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									Monte Carlo studies of Self-Avoiding Walks
               and Loops
                        Tom Kennedy


       Department of Mathematics, University of Arizona
           Supported by NSF grant DMS-0501168
              http://www.math.arizona.edu/e tgk




                                 Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.1/41
       Preface
The organizers wrote
“We would like to encourage you to speak on ongoing or recent
developments rather than to deliver a lecture that you have already
given on several occasions.”

I have not given this talk before.




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.2/41
        Preface
The organizers wrote
“We would like to encourage you to speak on ongoing or recent
developments rather than to deliver a lecture that you have already
given on several occasions.”

I have not given this talk before.

I will not give this talk again.




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.2/41
        Preface
The organizers wrote
“We would like to encourage you to speak on ongoing or recent
developments rather than to deliver a lecture that you have already
given on several occasions.”

I have not given this talk before.

I will not give this talk again.

I will mainly talk about ongoing work with an emphasis on things I do
not understand.




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.2/41
    Outline
• 0. Review: Def of SAW, conjectured relation to SLE8/3
• 1. Bond avoiding random walk
• 2. Two saw’s - comparison with Cardy/Gamsa formula
• 3. Distribution of points on SAW and SLE
         interior point vs. endpoint
         SAW vs. SLE
• 4. Bi-infinite SAW as a self-avoiding loop
         comparison with Werner’s measure on self-avoiding loops




                                 Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.3/41
       Definition of SAW
Take all N step, nearest neighbor walks in the upper half plane,
starting at the origin which do not visit any site more than once.
Give them the uniform probability measure.
Let N → ∞. Then let lattice spacing go to zero.




                                      Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.4/41
       Relation to SLE8/3
Do this in the upper half plane

Conjecture (LSW) : the scaling limit of the SAW is chordal SLE8/3
Simulations of SAW support the conjecture

SAW in other geometries typically requires a variable number of steps
N with a weight β N .
Note: All simulations supporting the conjecture have been for SAW’s
with a fixed number of steps.




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.5/41
       1. Bond-avoiding walk
Take all nearest neighbor walks of length N that self-avoid in the sense
that they do not traverse the same bond more than once.
Large loops are allowed, but entropically suppressed.
Everyone expects this model to have same scaling limit as usual SAW.


             1220



             1200



             1180



             1160



             1140



             1120



                    1700   1720   1740   1760       1780      1800
                                                Tom Kennedy     MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.6/41
       Bond and site avoiding SAW’s
Picture of two bond avoiding walks and two site avoiding walks
                                                               "1"
                                                               "2"
                                                               "3"
                                                               "4"




                                    Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.7/41
         Bond and site avoiding SAW’s
Blow up of previous picture showing small loops in bond SAW
  5640


  5620


  5600


  5580


  5560


  5540


  5520


  5500


  5480


  5460


         650    700       750       800              850                      900




Answer: 1=bond, 2=bond, 3=site, 4=site
                                   Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.8/41
                                         Probability of passing right
Schramm gave an explicit formula for the probability the SLE curve
passes to the right of a fixed point. Difference is about 0.1%.


                                1



                               0.8
probability of passing right




                               0.6



                               0.4



                               0.2

                                                                        bond SAW simulation
                                                                          Schramm’s formula
                                0
                                     0        0.2      0.4              0.6                 0.8                   1
                                                             theta/pi


                                                                              Tom Kennedy     MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.9/41
       RV tests
Lawler, Schramm and Werner gave an explicit formula for the
probability of certain events for SLE8/3 .




                                   Y

                            X

      0



                                   Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.10/41
                  Test using X RV
         1
              bond SAW simulation
                 Exact LSW result

        0.8



        0.6
CDF’s




        0.4



        0.2



         0
              0     0.1   0.2   0.3   0.4    0.5    0.6       0.7         0.8           0.9            1
                                        Distance to 1




                                                Tom Kennedy    MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.11/41
                               Test using X RV

                      0.005


                          0
difference in CDF’s




                      -0.005


                       -0.01


                      -0.015


                       -0.02


                      -0.025
                               0   0.1   0.2   0.3   0.4   0.5    0.6     0.7     0.8      0.9          1
                                                      Distance to 1




