2.6.2011 BIRS Ron Pele (Tel Aviv University) Portions joint with Oha N. Felheim (Tel Aviv University)
Surface: f : Λ Hamiltonian: H(f) DGFF: V f(x) f(y) x~y V(x) x 2. R or for f a Λ a box Ranom surface: sample with probability ) (ensity) proportional to e H ( f with parameter 0 representing inverse temperature. Bounary conitions: zero on bounary, zero at one point, slope bounary conitions, etc. : Λ potential Z. V : R R. in Z.
Properties preicte to be universal. Inepenent of potential V uner minor assumptions. When Λ is a box of sie length n, have: n fluctuations in 1 imensions. log n fluctuations in 2 imensions. Boune fluctuations in 3 imensions. When V is strictly convex, many universal properties establishe by a long list of authors. Some recent progress also for continuous, non-convex potentials (Aams, Biskup, Cotar, Deuschel, Kotecký, Müller, Spohn).
When the surface takes integer values f : Λ a new transition is expecte: a twoimensional roughening transition. Transition from previous logarithmic fluctuations to boune fluctuations as the temperature ecreases. Establishe only in 2 moels! The integervalue DGFF: V(x) x an the Soli-On-Soli 2 moel: V(x) x (Fröhlich an Spencer 81) Z,
A ranom Lipschitz function is the case 0 V(x) 1 x 1 otherwise The so-calle hammock potential. Here, the parameter β is irrelevant an the function is a just a uniformly sample Lipschitz function on Λ. Natural also from an analytic point of view. Analysis of this case is wie open for all 2!
A ranom M-Lipschitz function is an sample with the potential V(x) 0 M x Almost no analysis available for these functions. A ranom (graph) homomorphism function is the case V(x) 0 otherwise x{ 1,1} otherwise M isaninteger Investigate on general graphs as a generalization of SRW. 2 In Z, it is the height function for the square ice moel. Benjamini,Yain, Yehuayoff 07:lower boun (logn) 1/ on maximum. Galvin, Kahn 03-04: On hypercube 0,1, takes 5 values with high probability as. f : Λ Z
Outermost 0 1 level sets of homomorphism functions Trivial level sets (having exactly 4 eges) not rawn
Equals 0 (Green in the picture) on nearly all of the even sublattice (when the bounary values are 0 on even bounary vertices). Obtain 2 values for 1-Lipschitz functions (M=1 case).
Z Given a box of., uniformly sample a proper q-coloring How oes this coloring look? Does it have long-range orer? Are there multiple Gibbs measures as Z? Preicte interplay between an q as follows: No structure when q is large compare to a unique Gibbs measure. Proven when q 11/3 (Vigoa 00). Rigiity when is large compare to q most colorings use only about half of the colors on the even sub-lattice an about half of the colors on the o sub-lattice. Little is proven about this regime. Proper q-colorings are the same as the anti-ferromagnetic q-state Potts moel at zero temperature. Transition from above behavior to unique Gibbs measure as temperature increases.
Normalize homomorphism height functions: f : Z Z, f(u)-f(v) 1for u ~ v an f(0) 0 are in bijection with proper 3-colorings taking the color zero at the origin. The bijection is given simply by f f mo 3. Works also for f : Λ Z when Λ is a box in Z. Does not work when Λ Z n is a torus ue to existence of non-trivial cycles (there exist 3- colorings on Z n which are not moulo 3 of homomorphisms).
0 0 Theorem(P.): such that if an f is a uniformly sample homomorphism function on then P f(x) t exp( c t ) for all x, 1/ max x f(x) C log(n) with high probability as n. The bounary values for the above theorem are fixing f to be 0 on an arbitrary subset of Z n. Matching lower boun on maximum follows from results of BYY in the case of a one-point bounary conition. The results exten to 1-Lipschitz functions via a bijection of Yain. Z n
A level set of f is a (minimal) set of eges separating the bounary set from a point an having 0 s on one sie an 1 s on the other. Main Theorem(P., P. & Felheim): 0 such that if 0, x Z n an f is a uniformly sample homomorphism function on Z n then P( level set of length L aroun x) exp(-c L) 4 We prove c c/log. 0 1 0-1 0 1-1 0 1 0 1 0 0 1 2 1 2 1 1 2 3 2 1 0 0 1 2 1 0-1 -1 0 1 0 1 0
It follows that if f is sample with zeros on the even bounary of a large box, then it will be zero on nearly all even vertices insie the box, with the largest breakup of the pattern having logarithmic bounary length. By taking moulo 3, we obtain the same rigiity also for proper 3-colorings. This establishes for the first time the existence of a phase transition in the 3-state anti-ferromagnetic Potts moel.
