Fixation for Two-Dimensional ${\mathcal {U}}$-Ising and ${\mathcal {U}}$-Voter Dynamics

Abstract

Given a finite family ${\mathcal {U}}$ of finite subsets of ${\mathbb {Z}}^d\setminus \{0\}$, the ${\mathcal {U}}$-voter dynamics in the space of configurations $\{+,-\}^{{\mathbb {Z}}^d}$ is defined as follows: every $v\in {\mathbb {Z}}^d$ has an independent exponential random clock, and when the clock at v rings, the vertex v chooses $X\in {\mathcal {U}}$ uniformly at random. If the set $v+X$ is entirely in state $+$ (resp. −), then the state of v updates to $+$ (resp. −), otherwise nothing happens. The critical probability $p_c^{\mathrm{vot}}({\mathbb {Z}}^d,{\mathcal {U}})$ for this model is the infimum over p such that this system almost surely fixates at $+$ when the initial states for the vertices are chosen independently to be $+$ with probability p and to be − with probability $1-p$. We prove that $p_c^{\mathrm{vot}}({\mathbb {Z}}^2,{\mathcal {U}})<1$ for a wide class of families ${\mathcal {U}}$. We moreover consider the ${\mathcal {U}}$-Ising dynamics and show that its corresponding critical probability $p_c^{\mathrm{Is}}({\mathbb {Z}}^2,{\mathcal {U}})$ is also less than 1, for many families ${\mathcal {U}}$, so that this model exhibits the same phase transition.

Appendix A: On the Unfair Directions

In this appendix we provide many examples of families with (and without) unfair directions. By using Proposition 3.12, we are able to construct a large class of families having $e_2$ (wlog) as an unfair direction, as follows.

Example A.1

Fix a two-dimensional family ${\mathcal {U}}'$ satisfying

• (I) $\forall X\in {\mathcal {U}}', X\cap {\mathbb {H}}_{-e_2}\ne \emptyset$ and 0 is in the convex hull of X,

and let ${\mathcal {V}},{\mathcal {W}}$ be 1-dimensional families such that

• (II) $0<\nu _- \leqslant w_+$ and $w_-\leqslant \nu _+$,

where $\nu _\pm$ (resp. $w_\pm$) denotes the number of rules of ${\mathcal {V}}$ (resp. ${\mathcal {W}}$) entirely contained in ${\mathbb {Z}}_\pm$ (${\mathbb {Z}}_+={\mathbb {N}}$ and ${\mathbb {Z}}_-=-{\mathbb {N}}$). Then, for every $i\in {\mathbb {Z}}_+$, the following induced two-dimensional family is voter-eroding:

$$\begin{aligned} {\mathcal {U}}_i({\mathcal {U}}',{\mathcal {V}},{\mathcal {W}}) :={\mathcal {U}}' \cup \bigcup _{R\in {\mathcal {V}}}\{X(i,R)\} \cup \bigcup _{R\in {\mathcal {W}}}\{X(-i,R)\}, \end{aligned}$$

where $X(\pm i,R)$ denotes the rule $\{\pm ie_1\}\cup \{re_2:r\in R\}$.

In fact, ${\mathcal {U}}_i({\mathcal {U}}',{\mathcal {V}},{\mathcal {W}})$ satisfies (a), since either $ie_1$, or $-ie_1$, is the only vertex in $l_{e_2}$ and in some rule $X(\pm ie_1, R)$ at the same time. Condition (b) is also satisfied, since we can take $g(X)=X$ for $X\in {\mathcal {U}}'$, and it is easy to see that condition (II) ensures that we can make an one to one assignment to those rules $X\notin {\mathcal {U}}'$.

One specific example of a family satisfying (I) and (II) is

$$\begin{aligned} {\mathcal {U}}_1(\{\{e_2,-e_2\}\},\ \{\{1\},\{-1\}\},\ \{\{1\},\{-1\}\}) ={\mathcal {N}}_2^2\setminus \{\{e_1,-e_1\}\}. \end{aligned}$$

Moreover, note that Proposition 3.12 also implies that given $i_1,\dots ,i_k\in {\mathbb {Z}}_+$, the family

$$\begin{aligned} {\mathcal {U}}_{i_1}({\mathcal {U}}'_1,{\mathcal {V}}_1,{\mathcal {W}}_1) \cup \cdots \cup {\mathcal {U}}_{i_k}({\mathcal {U}}'_k,{\mathcal {V}}_k,{\mathcal {W}}_k) \end{aligned}$$

is voter-eroding, as long as ${\mathcal {U}}'_j,{\mathcal {V}}_j,{\mathcal {W}}_j$ satisfy (I) and (II) for each $j\leqslant k$. For instance, the family ${\mathcal {U}}_2$ considered in Example 1.3 is voter-eroding, since

$$\begin{aligned} {\mathcal {U}}_2={\mathcal {U}}_1(\{(1,1), (-1,4), -e_2\},\{\{-3,-2,-1\}\},\{\{1\}\}) \cup {\mathcal {U}}_2(\emptyset ,\{\{3,5\}\},\{\{-3\}\}). \end{aligned}$$

On the other hand, we are free to construct plenty of examples of different nature which satisfy conditions (a) and (b) of Proposition 3.12, by using the following trivial observation.