                                                                    Tom Kennedy    MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.12/41
                  Test using Y RV

         1
                                                 bond SAW simulation
        0.9                                          LSW exact result

        0.8

        0.7

        0.6
CDF’s




        0.5

        0.4

        0.3

        0.2

        0.1

         0
              0     2   4   6     8      10     12       14      16       18          20
                            min height along vertical line




                                                   Tom Kennedy    MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.13/41
                               Test using Y RV

                       0.01




                      0.005
difference of CDF’s




                          0




                      -0.005




                       -0.01
                               0   2   4   6      8     10     12       14      16        18         20
                                           min height along vertical line




                                                                  Tom Kennedy    MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.14/41
       2. Two SAW’s
Consider two SAW’s in the upper half plane both starting at the origin
with the condition that they do not intersect each other.
Uniform probability measure




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.15/41
       Two SLE’s
Cardy and Gamsa considered two SLE curves starting at the origin.

Used boundary CFT to derive the probabilities that a given point is left
of both curves, in the middle of the curves or to the right of both
curves. For κ = 8/3 their formulae are

Let t = cot(θ).

               −2t(13 + 15t2 ) + (3π − 6 arctan(t))(1 + 6t2 + 5t4 )
    Plef t   =
                                  30π(1 + t2 )2


                                           4
                           Pmiddle   =
                                       5(1 + t2 )


                2t(13 + 15t2 ) + (3π + 6 arctan(t))(1 + 6t2 + 5t4 )
     Pright   =
                                   30π(1 + t2 )2

                                          Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.16/41
          SAW simulation vs. CFT formula
 1
                                                   SAW’s
                                                    exact
0.8



0.6



0.4



0.2



 0
      0        0.2      0.4              0.6                    0.8                           1
                              theta/pi




                                     Tom Kennedy      MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.17/41
         Differences - SAW’s vs exact
0.004
                                                                middle
0.003                                                               left
                                                                  right

0.002

0.001

    0

-0.001

-0.002

-0.003

-0.004
         0      0.2     0.4              0.6                 0.8                          1
                              theta/pi




                                    Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.18/41
       3. Distribution of points on SAW and SLE
SAW with N steps is not the same as the first N steps of a SAW with
2N steps.
SLE on time interval [0, 1] is the same as SLE on [0, 2] restricted to
[0, 1].
                                             "100K steps"
                                             "200K steps"




                                      Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.19/41
       SAW vs SLE
We consider the distribution of the random variable which is the
distance from the origin to various points on the random curves.

We compare the following
 • The endpoint of SAW with N steps
 • The point after N/2 steps for a SAW with N steps.
 • SLE at time (half-plane capacity) 1.
All RV’s are normalized to have mean 1.




                                    Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.20/41
        SAW vs SLE
6
                                             midpoint of SAW
                                             endpoint of SAW
5                                         SLE at fixed capacity


4


3


2


1


0
    0         0.5             1                     1.5                                  2
                     Distance from origin



                                   Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.21/41
        SAW vs SLE
Now compare the following
  • The endpoint of SAW with N steps
  • The point after N/2 steps for a SAW with N steps.
  • SLE at a fixed “length”


Length is p-variation or fractal variation:
Let ∆x > 0. Let ti be first time after ti−1 with

                        |γ(ti ) − γ(ti+1 )| = ∆x

Stop when tj > t. Then

               length[0, t] = lim          |γ(tj ) − γ(tj−1 )|1/ν
                              ∆x→0
                                      j




                                          Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.22/41
          SAW vs SLE
1.8
                                                midpoint of SAW
1.6                                            endpoint of SAW
                                              SLE at fixed length
1.4

1.2

 1

0.8

0.6

0.4

0.2

 0
      0         0.5             1                    1.5                                  2
                       Distance from origin



                                    Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.23/41
       SAW endpoint and radial SLE?
Fixed length SAW gives a curve from boundary to interior.

Is it related to radial SLE?

Proposal: apply conformal map (random) of half plane to itself that
takes endpoint of SAW to i.
Do you get radial SLE (in the half plane) in the scaling limit?

