A. Telcs AFRT December 2012 Hong Kong
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1 Supporting research schools at the University of Pannonia Reg.No.:TÁMOP-4.2.2/B-10/ Project period: Name and address of Contractor: University of Pannonia Egyetem utca 10, Veszprém H-8200 Name and Address of Subcontractor: A. Telcs AFRT December 2012 Hong Kong Agricultural Research Institute of the Hungarian Academy of Sciences Brunszvik utca 2, Martonvásár H-2462 Total funding granted by the European Union and the Hungarian Government: HUF.
2 Di usive limits on the Penrose tiling András Telcs Hong Kong December 2012.
3 The beginings Figure: Islamic tiling from the 15th century Peter J. Lu and Paul J. Steinhardt "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture," Science 315, 1106 (2007).
4 The beginings
5 The Penrose tiling The Penrose tiling Penrose 1978
6 Random walk on the Penrose graph Random walk on the Penrose graph The graph and the net (the dual of the Penrose tiling)
7 Random walk The graph and the net (the dual of the Penrose tiling) countably in nite graph, with edges x y; and trivial edge weight 1, degree of nodes, d (x) = P y I (x y) = 1 4 P (x; y) = 1 d (x) if x y P (X n = yjx n 1 = x) = P (x; y) de nes a d (x)-reverzibe Markov chain
8 Random walk Conventions with d (x; y) ; graph distance called Penrose graph with jx yj embedded into R 2 called Penrose net Z 2 with d (x; y) the graph on the integer lattice Z 2 with jx yj embedded into R 2 called integer net
9 Questions Questions Random walks on graphs x;y = 1; d (x; y), p n (x; y)? 1 Cn e C d 2 (x;y ) n p n (x; y) + p n+1 (x; y) C n e d 2 (x;y ) Cn
10 Questions Questions Domokos Szász problem: Do we have invariance principle on the Penrose net? x;y = 1; jx yj as a! 1: D pos. def. 1 p a X batc =) W D (t) as a! 1
11 Questions Why is this a question? put Z d grid on R d then the invariance principle holds for the ssnn RW Put any periodic (shift invariant) grid on R d, still we have the IP. What if we put a non-periodic net on it? What if... (comes later).
12 Pentagrid The construction of the Penrose tiling (debruijn) Pentagrid
13 Pentagrid The construction of the Penrose tiling (debruijn) Pentagrid
14 Pentagrid
15 Rough isometry Rough isometry Given ; 0 ; rough isometric if, there are :! 0 ; a; b; c; M > 0 : 1 a d (x; y) b d0 ( (x) ; (y)) ad (x; y) + b; 1 c d0 ( (x)) d (x) cd 0 ( (x)) ; for all y : d 0 ( ( ) ; y 0 ) < M: Let us use 1.
16 Stability Stability against rough isometry Stability of (GE ;2 ) c 1 exp n=2 C d2 (x; y) t ep n (x; y) C 1 exp n=2 c d2 (x; y) t ep n = p n + p n 1, Theorem (Delmotte 1997.) Yes. On Z d we have,(ge d ;2 )
17 Random walk on the Penrose graph Random walk on the Penrose graph Penrose graph If roughly isometric to Z 2, then c 1 n exp C d2 (x; y) ep n (x; y) C 1n t exp c d2 (x; y) t on the Penrose graph.
18 Rough isometry Rough isometry Theorem Solomon The Penrose graph is bi-lipschitz to Z 2.
19 Rough isometry Corollary A Penrose graph is rough isometric to Z 2. Direct proof. :
20 Rough isometry Random walk on the Penrose graph Theorem c 1 n exp C d2 (x; y) ep n (x; y) C 1n t exp c d2 (x; y) t
21 Invariance principle Preparation c 1 n exp C d2 (x; y) ep n (x; y) C 1n t exp c d2 (x; y) t
22 Invariance principle On the Penrose net the random walk X n = P n i=1 x i :x i vectors between centers of tiles.! n the environment seen from X n Z n = (! n ; X n ) ; nx X n = V (Z i 1 ; Z i ) i=1 V (Z i 1 ; Z i ) = X i X i 1 = x i :
23 Invariance principle Penrose graph Theorem (demasi, Ferrari, Goldstein, Wick) Z n reversible ergodic Markov chain with stationary probability measure, X n as above, anti-symmetric function of Z then 1 p A X batc =) W D (t) : D???.
