Uniform spanning trees and loop-erased random walks on Sierpinski graphs
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1 Uniform spanning trees and loop-erased random walks on Sierpinski graphs Elmar Teufl Eberhard Karls Universität Tübingen 12 September 2011 joint work with Stephan Wagner Stellenbosch University Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
2 Sierpiński gasket Self-similar set K with respect to the similitudes: ψ i (x) = 1 2 (x w i) + w i for i {1, 2, 3} and w 1 = (0, 0), w 2 = (1, 0), w 3 = 1 2 (1, 3). Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
3 Sierpiński gasket Self-similar set K with respect to the similitudes: ψ i (x) = 1 2 (x w i) + w i for i {1, 2, 3} and w 1 = (0, 0), w 2 = (1, 0), w 3 = 1 2 (1, 3). w 3 Hausdorff dimension: dim H K = log 3 log 2 = K w 1 w 2 Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
4 Sierpiński graphs G 0, G 1,... sequence of graphs, discrete approximations of the Sierpińksi gasket. Recursive definition: G n+1 is the amalgam of three copies of G n. VG 0 = {w 1, w 2, w 3 }, EG 0 = {w 1 w 2, w 2 w 3, w 3 w 1 } VG n+1 = ψ 1 (VG n ) ψ 2 (VG n ) ψ 3 (VG n ), EG n+1 = ψ 1 (EG n ) ψ 2 (EG n ) ψ 3 (EG n ). G 0 G 1 G 2 G 3 Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
5 Loop erasure and LERW loop erasure: x = (x 0,..., x n ) a walk. LE(x) is obtained from x by chronologically deleting cycles. LE(x) is a self-avoiding walk from x 0 to x n. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
6 Loop erasure and LERW loop erasure: x = (x 0,..., x n ) a walk. LE(x) is obtained from x by chronologically deleting cycles. LE(x) is a self-avoiding walk from x 0 to x n. Formally: LE(x) k = x ι(k), where ι(0) = max{j n : x j = x 0 }, ι(k + 1) = max{j n : x j = x ι(k)+1 }. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
7 Loop erasure and LERW loop erasure: x = (x 0,..., x n ) a walk. LE(x) is obtained from x by chronologically deleting cycles. LE(x) is a self-avoiding walk from x 0 to x n. Formally: LE(x) k = x ι(k), where ι(0) = max{j n : x j = x 0 }, ι(k + 1) = max{j n : x j = x ι(k)+1 }. loop erased random walk (LERW): (X n ) n a simple random walk with start in z. s = s B the hitting time on some vertex set B. LE((X n ) 0 n s ) is a random self-avoiding walk from z to B. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
8 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
9 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
10 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
11 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
12 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
13 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
14 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
15 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
16 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
17 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
18 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
19 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
20 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
21 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
22 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
23 Example: LERW start stop Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
24 LERW introduced by G. Lawler (1980). comparison with self-avoiding walks (SAW): SAW is a random self-avoiding walk with respect to the uniform distribution. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
25 LERW introduced by G. Lawler (1980). comparison with self-avoiding walks (SAW): SAW is a random self-avoiding walk with respect to the uniform distribution. LERW on Z d : d = 2: convergence to SLE 2. d 4: convergence to Brownian motion. d = 3: difficult? Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
26 LERW introduced by G. Lawler (1980). comparison with self-avoiding walks (SAW): SAW is a random self-avoiding walk with respect to the uniform distribution. LERW on Z d : d = 2: convergence to SLE 2. d 4: convergence to Brownian motion. d = 3: difficult? Wilson s algorithm (1996) provides a connection between LERW and uniform spanning trees (UST), a random spanning tree with respect to the uniform distribution. (spanning tree: subgraph, which is a tree and contains all vertices) Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
27 Wilson s algorithm G a finite graph and x an arbitrary vertex in G. VT 0 = {x} and ET 0 =. {(X z n ) n : z VG} a family of independent simple random walks. (X z n ) n starts in z. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
28 Wilson s algorithm G a finite graph and x an arbitrary vertex in G. VT 0 = {x} and ET 0 =. {(X z n ) n : z VG} a family of independent simple random walks. (X z n ) n starts in z. recursive definition of T k (if VT k 1 VG): choose z / VT k 1 arbitrary and set B = VT k 1, s = s B. set T k = T k 1 LE((X z n ) 0 n s ). Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