Test: X=max of real part of points on walk.
Exact distribution is known:
                                                 1
            P (X ≤ x) = x55/48 [x2 + 1]5/96 [x2 + ]−5/8
                                                 4




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.24/41
          Image of SAW is not radial SLE
                      SLE sim only has 5K samples, dx=0.01
 1

0.9

0.8

0.7


0.6

0.5

0.4

0.3

0.2


0.1                                                             Exact radial SLE
                                                                 SLE simulation
                                                                SAW 200K steps
 0
      0         0.5           1                  1.5                     2                          2.5
                                       X




                                            Tom Kennedy      MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.25/41
       4. Conformally invariant self-avoiding loops
Werner (building on work of Lawler, W and Lawler, Schramm, W.):
There is an essentially unique measure µ on self-avoiding loops in the
plane with following property
Let D,D′ be simply connected, φ conformal map D → D′ .
Let µD be the restriction of µ to loops inside D.
Then µD , µD′ related by φ.
µ is a measure on single loops, not ensembles of loops.
µ must be an infinite measure.

Proposition: Let 0 ∈ D′ ⊂ D be simply connected. Let Φ : D′ → D,
Φ(0) = 0, Φ′ (0) > 0. Then

          µ({γ : 0 ∈ int(γ), γ ⊂ D, γ ⊂ D′ } = c log(Φ′ (0))




                                      Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.26/41
       Bi-infinite SAW as self-avoiding loop
Bi-infinite SAW: Take SAW’s starting at origin with 2N steps, shift so
midpoint is at origin.
Let N → ∞, lattice spacing → 0.
This walk ω is a loop passing through 0 and ∞.
Let φ(z) = 1/(z − a).
φ(ω) is a loop through 0 and −1/a.
Probability measure on SAW gives probability measure on these loops.
But these loops go through two fixed points.
How is it related to Werner’s infinite measure?




                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.27/41
Bi-infinite SAW as self-avoiding loop




                      Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.28/41
       Werner’s measure on shapes
Define two loops to be equivalent if they are related by a translation,
dilation and rotation.
Call equivalence classes shapes.
Use µ to define a probability measure P on shapes.
Let Bǫ = {γ : ǫ < |γ| < 1/ǫ, 0 ∈ int(γ)}
µ(Bǫ ) < ∞, so µ restricted to Bǫ can be normalized.
This gives a probability measure on shapes.
Let P be the probability measure we get when ǫ → 0.

Easy propositon: With γ = reiθ (γ − w), A(γ) = area of interior,
                      ˆ

                                    dθ d2 w dr
                       γ
                    dµ(ˆ ) = dP (γ)
                                    2π A(γ) r




                                      Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.29/41
        Bi-infinite SAW as self-avoiding loop
Is the bi-infinite SAW probability measure equal to Werner’s P ?
Test using the explicit formula µ(E) = c log(Φ′ (0)), for

                E = {ˆ : 0 ∈ int(ˆ ), γ ⊂ D, γ ⊂ D′ }
                     γ           γ ˆ         ˆ

Recall γ = reiθ (γ − w), so 0 ∈ int(ˆ ) iff w ∈ int(γ).
       ˆ                            γ


                                 1                    dθ
           µ(E) =      dP (γ)          d2 w              F (γ, w, θ)
                                A(γ)                  2π

where

                      dr
      F (γ, w, θ) =      1(reiθ (γ − w) ⊂ D, reiθ (γ − w) ⊂ D′ )
                       r

Take D to be unit disc, D′ a disc containing 0 contained in D.


                                        Tom Kennedy      MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.30/41
            Bi-infinite SAW as self-avoiding loop
 4



3.5



 3



2.5



 2



1.5



 1



0.5



 0
      0.1     0.2   0.3   0.4   0.5   0.6           0.7           0.8             0.9             1




                                      Tom Kennedy         MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.31/41
             Bi-infinite SAW as self-avoiding loop


1.04




1.02




  1




0.98




0.96



       0.1      0.2   0.3   0.4   0.5   0.6           0.7           0.8             0.9             1




                                        Tom Kennedy         MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.32/41
       Confusion
There are other ways to get a probability measure on shapes from µ.
Let C(γ) denote the center of mass of the interior of γ.
Let d(γ) be the diameter of γ.
Define Bǫ,δ = {γ : |C(γ)| < δd(γ),    ǫ < d(γ) < 1/ǫ}
µ(Bǫ,δ ) < ∞, so µ restricted to Bǫ,δ can be normalized.
This gives another probability measure on shapes.
Let P d be the probability measure we get when ǫ → 0.