24 The Penrose graph Theorem For the random walk on the Penrose net the invariance principle holds. Proof. 1. Z n egrodic stac. (M. Kunz,A. Robinson) 2. D pos. def. so the limiting process is non-degenerate.
25 Proof 1. The ve directions e k = cos + (k 1) ; sin + (k 1) k = 1::5 and the grids G k = fx 2 R 2 : xe k? = z + k ; z 2 Zg 2 + (k 1) 2 5
26 A cross-points G i \ G j are center of a tiles, the position is defned by the phases l ; m mod 1 so that and z: 5X i = 0 i=1
27 Proof 1. X n is at a crosspoint of the pentagrid G i \ G j : i 6= j 2 f1; 2; 3; 4; 5g that is the reference point for!, i.e.: i = j = 0. For the pentagrid we know that 5X i = 0: i=1 ( i mod 1 ) Still we have two "free" : So! $ (i; j; k ; l ) identi es our position. We get 10 tori i;j : Let = [ i;j : The dinamics:! n!! n+1 on.
28 Proof 1. f! n g is dense with null Lebesgue measure on the compact. Let! 2 i;j ; ji jj = 1;! 0 2 i;j ; ji jj = 2; d (!) = d (!) d (! 0 ) = d (! 0 ) the golden ratio. is nite, can be normalised to probability measure. From the density theorem of topological groups it follows that if A invarian and (A) > 0 then (A) = 1,i.e. ergodic.
29 Proof 2. D non-degenerate. 0 6= e 2 R 2 e De > 0:
30 Proof 2. D non-degenerate. 0 6= e 2 R 2 e De > 0:
31 Proof 2. D non-degenerate. 0 6= e 2 R 2 e De > 0: Let A = B x 0 ; C 2 p n nb x0 ; C 1 p n ; C cone at e with angle : =2 > > 0. The intersection H = A \ C. E (e X n Xn EejX 0 = x 0 ) = E (e X n ) 2 jx 0 = x 0
32 Proof 2. Let A = B x 0 ; C 2 p n nb x0 ; C 1 p n ; C cone about x0 ;with =2 > > 0. H = A \ C. P N ' P G! E (e X n Xn ejx 0 = x 0 ) = E (e X n ) 2 jx 0 = x 0 X (ex) 2 P n (x 0 ; x) x2h c cos () p n c 0 exp C (ac p 2 n+b) 2 n 2 c > 0 n
33 Other tilings? What conditions needed for the IP?
34 Quenched invariance principle Is the invariance principle true for almos all starting point?
35 Projection The construction of the Penrose tiling (debruijn) Pentagrid
36 Projection The construction of the Penrose tiling (debruijn) Pentagrid
37 Projection The construction of the Penrose tiling (debruijn) Pentagrid Z 5 R 5 Let = exp (2i=5) ; the plain P consists of points x; z 2 R 5 : orthogonal to 1 = (1; 1; 1; 1; 1) and a = 1; 2 ; 4 ; 6 ; 8 K is the set of unit cubes in R 5 identi ed with their centre: C = fk 2 K : K \ P 6= ;g and project vertices and edges to P.
38 Projection Thanks for the attention!
39 Local modi cation Local modi cation Let 0 a local modi cation of the Penrose graph, Clearly and 0 rough isometric,(ge 2;2 ) holds: c n exp C d2 (x; y) p 0n (x; y) Cn n exp c d2 (x; y) n
40 Local modi cation Local modi cation Does the invariance hold? (Doma Szász s question)
41 Further questions Further questions 2 0 Is it true that D = 0 2? For local modi cation is D invariant? Is the invariance principle stabil agains rough isometry? How the percolation behaves for the Penrose graph? And the RW on it??
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