29 Wilson s algorithm G a finite graph and x an arbitrary vertex in G. VT 0 = {x} and ET 0 =. {(X z n ) n : z VG} a family of independent simple random walks. (X z n ) n starts in z. recursive definition of T k (if VT k 1 VG): choose z / VT k 1 arbitrary and set B = VT k 1, s = s B. set T k = T k 1 LE((X z n ) 0 n s ). all graphs T k are trees. the last graph in this sequence is a uniform spanning tree. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
30 Example: Wilson s algorithm T 0 : green Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
31 Example: Wilson s algorithm T 0 : green starting point for LERW: red Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
32 Example: Wilson s algorithm T 0 : green LERW: red Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
33 Example: Wilson s algorithm T 1 : green Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
34 Example: Wilson s algorithm T 1 : green starting point for LERW: red Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
35 Example: Wilson s algorithm T 1 : green LERW: red Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
36 Example: Wilson s algorithm T 2 : green Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
37 Example: Wilson s algorithm T 2 : green starting point for LERW: red Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
38 Example: Wilson s algorithm T 2 : green LERW: red Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
39 Example: Wilson s algorithm T 3 : green, sample of a uniform spanning tree Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
40 Reversed use of Wilson s algorithm G a finite graph, z, x two vertices, T a spanning tree of G. ztx = (z,..., x) the unique self-avoiding walk in T from z to x. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
41 Reversed use of Wilson s algorithm G a finite graph, z, x two vertices, T a spanning tree of G. ztx = (z,..., x) the unique self-avoiding walk in T from z to x. Wilson s algorithm: If T is a UST, then ztx has the same distribution as LERW from z to x. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
42 Reversed use of Wilson s algorithm G a finite graph, z, x two vertices, T a spanning tree of G. ztx = (z,..., x) the unique self-avoiding walk in T from z to x. Wilson s algorithm: If T is a UST, then ztx has the same distribution as LERW from z to x. T : green, a sample of a UST. z, x: red, two vertices. z x Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
43 Reversed use of Wilson s algorithm G a finite graph, z, x two vertices, T a spanning tree of G. ztx = (z,..., x) the unique self-avoiding walk in T from z to x. Wilson s algorithm: If T is a UST, then ztx has the same distribution as LERW from z to x. T : green, a sample of a UST. z, x: red, two vertices. ztx: red, a sample of LERW from z to x. z x Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
44 On the number of spanning trees I n: family of spanning trees in G n. w 3 ψ 3 ψ 1 ψ 2 w 1 w 2 Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
45 On the number of spanning trees I n: family of spanning trees in G n. n: family of spanning forests with two components; one contains w 1, the other w 2, w 3. n, n analogous. w 3 ψ 3 ψ 1 ψ 2 w 1 w 2 Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
46 On the number of spanning trees I n: family of spanning trees in G n. n: family of spanning forests with two components; one contains w 1, the other w 2, w 3. n, n analogous. n: family of spanning forests with three components; each component contains one of the vertices w 1, w 2, w 3. w 3 ψ 3 ψ 1 ψ 2 w 1 w 2 Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
47 On the number of spanning trees II τ n = n, σ n = n = n = n, ρ n = n. 6 } {{ } τ n+1 = 6τn 2 σ n }{{} σ 7τ n σn 2 n+1 = + 1 } {{ } τn 2 ρ n }{{} ρ 14σn 3 n+1 = } {{ } 12τ n σ n ρ n Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
48 Inverse limits There are natural projections n+1 n, n+1 n, etc. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
49 Inverse limits There are natural projections n+1 n, n+1 n, etc. These projections are consistent with the uniform distributions on the respective sets. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
50 Inverse limits There are natural projections n+1 n, n+1 n, etc. These projections are consistent with the uniform distributions on the respective sets. Hence there are inverse limits of ( 0, Unif 0) ( 1, Unif 1) (, P), etc. Write T n to denote the map T n : n. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
51 A multi-type Galton-Watson tree Each G n consists of 3 n upright triangles, each of which belongs to exactly one upright triangle of G n 1. Hence all triangles of all levels are equipped with the structure of a 3-ary tree. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