The results o not apply to the homomorphism moel in 2 imensions (the square ice moel). However, they may be applie to tori with non-equal sie lengths: n1 n 2 n. Nee only that is large an that the torus is non-linear: 1 n exp 3 log ( ) n i i1 Thus the moel on the n x n x 2 x 2 x x 2 torus (with a fixe number of 2 s) has boune fluctuations. I.e., the roughening transition occurs when aing a critical number of such 2 s! We prove the full roughening transition occurs between the moel on an n x log(n) torus an the moel on an n x log(n) x 2 x x 2 torus. Refutes a conjecture of Benjamini, Yain, Yehuayoff an answers a question of Benjamini, Häggström an Mossel. 1
Infinite volume limit: We prove that as Z with zero bounary conitions, the moel converges to a limiting Gibbs measure gives a meaning to a uniformly sample homomorphism or 1-Lipschitz function on the whole Z. For 3-colorings prove existence of 6 ifferent Gibbs measures (in each, one of the 3 colors is ominant on one of the two sub-lattices). Scaling limit: Embeing in 0,1 an taking finer an finer mesh, we fin that the scaling limit of the moel is white noise. Integrals over isjoint regions converge to inepenent Gaussian limits.
Z n Work on. Place zeros on the even bounary an uniformly sample a homomorphism function. Fix a point x an consier outermost level set aroun x. Denote it by LS(f). We want to show that P LS(f) L exp( c L). 0 1 0-1 0 1-1 0 1 0 1 0 0 1 2 1 2 1 1 2 3 2 1 0 0 1 2 1 0-1 -1 0 1 0 1 0
0 1 0-1 0 1-1 0 1 0 1 0 0 1 2 1 2 1 1 2 3 2 1 0 0 1 2 1 0-1 -1 0 1 0 1 0 Shift Transformation 0 1 0-1 0 1-1 0-1 0-1 0 0 1 0 1 0-1 1 2 1 0-1 0 0 1 0-1 0-1 -1 0-1 0 1 0 The value at each vertex insie the level set is replace by the value to its right minus 1. Remain with a homomorphism height function!
0 1 0-1 0 1-1 0 1 0 1 0 0 1 2 1 2 1 1 2 3 2 1 0 0 1 2 1 0-1 -1 0 1 0 1 0 Shift Transformation 0 1 0-1 0 1-1 0-1 0-1 0 0 1 0 1 0-1 1 2 1 0-1 0 0 1 0-1 0-1 -1 0-1 0 1 0 Vertices with level set on right are surroune by zeros after the shift! Can change their values arbitrarily to ±1. There are exactly LS(f) /2 such vertices. Transformation still invertible given the level set.
Thus, given a contour, or cutset, Γ, we associate to each f with LS(f) = Γ a set of 2 other homomorphisms in an invertible way. Γ /2 It follows that P LS(f) Γ 2. Can we conclue by a union boun (Peierls / 2 argument), by enumerating 2 all possible / 2 2 / 2 contours? 2 The union boun fails!!! There are too many possible contours. LS(f)=Γ / 2 2 / 2
Instea of using a union boun, we use a coarse graining technique, grouping the cutsets into sets accoring to common features an bouning the probability of each set. We note that our cutsets have a istinguishing feature they are o cutsets. Cutsets whose interior bounary is on the o sub-lattice. Denote by OCut L the set of all o cutsets with exactly L eges. L/ 2 It turns out that while OCutL 2, there are much fewer global shapes for cutsets an most cutsets are minor perturbations of some shape.
Say that A Z n to ΓOCut L if in is an interior approximation Γ expose( ) A interior(γ) where expose(γ) is the set of vertices ajacent to many eges of Γ ( 2 eges).
Theorem: For any L, there exists a set Ω containing an interior approximation to every cutset in with Clog Ω exp 3/2 L. ΓOCut L L/2 Much smaller if is large than OCutL 2. Thus, to conclue the proof it is sufficient to give a goo boun on the probability that LS(f) is any of the cutsets with a given interior approximation. This is what we en up oing, but our boun is sufficiently goo only for cutsets having a certain regularity. That they o not have almost all their eges on expose vertices. 2
We conclue by proving that there are relatively few irregular cutsets. Fewer than exp(l/100) of them in OCut L. Thus, we may separately apply a union boun using our previous estimate P Γ /2 LS(f) Γ 2 to show that none of these cutsets occur as a level set of f.
1. Analyze other Lipschitz function moels. Is the behavior similar? As mentione, analysis of the 1-Lipschitz moel follows from our results by a bijection of Yain. Analyze slope bounary values. 2. Analyze proper q-colorings for q>3. Show rigiity of a typical coloring when is sufficiently high. There oes not seem to be a similar connection to height functions any more. 3. Analyze regular (non o) cutsets an prove similar structure theorems. Can be useful in many moels (Ising, percolation, colorings, etc.) as the cutsets form the phase bounary between two pure phases. 4. Improve structure theorems for o cutsets will reuce the minimal imension for our theorems an will help in analyzing other moels. With W. Samotij we use these theorems to improve the bouns on the phase transition point in the har-core moel.