More generally, d(γ) can be any function of γ which is invariant under
rotations and translations and d(λγ) = λd(γ).
Easy proposition: With γ = reiθ (γ − w),
                        ˆ

                                 d dθ d2 w dr
                      γ
                   dµ(ˆ ) = dP (γ)
                                   2π d(γ)2 r


                                     Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.33/41
       Bi-infinite SAW as self-avoiding loop
Which probability measure on shapes does bi-infinite SAW give?
Can you study this by simulations?
Simulate random walk starting at origin, conditioned to end at origin
(Brownian bridge). Take its outer boundary.
This gives a probability measure on loops which is known to be P .
                            A(γ)
Can you distinguish P and   d(γ)2 P   ?

Let γ = reiθ (γ − w), compare
    ˆ

                                    dθ d2 w dr
                       γ
                    dµ(ˆ ) = dP (γ)
                                    2π A(γ) r


                                     dθ d2 w dr
                   dµd (ˆ ) = dP (γ)
                        γ
                                     2π d(γ)2 r



                                          Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.34/41
The D′ ’s




            Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.35/41
            RW loop using area (1/A(γ) formula)
Note vertical scale
  1.1
                                                     disc at 0.15
                                                               slit
                                         crescent, center at 1.25
                                          crescent, center at 1.5
                                          crescent, center at 2.0
 1.05                                                            1




    1




 0.95




  0.9
        0     0.2   0.4   0.6   0.8    1            1.2           1.4            1.6            1.8



                                      Tom Kennedy         MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.36/41
           RW loop using diameter (1/d(γ)2 formula)
                                                    disc at 0.15
0.32                                                          slit
                                        crescent, center at 1.25
                                         crescent, center at 1.5
                                         crescent, center at 2.0
0.31                                                   0.29555


 0.3


0.29


0.28


0.27

       0     0.2   0.4   0.6   0.8    1            1.2           1.4            1.6            1.8




                                     Tom Kennedy         MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.37/41
       What about SAW?

Return to the bi-infinite SAW.
Look at the same five D′ ’s
Look at µ using
  • 1/Area(γ) formula
  • 1/d(γ)2 formula

200K steps in SAW
77 CPU-days




                                Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.38/41
           SAW loop using area (1/A(γ) formula)
 1.1
                                                    disc at 0.15
                                                              slit
                                        crescent, center at 1.25
                                         crescent, center at 1.5
                                         crescent, center at 2.0
1.05                                                            1




  1




0.95




 0.9
       0     0.2   0.4   0.6   0.8    1            1.2           1.4            1.6            1.8




                                     Tom Kennedy         MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.39/41
           SAW loop using diameter(1/d(γ)2 formula)
                                                    disc at 0.15
                                                              slit
                                        crescent, center at 1.25
 0.3                                     crescent, center at 1.5
                                         crescent, center at 2.0
                                                         0.2809
0.29



0.28



0.27



0.26


       0     0.2   0.4   0.6   0.8    1            1.2           1.4            1.6            1.8




                                     Tom Kennedy         MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.40/41
     Conclusions/Homework for the week
• Simulations of bond SAW agree with chordal SLE8/3 .
• Simulate variable length SAW to check it agrees.
• Simulations of two mutually avoiding SAW’s agree with
  Cardy-Gamsa formulae.
• Simulations indicate that fixed length SAW is not related to radial
  SLE by maping the endpoint to i.
• Does SLE tell us anything about the distribution of the endpoint of
  a fixed length SAW?
• What probability measure on shapes does the bi-infinite SAW
  give? What is the RN derivative with respect to P?
• Caution: above is hard to study by simulation.




                                  Tom Kennedy   MC studies of self-avoiding walks and loops, Aug 4-8, Montreal – p.41/41

								
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