k,..., R n k = ln k ) c(n, w) n i=1 w(li k + l 0) n i=1 w(li k + l 0). Take the simplest case with n = 3. We get from the above for 0 l 2 k [k/2] ( P(R 1 k = l 1 k, R 2 k = l 2 k, R 3 k = l 3 k) c(w) + 1 w(l 1 k + l 0)w(l 2 k + l 0) ). 1 w(l 1 k + l 0)w(l 3 k + l 0) + 1 w(l 2 k + l 0)w(l 3 k + l 0)

22 Main ideas of the proof In the general case, we can similarly show [ P G (R 2 k = l 2 1 k) C(w, n, V(G), l 0 ) w(l 2 k + l 0) + k l 2 k i=max([k/n],l 2 k) 1 w(i + l 0 ) Q n 2 (k i l 2k; )] l 0.

23 Main ideas of the proof In the above, we have for all a, b N, m 1 Q m (a; b) := Moreover, for all j N (h 1,...,h m ) N m : 0 h m... h 1 h h m =a j Q m (s + a; b) (c(b)) m, with c(b) := s=0 1 m j=1 w(b + hj ). l=0 1 w(l + b).

24 Main ideas of the proof We will show next that R 2 < a.s.. There exists a function g : N [0, ) such that (i) g ( ) is increasing and lim l g (l) = (ii) M := l 1 g(l+l 0) w(l+l 0) <. Since g is an increasing function, we have by the monotone convergence theorem, that E G (g(r 2 )) = lim k EG (g(r 2 k)) = lim k [k/2] g(l 2 k)p G (R 2 k = l 2 k) l 2 k =0 [k/2] [ C(w, n, V(G), l 0 ) lim g(l 2 k + l 0 ) k l 2 k =0 + k l 2 k i=max([k/n],l 2 k) 1 w(l 2 k + l 0) 1 w(i + l 0 ) Q n 2 (k i l 2k; )] l 0.

25 Main ideas of the proof By property (ii) of g, we have for all k 0 g(l 2 k + l 0) w(l 2 k + l 0) < M <. l 2 k =0 To bound the second sum, we have [k/2] l 2 k =0 g(l 2 k + l 0 ) [k/2] l 2 k =0 k i=0 k l 2 k i=l 2 k min(i,k i) l 2 k =0 (c(l 0 )) n 2 M. k l 2 k i=max([k/n],l 2 k) 1 w(i + l 0 ) Q n 2 (k i l 2k; ) l 0 g(l 2 k + l 0) w(i + l 0 ) Q ( ) n 2 k i l 2 k ; l 0 g(l 2 k + l 0) w(i + l 0 ) Q ( ) n 2 k i l 2 k ; l 0

26 Main ideas of the proof To pass to infinite connected graphs of bounded degree, we will proceed as follows For all n 1, let G n be the graph centred at v 0 formed of the vertices in G at graph distance n. Let Since A n = { G = 1, the walk never leaves G n }. A n A n+1, n 1, and { G = 1} n 1 A n, we have P( G = 1) lim n P(A n).

27 Main ideas of the proof Moreover P(A n ) = P(the walk never leaves G n ) P( G = 1 the walk never leaves G n ) (1 θ [n/2] )P( G = 1 the walk never leaves G n ) 1 θ [n/2], for some given 0 < θ < 1.

28 Vertex-reinforced random walks We re-label the number of vertex visits at time k 0 in increasing order. More precisely, we define the order statistics at time k as a vector R k = (R 1 k,..., Rn k ). The components of this vector are the values of v X v k lv 0 put in non-increasing order; this defines the vector R k uniquely. Then, for all k 0 we have 0 R n k Rn 1 k... R 1 k, R i k Ri k+1 for all i = 1, 2,..., n, Moreover [k/n] R 1 k [(k + 1)/2] and [(k 1)/(2(n 1)] R 2 k [k/2] and n R j k = k. j=1

29 Vertex-reinforced random walks P G (R 3 k = l 3 k) 6 3! w(l 0 ) (l 1 k,l2 k ) N2 :0 l 3 k l2 k l1 k [(k+1)/2] + l 1 k +l2 k =k l3 k ( 1 w(l 1 k + l 0) ) 1 w(l 2 k + l 0) + 1 w(l 3 k + l. 0)

30 Vertex-reinforced random walks THANK YOU!