Augusto Teixeira (IMPA)

Transcription

1 Augusto Teixeira (IMPA)

2 partially joint with Elisabetta Candellero (Warwick) Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

3 Percolation Physical phenomenon: Fluid through porous medium Material sciences, epidemics, networks Main motivations: Phase transition Universality Challanges Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

4 Mathematical model Broadbent, Hammersley 57 Infinite graph G = (V, E), e.g. Z d Parameter p [0, 1] Open each vertex x V independently with probability p Connectivity properties of the induced subgraph Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

5 Phase transition p = p = p = Is the origin connected to infinity? θ(p) 1 θ(p) = P[o ] p c (G) = sup{p; θ(p) = 0} 0 p c 1 p Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

6 On Z d A lot is known: d 2 non-trivial transition (p c (0, 1)) not true for d = 1 d 2 θ smooth for p > p c d 11 θ continuous + critical behavior d = 2 θ continuous + critical behaviour Still a lot to learn: θ continuous for all d? critical behavior for all d? Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

7 Other graphs Is p c (G) (0, 1)? If degree G bounded by p c 1/. What about p c (G) < 1? The first step in a study of percolation on other graphs ( ) will be to prove that the critical probability on these graphs is smaller than one. Benjamini, Schramm Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

8 Other graphs We know p c (G) < 1 for: Regular trees (strong symmetries) Expanders [Benjamini, Schramm] Cayley: with exponential growth [Lyons] Cayley: finitely generated, with one end [Babson, Benjamini] Cayley: intermediate growth [Muchnik, Pak] Techniques range from Entropy vs energy Transport principle Analytical tools Homology... Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

9 Isoperimetric inequalities p c (Z 1 ) = 1, does dimension play a role? On R d, every compact A Z d with smooth boundary: A c d A (d 1)/d (recall B = c d B (d 1)/d ) Definition dim(g) d def inf A V ; finite A A (d 1)/d > 0 Examples: Z d, Sirpinsky graphs, regular trees... Very useful concept to study random walks on G Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

10 Benjamini-Schramm s question Question Is it true that dim(g) > 1 implies p c (G) < 1? Two results in this direction: Theorem (Kozma) If G is planar, with polynomial growth, no accumulation points then: dim(g) > 1 p c (G) < 1. Theorem (Candellero, T.) If G is transitive, with polynomial growth, then: dim(g) > 1 p c (G) < 1. Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

11 Local isoperimetric inequalities Definition - local isoperimetric dimension dim l (G) d def inf B=B(x,r) inf A B; A B /2 B A A (d 1)/d > 0 Counterexamples: regular trees, two Z d s glued... Theorem (T.) If G has polynomial growth, then: dim l (G) > 1 p c (G) < 1. Moreover, p u (G) < 1, dependent models, Ising [Häggström]. Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

12 Renormalization Multi-scale, in a nutshell: Coarse-graining procedure Process dynamics Geometry aware Robust to changes Well adapted for perturbative systems Well adapted for Z d Various models: rwre, sand-piles, interface motion... Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

13 Renormalization example Let us show p c (Z 2 ) > 0: L k = 5 k, for k 0 A k+1 A k A k q k q 2 k A k = L k f (p) = cp 2 3L k q k (p) = P[A k ] 0 p 3L k+1 L k Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

14 Renormalization steps Multi-scale, a perturbative recipe: Find suitable scale sequence L k Get a paving structure on G Define bad events A k Show A k cascades Chose parameter to start induction A few advantages: Very resilient to changes in the model Provides quantitative estimates Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

15 Hypotheses Polynomial growth Allows paving: Local isoperimetric inequality Big flow between sets: Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

16 Separation events A A Lemma The separation events are cascading. S(x, L) = there exist sets A, A in B(x, L) with diameters L/100 and which are not connected

17 Separation events A A Lemma The separation events are cascading. S(x, L) = there exist sets A, A in B(x, L) with diameters L/100 and which are not connected

18 Separation events A A Lemma The separation events are cascading. S(x, L) = there exist sets A, A in B(x, L) with diameters L/100 and which are not connected Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

19 Classic isoperimetric inequality The arms from A and A need not meet: A A Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

20 Embedding a tree Transitivity implies a tree could be embedded in G: Contradicting polynomial growth. Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

21 Bibliography Itai Benjamini and Oded Schramm, Percolation beyond Z d, many questions and a few answers, Electron. Comm. Probab. 1 (1996), no. 8, Gady Kozma, Percolation, perimetry, planarity., Rev. Mat. Iberoam. 23 (2007), no. 2, (English). Russell Lyons, Random walks and the growth of groups, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 11, Russell Lyons and Yuval Peres, Probability on trees and networks, Cambridge University Press, Cambridge, A. Teixeira, Percolation and local isoperimetric inequalities, ArXiv e-prints (2014). E. Candellero and A. Teixeira, Percolation and isoperimetry on transitive graphs, preprint (2015). Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19

22 Thank you! Augusto Teixeira (IMPA) Percolation and isoperimetry SPA - Oxfort, / 19