Remark A.2

(Adding rules) Infinitely many families with an unfair direction y can be constructed from any family ${\mathcal {U}}$, just by properly adding new rules $X\subset {\mathbb {H}}_{-y}$. Moreover, if ${\mathcal {U}}$ is critical, then the new families can be chosen critical as well, we just need to add new rules carefully, without modifying (too much) the stable set; an example is going to be given below (see Example A.3).

Fig. 5

The 1-dimensional dynamics associated to $e_2$ for the family ${\mathcal {U}}_{3,8}$, with $L=100$ and run time 1000. Time is oriented downwards

1.1 No Unfair Directions

For a concrete example of critical families without unfair directions, consider the collection ${\mathcal {U}}_{3,8}$ of all subsets of size 3 of

$$\begin{aligned} \{\pm e_1,\pm e_2,\pm 2e_1,\pm 2e_2\}. \end{aligned}$$

In Example 3.5 we realized that ${\mathcal {S}}({\mathcal {U}}_{3,8}) =\{\pm e_1,\pm e_2\}$. In order to check that no direction in ${\mathcal {S}}({\mathcal {U}}_{3,8})$ is unfair, by symmetry it is enough to consider $y=e_2$; in the 1-dimensional setting, straightforward calculations show that configurations $\eta '$ of the form

$$\begin{aligned} \eta '=[-,+,+,+,+,-,+,+,+,+,-,+,+,+,+,-, \cdots , +,+,+,+,-], \end{aligned}$$

which alternate four $+$s and one −, do not satisfy (11). However, simulations (see Fig. 5) indicate that the 1-dimensional dynamics look like a diffusive process (configurations like $\eta '$ are very unlikely), and we could erode the segment $Y(e_2,L)$ in time $O(L^{2})$, which would mean that ${\mathcal {U}}_{3,8}$ is voter-eroding. This family could be a good starting point for future research, in order to prove that Conjecture 1.14 holds for symmetric critical families.

Another such family, which is special since its droplets are triangular, is

$$\begin{aligned} {\mathcal {U}}_\rhd =\{\{(-1,1),(-1,-1)\}, \{(0,1),(1,1)\}, \{(0,-1),(1,-1)\}\}, \end{aligned}$$

with ${\mathcal {S}}({\mathcal {U}}_\rhd )=\left\{ -e_1, \frac{1}{\sqrt{2}}(1,1), \frac{1}{\sqrt{2}}(1,-1)\right\}$ (see Fig. 6).

Fig. 6

A voter-eroding critical family without unfair directions, its stable set and a failing configuration

This family only has 3 candidates to be y and all of them fail Condition (11). In fact, say we choose $y=-e_1$ and take $L=2n+1$ for some fixed n. The configuration $\eta$ given by

$$\begin{aligned} \eta ^+=\{(0,2k):k=0,\ldots ,n\}, \end{aligned}$$

yields $\sum _{u\in \eta ^-}r_u(\eta )=n$, while $\sum _{v\in \eta ^+}r_v(\eta )=2n$. An analogous situation happens for the other 2 candidates. However, ${\mathcal {U}}_{\rhd }$ is voter-eroding (see Proposition A.4).

Now, we illustrate how to construct a voter-eroding family from ${\mathcal {U}}_\rhd$, in the sense of Remark A.2.

Example A.3

The direction $y=-e_1$ is unfair for the family

$$\begin{aligned} {\mathcal {U}}_\rhd \cup \{\{(-1,2),(-1,-1)\}\}, \end{aligned}$$

this is just because we added a rule in the + side to enforce inequality (15); since it has the same stable set as ${\mathcal {U}}_\rhd$, then it is critical and voter-eroding so our main theorem holds for this new family.

In general, if we consider any rule $X_0\subset {\mathbb {H}}_{e_1}$, $X_0\ne \{(-1,1), (-1,-1)\}$ such that there exist $x,x'\in X_0$ with $x_2\geqslant -x_1$ and $x_2'\leqslant x_1'$, we can check that $-e_1$ is an unfair direction for the family ${\mathcal {U}}_\rhd \cup \{X_0\}$, and the latter has the same stable set as ${\mathcal {U}}_\rhd$. Of course, we can construct infinitely many such families in this way.

As a last example, we will show that ${\mathcal {U}}_{\rhd }$ is $(2+\varepsilon ,{\mathcal {S}}({\mathcal {U}}_\rhd ))$-eroding for any $\varepsilon >0$, by using an argument similar to the one given in the proof of Proposition 3.4, and coupling with the 1-dimensional contact process.

Proposition A.4

The family ${\mathcal {U}}_\rhd$ is voter-eroding.

Proof

The droplet erosion time $\tau$ in the 1-dimensional dynamics associated to $y=-e_1$ is stochastically dominated by the extinction time $\tau _{CP}$ of a (slowed down by a 1/3 factor) contact process on $\{1,\dots , L\}$ with parameter 1. This process is subcritical, so by Theorem 1 in [9], $\tau _{CP}/\log L$ converges in probability to a positive constant, as $L\rightarrow \infty$. Now, by following the same lines along the proof of Proposition 3.4, (10) and (9) are straightforward. $\square$