1 Introduction
Abstract

We consider a long-range growth dynamics on the two-dimensional integer lattice, initialized by a finite set of occupied points. Subsequently, a site becomes occupied if the pair consisting of the counts of occupied sites along the entire horizontal and vertical lines through lies outside a fixed Young diagram . We study the extremal quantity , the maximal finite time at which the lattice is fully occupied. We give an upper bound on that is linear in the area of the bounding rectangle of , and a lower bound , where is the side length of the largest square contained in . We give more precise results for a restricted family of initial sets, and for a simplified version of the dynamics.

Maximal spanning time for neighborhood growth on the Hamming plane111Version 1, July 20, 2019

Janko Gravner

Mathematics Department

University of California

Davis, CA 95616, USA

gravner@math.ucdavis.edu

J.E. Paguyo

Mathematics Department

University of California

Davis, CA 95616, USA

jepaguyo@ucdavis.edu

Erik Slivken

Mathematics Department

University of California

Davis, CA 95616, USA

erikslivken@math.ucdavis.edu

AMS 2000 subject classification. 05D99Key words and phrases. Hamming plane, growth dynamics, spanning time, Young diagram.

1 Introduction

The Hamming plane is the Cartesian product of two complete graphs on and so it has vertex set with an edge between any pair of points that differ in a single coordinate. We refer to the points in as sites. Investigation of percolation and growth models on the Hamming plane and related highly connected graphs is a recent development [Siv, GHPS, BBLN, Sli], and this paper addresses an extremal quantity associated with a growth process introduced in [GSS].

We keep the terminology and notation from [GSS]. For , we let be the discrete rectangle. A union of rectangles over some finite set is called a zero-set. Note that it is possible that . Also note that we restrict our consideration to finite zero-sets.

For a site , we denote by and the horizontal and vertical lines through , respectively. The neighborhood of then is . The row and column counts of in a set are given by

A zero-set determines a neighborhood growth transformation as follows. Fix . If , then . If , then if and only if the pair of row and column counts of lies outside the zero-set, i.e., . The neighborhood growth dynamics is given by the discrete time trajectory obtained by iteration of : , for . We call sites in occupied and sites in empty. See Figure 1 for an example of neighborhood growth dynamics.

(a)

(b)
Figure 1: An example of neighborhood growth dynamics with zero-set given in (a) and the initial set in the leftmost panel of (b). The next two panels of (b) depict the subsequent two iterations. Observe that , thus does not span.

The simplest example, line growth, introduced as line percolation in [BBLN], is given by a rectangular zero-set for some . Another special case, perhaps the most important one, is threshold growth, which is determined by an integer threshold . This natural growth rule is defined on an arbitrary graph as follows: a site becomes occupied when the number of already occupied sites among its neighbors is at least . This dynamics was first introduced on trees in [CLR] and is typically called bootstrap percolation. The most common setting, with many deep and surprising results, is a graph of the form , a Cartesian product of path graphs of points, and thus with standard nearest neighbor lattice connectivity [AL, GG, Hol, BB, GHM, BBDM]. On the Hamming plane, threshold growth is given by the triangular zero-set ; see [GHPS] and [GSS] for further background.

Define the set of eventually occupied sites by . The set spans if . For a fixed zero-set , we let denote the collection of all finite spanning subsets of . It follows from Theorem 2.8 in [GSS] that, for any , the spanning time

is finite. Our main focus of attention is the maximal spanning time, the extremal quantity defined by

Theorem 2.8 in [GSS] shows that by providing a very large upper bound (essentially the product of lengths of all rows of , multiplied by their number). One of our results is a substantial improvement of that bound (see Theorem 1.1 below). Before we give further definitions and state our results, we give a brief review of related results in the literature.

Arguably, the topic that most closely resembles ours is the following classic problem on matrix powers. Let be the set of all primitive non-negative matrices. Then let be the smallest power that makes all elements of the power nonzero for every . See [HV] for the solution of this problem, and related results for analogous extremal quantities obtained by replacing by some natural subsets of .