52 A multi-type Galton-Watson tree Each G n consists of 3 n upright triangles, each of which belongs to exactly one upright triangle of G n 1. Hence all triangles of all levels are equipped with the structure of a 3-ary tree. On each of these 3 n triangles of G n a spanning tree of G n induces one of the 7 subgraphs {,,,,,, }. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
53 A multi-type Galton-Watson tree Each G n consists of 3 n upright triangles, each of which belongs to exactly one upright triangle of G n 1. Hence all triangles of all levels are equipped with the structure of a 3-ary tree. On each of these 3 n triangles of G n a spanning tree of G n induces one of the 7 subgraphs {,,,,,, }. Under P the assignment of one of these 7 subgraphs induced by the spanning tree T n to each of the 3 n triangles of G n yields a multi-type Galton-Watson tree with 7 types, where each individual produces exactly 3 offsprings. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
54 Example: multi-type Galton-Watson tree Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
55 LERW on Sierpińksi graphs Using the description of UST by a multi-type Galton-Watson tree one obtains, that LERW can be described by a multi-type Galton-Watson tree, too. (12 types, 2 or 3 offsprings, non-singular, positively regular, supercritical; the multi-type Galton-Watson tree contains more information than the LERW) Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
56 LERW on Sierpińksi graphs Using the description of UST by a multi-type Galton-Watson tree one obtains, that LERW can be described by a multi-type Galton-Watson tree, too. (12 types, 2 or 3 offsprings, non-singular, positively regular, supercritical; the multi-type Galton-Watson tree contains more information than the LERW) The length of LERW on G n from w 1 to w 2 has the same distribution as the length L n of the self-avoiding walk w 1 T n w 2 connecting w 1 and w 2 in T n. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
57 LERW on Sierpińksi graphs Using the description of UST by a multi-type Galton-Watson tree one obtains, that LERW can be described by a multi-type Galton-Watson tree, too. (12 types, 2 or 3 offsprings, non-singular, positively regular, supercritical; the multi-type Galton-Watson tree contains more information than the LERW) The length of LERW on G n from w 1 to w 2 has the same distribution as the length L n of the self-avoiding walk w 1 T n w 2 connecting w 1 and w 2 in T n. D. Dhar & A. Dhar (1997): E(L n ) ( ) α n with α = Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
58 LERW on Sierpińksi graphs Using the description of UST by a multi-type Galton-Watson tree one obtains, that LERW can be described by a multi-type Galton-Watson tree, too. (12 types, 2 or 3 offsprings, non-singular, positively regular, supercritical; the multi-type Galton-Watson tree contains more information than the LERW) The length of LERW on G n from w 1 to w 2 has the same distribution as the length L n of the self-avoiding walk w 1 T n w 2 connecting w 1 and w 2 in T n. D. Dhar & A. Dhar (1997): E(L n ) ( ) α n with α = T. & Wagner (2011): α n L n converges almost surely to a positive random variable with continuous density. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
59 The limit process Write (X n t ) t 0 for the continuous-time process with state space K obtained from the self-avoiding walk w 1 T n w 2 by linear interpolation and constant extension. Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
60 The limit process Write (X n t ) t 0 for the continuous-time process with state space K obtained from the self-avoiding walk w 1 T n w 2 by linear interpolation and constant extension. T. & Wagner (2011): (X n α n t) t 0 converges almost surely to a process (X t ) t 0 in K. Proof is based on previous work by M. Barlow & E. Perkins (1988, simple random walk), K. Hattori & T. Hattori (1991, self-avoiding walk), T. Kumagai (1993, p-stream diffusions ). Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
61 The limit process Write (X n t ) t 0 for the continuous-time process with state space K obtained from the self-avoiding walk w 1 T n w 2 by linear interpolation and constant extension. T. & Wagner (2011): (X n α n t) t 0 converges almost surely to a process (X t ) t 0 in K. Proof is based on previous work by M. Barlow & E. Perkins (1988, simple random walk), K. Hattori & T. Hattori (1991, self-avoiding walk), T. Kumagai (1993, p-stream diffusions ). Almost sure properties of the limit process (X t ) t 0 : self-avoiding Hausdorff dimension = log α log Hölder continuous to any exponent < log 2 log α Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
62 Example: limit process Elmar Teufl (Uni Tübingen) UST & LERW 12 September / 17
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