A family of self-avoiding walks on the Sierpi´ nski gasket Takafumi Otsuka Departmant of Mathematics and Information Scienses Tokyo Metropolitan University 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals Cornell University, Ithaca June 2017 Takafumi Otsuka (TMU) SAW on the SG 1 / 13
Outline Pre-Sierpinski gasket 1 A set of probability measures with multi parameter 2 Scaling limit 3 Self-avoiding property 4 Takafumi Otsuka (TMU) SAW on the SG 2 / 13
1. Pre-Sierpi´ nski gasket √ a = ( 1 3 2 , 2 ) a a a F 0 F 1 F N F 2 − N 2 − 1 1 O b O b O = (0 , 0) b = (1 , 0) O b Let F 0 be a triangle △ Oab , F N be the graph with edges of length 2 − N , and F = cℓ ( ∪ ∞ N =0 F N ) be a (finite) Sierpi´ nski gasket. Next, define self-avoiding paths on F N and probability measures on the path spaces inductively. Takafumi Otsuka (TMU) SAW on the SG 3 / 13
Self-avoiding paths w : self-avoidng path on F N if w (0) = O , ( w ( i ) , w ( i + 1)) ∈ { edges on F N } , w ( i ) ∈ { vertices on F N } , w ( i ) ̸ = w ( j ) ( i ̸ = j ) , and w ( ℓ ( w )) = a . √ a = ( 1 3 2 , 2 ) 2 − N b = (1 , 0) O = (0 , 0) Takafumi Otsuka (TMU) SAW on the SG 4 / 13
2. A set of probability measures with multi parameter Self-avoiding paths on F 1 : a w 1 w 2 w 3 w 4 w 5 O b w 7 w 8 w 9 w 10 w 6 Given p i , q i ≥ 0 , ∑ 10 i =1 p i = 1 , ∑ 10 i =1 q i = 1 , p 8 , p 9 , p 10 = 0 , define P 1 P 2 1 [ w i ] = p i , 1 [ w i ] = q i . Takafumi Otsuka (TMU) SAW on the SG 5 / 13
Branching Idea: P 1 N +1 and P 2 N +1 are obtained by the following branching. P 1 1 w 1 , w 2 , . . . , w 7 w 1 , w 2 , . . . , w 7 , w 8 , w 9 , w 10 P 2 1 Following this idea, define X N ( w )( j ) = w ( j ) , j = 0 , 1 , . . . , ℓ ( w ) , where w is a self-avoiding path on F N . (Suppose that the first branching follows P 1 1 . ) Takafumi Otsuka (TMU) SAW on the SG 6 / 13
Example Loop-erased random walk (defined by erasing loops in descending order of the size of loops from a simple random walk ) on F N (Hattori, Mizuno 14’) is included as a special case. p 1 = 1 2 , p 2 = p 3 = p 7 = 2 15 , p 4 = p 5 = p 6 = 1 30 , p 8 = p 9 = p 10 = 0 , q 1 = 1 9 , q 2 = q 3 = q 7 = 11 90 , q 4 = q 5 = q 6 = 2 45 , q 7 = 8 45 , q 8 = 2 9 , q 9 = q 10 = 1 18 . Additionally, ‘standard’ self-avoiding walk (HHK 91’), loop-erased self-repelling walk (HOO, 17’) are included as special cases respectively (we omit details here). Assigning different values to p i , q i gives different SAWs. Takafumi Otsuka (TMU) SAW on the SG 7 / 13
3. Scaling limit Let p = ( p 1 , . . . , p 10 , q 1 , . . . , q 10 ) and X N ( w )( t ) = w ( t ) , t ∈ [0 , ∞ ) : a self avoiding walk on F N (linear interpolated). Theorem (scaling limit) For any p , there exists λ = λ ( p )(2 ≤ λ ≤ 3) such that X N ( λ N t ) → X ( t ) a.s. as N → ∞ . and d H = log λ/ log 2 (Hausdorff dimension of the path) with probability 1 . d H takes any values from 1 to log 3 / log 2 . X N is self-avoiding, but X = X ( p ) is not neccesarily self-avoiding and the number of triangles produced at each branching affects the self-avoiding property. Takafumi Otsuka (TMU) SAW on the SG 8 / 13
Two extream cases w 2 w 3 w 4 w 5 w 1 w 7 w 8 w 9 w 10 w 6 If p 1 + p 2 + p 3 + p 4 = q 1 + q 2 + q 3 + q 4 = 1 , then, with probability 1 , d H = 1 and X is an uniform linear motion. If p 5 + p 6 + p 7 = q 5 + · · · + q 10 = 1 , then, with probability 1 , d H = log 3 / log 2 , that is, X fills the state space(the SG), and the speed of the motion is constunt. We call it Peano curve. Takafumi Otsuka (TMU) SAW on the SG 9 / 13
4. Self-avoiding property Theorem (Self-avoiding) If p 5 + p 6 + p 7 < 1 , q 5 + · · · + q 10 < 1 , then the scaling limit X is self-avoiding. a O b Takafumi Otsuka (TMU) SAW on the SG 10 / 13
Asymptotic behavior Suppose that p 5 + p 6 + p 7 < 1 , q 5 + · · · + q 10 < 1 . In this case, we have some results about asymptotic behavior. Theorem (Short time behavior) Let γ = log 2 / log λ . There exist positive constants C 1 , C 2 such that for all s > 0 E [ | X ( t ) | s ] C 1 ≤ lim ≤ C 2 . t γs t → 0 Theorem (Laws of the iterated logarithm) There exist positive constants C 3 , C 4 such that | X ( t ) | C 3 ≤ lim sup t γ (log log(1 /t )) 1 − γ ≤ C 4 a.s. t → 0 Takafumi Otsuka (TMU) SAW on the SG 11 / 13
Self-intersections Theorem If p 5 + p 6 + p 7 = 1 , then, with probability 1 , the scaling limit X has infinitely many self-intersections. If p 7 < 1 and q 5 + · · · + q 10 < 1 , then d H < log 3 / log 2 . Example: If the branching is w 6 → ( w 5 , w 6 , w 7 ) → . . . , then following every branchings in the sets of triangles including y follows P 1 1 . Iterating this, therefore, X reaches y from opposite side of x . a y y y O b Takafumi Otsuka (TMU) SAW on the SG 12 / 13
Conclusion ● Using the branching method, we have studied a family of self-avoiding walks on the SG. ● The limit is divided into four types in the meaning of shape. ● The case that the limit have intersections but does not fill the state space is not included in previous models. ● For the self-avoiding case, we have studied asymptotic behavior. References [1 ]Hattori, K., Mizuno, M. : Loop-erased random walk on the Sierpinski gasket , Stoch. Proc. Applic, 124 , (2014) 566–585. [2 ]Hattori, K., Ogo, N., Otsuka, T. : A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpinski gasket , Discrete Contin. Dyn. Syst. S, 10 , (2017) 289–312. Takafumi Otsuka (TMU) SAW on the SG 13 / 13
Recommend
More recommend
