Coarsening with a frozen vertex

In the standard nearest-neighbor coarsening model with state space $\{-1,+1\}^{\mathbb{Z}^2}$ and initial state chosen from symmetric product measure, it is known (see~\cite{NNS}) that almost surely, every vertex flips infinitely often. In this paper, we study the modified model in which a single vertex is frozen to $+1$ for all time, and show that every other site still flips infinitely often. The proof combines stochastic domination (attractivity) and influence propagation arguments.


Introduction
As in our earlier paper [1], we study and compare the long time behavior of two continuous time Markov coarsening models with state space Ω = {−1, +1} Z d . One, σ(t), is the standard model in which at time zero {σ x (0) : x ∈ Z d } is an i.i.d. set with θ ≡ P (σ x (0) = +1) = 1/2 and then vertices update to agree with a strict majority of their 2d nearest neighbors or, in case of a tie, choose their value by tossing a fair coin. The modified model, σ ′ (t), is the same except that σ ′ at the origin (0, 0....0) is frozen to +1 for all t ≥ 0.
For d = 2, it is an old result [2] that in the standard σ(t) model, almost surely, every vertex changes sign infinitely many times as t → ∞. The main result of this paper (see Theorem 2.7) is that the same is true for the frozen model σ ′ (t) on Z 2 . It is believed (see, for example, Sec. 6.2 of [3]), but not proved, that the d = 2 behavior of σ remains valid at least for some values of d > 2. If this were so, then the arguments of this paper would show the same for the corresponding σ ′ model.
In the previous paper [1] we considered models with infinitely many frozen vertices and in this paper a model with a single frozen vertex. It would be of interest to study models with finitely many, but more than one, frozen vertices; in this regard, see the remark following the proof of Theorem 2.8 below.

Results
In this section we fix d = 2. We also use the standard convention that the updates are made when independent rate one Poisson process clocks at each vertex ring.
Let A T denote the event that the "right" neighbor of the origin (at x = (1, 0)) is −1 for some t ≥ T . Let A 1 T ⊂ A T denote the event that the right neighbor of the origin is the first neighbor to be −1 at some time t ≥ T (more precisely, that no other neighbor is −1 at an earlier time in [T, ∞]). Let B L,s for s ∈ {−1, +1} ΛL (where Λ L = {−L, −L + 1, ...., L} 2 ) denote the event that σ ′ (0)| ΛL = s and write B L,+ when s ≡ +1. We denote the probability measure for the frozen origin σ ′ (·) model by P ′ and that for the regular coarsening model σ(·) by P .
The result is an easy consequence of symmetry among the four neighbors of the origin and the fact that P (A 0 ) = 1 (indeed, for all T , P (A T ) = 1 -see [2]). Let Σ L T denote the sigma-field generated by the initial spin values and clock rings and coin tosses up to time T inside the box Λ L .
Proof. Letσ L T (.) denote the model with the spin values at all sites in Λ L frozen to +1 from time 0 up to time T and with the spin value at the origin remaining frozen at +1 thereafter. Denote the corresponding probability measure byP L T . Under the standard coupling,σ(·) stochastically dominates σ ′ (·), so we have To continue the proof, we will use the following result about the "propagation speed" of influence between different spatial regions: Proof. Let L ′ >> L and note that given B L ′ ,+ , (D L T ) c can occur only if there is a nearest neighbor (self-avoiding) path between the boundaries of the two sets, Z 2 \ Λ L ′ and Λ L , along which there are clock rings occurring in succession between times 0 and T . Any such path is at least of length L ′ − L (i.e., contains at least L ′ − L vertices besides the starting one).
Consider a particular path γ of length m ≥ L ′ − L. For each m there are no more than 3 m such paths from each boundary point and the time it takes for successive clock rings along γ is at least S m = m i=1 τ i where the τ i are i.i.d. exponential random variables with parameter 1. By the exponential Markov inequality, for any α > 0, Therefore, since there are at most CL ′ possible starting points (for some constant C), where C(α, T, L) is a constant depending on α, T and L. Taking α > 2 and the limit as L ′ → ∞ completes the proof of the lemma.
Proof. We continue the proof of Proposition 2.2. Pick ǫ > 0 and fix T and L. By Lemma 2.3, ∃ L ′ such that P (D L T |B L ′ ,+ ) ≥ 1 − ǫ . Therefore, given B L ′ ,+ , with probability at least 1−ǫ, σ t (·) positively dominatesσ T L (·) for 0 ≤ t < S, where S = inf{t > 0 | σ t (0, 0) = −1}, and sõ Taking the limit as ǫ → 0 completes the proof of Proposition 2.2. Now let Σ T denote the sigma field generated by the initial assignment of spins on Z 2 and the clock rings and coin tosses on Z 2 up to time T .
Proof. This is a straightforward consequence of the preceding corollary. It follows that with probability one, σ ′ (1,0) (t) changes sign infinitely many times as t → ∞.
Letting ǫ → 0 completes the proof of the first part of the theorem. The second part then follows because by stochastic domination (attractivity) and the results of [2], σ ′ (0,0) (t i ) equals +1 for an infinite sequence of t i → ∞.
The next theorem follows from a modified version of the proof of Theorem 2.7.
Proof. For any site z other than the origin, and for L much larger than say the Euclidean norm of z, we consider the unfrozen σ model in which at time zero all the vertex values are set to +1 in the box of side length 2L, centered at z/2 (so that the origin and z are located symmetrically with respect to this box). Then with probability 1/2 the vertex at z flips to −1 before the one at the origin flips and until just after that time, there is no difference between the frozen (at the origin) σ ′ model and the unfrozen σ model. Hence there is probability at least 1/2 in σ ′ that z will flip to minus. By applying the methods used in the proof of Theorem 2.7 (but with 1/4 now replaced by 1/2), we conclude that z will flip infinitely many times with probability one. We note that the line of reasoning in the proof of the last theorem could have also been used to give a modified proof of Theorem 2.7 with 1/4 replaced by 1/2. A more interesting remark is the following.
Remark 2.9. For the process σ ′′ with some finite set S of vertices frozen to +1, it is possible to show by an extension of the arguments used in this paper that there is a finite deterministic S ′ ⊇ S such that all sites in {Z 2 \S ′ } flip infinitely many times in σ ′′ (·) with probability one. In some cases, S ′ must be strictly larger than S -e.g., when S = {(−L, −L), (−L + L), (+L, −L), (+L, +L)}, S ′ includes all of Λ L . One may also consider processes where some vertices are frozen to −1 and some to +1. We expect to to pursue these issues in a future paper.