A family of self-avoiding walks on the Sierpi nski gasket Takafumi - - PowerPoint PPT Presentation

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

1

Pre-Sierpinski gasket

2

A set of probability measures with multi parameter

3

Scaling limit

4

Self-avoiding property

Takafumi Otsuka (TMU) SAW on the SG 2 / 13

• 1. Pre-Sierpi´

nski gasket

O = (0, 0) a = ( 1

2, √ 3 2 )

b = (1, 0) O a b F0 FN 2−N O a b O a b F1 F 1 2−1

Let F0 be a triangle △Oab, FN be the graph with edges of length 2−N, and F = cℓ(∪∞

N=0FN) be a (finite) Sierpi´

nski gasket. Next, define self-avoiding paths on FN 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 FN if w(0) = O, (w(i), w(i + 1)) ∈ { edges on FN}, w(i) ∈ { vertices on FN}, w(i) ̸= w(j) (i ̸= j), and w(ℓ(w)) = a. O = (0, 0) a = ( 1

2, √ 3 2 )

b = (1, 0) 2−N

Takafumi Otsuka (TMU) SAW on the SG 4 / 13

• 2. A set of probability measures with multi parameter

Self-avoiding paths on F1:

w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 O a b

Given pi, qi ≥ 0, ∑10

i=1 pi = 1, ∑10 i=1 qi = 1, p8, p9, p10 = 0, define

P 1

1 [wi] = pi,

P 2

1 [wi] = qi.

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

P 2

1

w1, w2, . . . , w7, w8, w9, w10 w1, w2, . . . , w7 Following this idea, define XN(w)(j) = w(j), j = 0, 1, . . . , ℓ(w), where w is a self-avoiding path on FN. (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 FN (Hattori, Mizuno 14’) is included as a special case. p1 = 1 2, p2 = p3 = p7 = 2 15, p4 = p5 = p6 = 1 30, p8 = p9 = p10 = 0, q1 = 1 9, q2 = q3 = q7 = 11 90, q4 = q5 = q6 = 2 45, q7 = 8 45, q8 = 2 9, q9 = q10 = 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 pi, qi gives different SAWs.

Takafumi Otsuka (TMU) SAW on the SG 7 / 13

• 3. Scaling limit

Let p = (p1, . . . , p10, q1, . . . , q10) and XN(w)(t) = w(t), t ∈ [0, ∞): a self avoiding walk on FN (linear interpolated).

Theorem (scaling limit)

For any p, there exists λ = λ(p)(2 ≤ λ ≤ 3) such that XN(λNt) → X(t) a.s. as N → ∞. and dH = log λ/ log 2(Hausdorff dimension of the path) with probability 1. dH takes any values from 1 to log 3/ log 2. XN 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

w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

If p1 + p2 + p3 + p4 = q1 + q2 + q3 + q4 = 1, then, with probability 1, dH = 1 and X is an uniform linear motion. If p5 + p6 + p7 = q5 + · · · + q10 = 1, then, with probability 1, dH = 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 p5 + p6 + p7 < 1, q5 + · · · + q10 < 1, then the scaling limit X is self-avoiding.

O a b

Takafumi Otsuka (TMU) SAW on the SG 10 / 13

Asymptotic behavior

Suppose that p5 + p6 + p7 < 1, q5 + · · · + q10 < 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 C1, C2 such that for all s > 0 C1 ≤ lim

t→0

E[|X(t)|s] tγs ≤ C2.

Theorem (Laws of the iterated logarithm)

There exist positive constants C3, C4 such that C3 ≤ lim sup

t→0

|X(t)| tγ(log log(1/t))1−γ ≤ C4 a.s.

Takafumi Otsuka (TMU) SAW on the SG 11 / 13

Self-intersections

Theorem

If p5 + p6 + p7 = 1, then, with probability 1, the scaling limit X has infinitely many self-intersections. If p7 < 1 and q5 + · · · + q10 < 1, then dH < log 3/ log 2. Example: If the branching is w6 → (w5, w6, w7) → . . . , 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.

y y O a b y

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