Extremal quantities in growth models have been of substantial interest. Perhaps the most natural one is the smallest cardinality of a spanning set. It is a famous folk theorem that this quantity equals exactly for the bootstrap percolation on with . For bootstrap percolation on with , the smallest spanning sets have size [BBM]. Not much was known for thresholds in such lattice setting until a recent breakthrough [MN]. Smallest spanning sets have also been studied for bootstrap percolation on trees [Rie] and on hypergraphs [BBMR]. For Hamming graphs, [BBLN] determines the size of the smallest spanning sets for line growth, while [GSS] gives bounds for general neighborhood growth.

Maximal spanning time results are comparatively scarce. In [BP2], it is shown that the maximal spanning time for bootstrap percolation on with is (see also [BP1]). For bootstrap percolation on the hypercube , the maximal spanning time is when [Prz] and when [Har]. A related clique-completion process is studied from this perspective in [BPRS]. It appears that in general the maximal spanning time is considerably more complex than the smallest size of a spanning set. In our case, a major difficulty is non-monotonicity of : if , then but for all , thus it is not clear how and compare. As announced, our first result is an improved upper bound from [GSS] on the maximal spanning time.

Theorem 1.1.

For any and zero-set ,

We are able to obtain a better upper bound for a special class of initial sets, which will also provide a lower bound for .

A finite set is thin if, for every site , either or . That is, any point has no other points of either on the horizontal line or on the vertical line through . We define to be the set of all spanning thin sets, and let

Arguably, thin sets are the simplest general family of initial sets and are for this reason used in [GHPS, GSS] and in Section 3. In certain circumstances, thin set constructions are close to optimal [GSS]; in the present context, the extent to which and are comparable remains unclear (see open problem 3 in Section 7). The utility of thin sets in part comes from their connection to a simplified growth dynamics, which we now introduce.

The row and column enhancements and are two weakly decreasing sequences of non-negative integers which increase row and column counts by fixed values. To be precise, the enhanced neighborhood growth dynamics [GSS] is given by the triple , which defines a growth transformation as follows:

By default, we initialize the enhanced growth by the empty set, and we say that the pair of enhancements ( spans for if . We let be the set of all pairs of enhancements that have finite support and span for . Next, we introduce the enhanced spanning time

and finally define the corresponding maximal quantity

We next state a comparison result between and .

Theorem 1.2.

For all zero-sets ,

For a zero-set , we let to be the side of the largest square included in , that is, the integer such that but . The upper bound we obtain for and is linear in , and is in this sense the best possible.

Theorem 1.3.

For any zero-set ,

Moreover,

Finally, we give the lower bounds on the maximal spanning times. We do not know whether these are in any sense optimal (see open problem 2 in Section 7).

Theorem 1.4.

For any zero-set ,

and

The paper is organized as follows. In Section 2 we introduce additional notation and definitions, and prove preliminary results. In Section 3, we address special cases of neighborhood growth. We prove Theorem 1.2 in Section 4, Theorems 1.1 and 1.3 in Section 5, and Theorem 1.4 in Section 6. Finally we conclude with a selection of open questions in Section 7.

2 Preliminaries

2.1 Notation and Terminology

We define the partial order on as follows. For two sites and , if and only if and .

A Young diagram is then a set of sites such that and implies for all sites . Observe that any zero-set is a Young diagram.

For , we denote the respective projections of onto the -axis and -axis by and .

Consider a vector with weakly decreasing entries. The size of is . The support of is the smallest interval such that for . We will often write as a finite vector, omitting its zero coordinates.

We say a set is covered at time by either regular or enhanced growth dynamics if every site of is occupied at time by the respective dynamics.

A line in is either (also called a row) or (also called a column) for some .

The dynamics given by the growth transformation is sometimes called the regular dynamics when it needs to be distinguished from the enhanced version.

2.2 Operations with Young diagrams

Let be a Young diagram and . We define reductions of obtained by removing the leftmost columns or bottommost rows of ,

and the diagonal shift of ,

Given two Young diagrams , we define the infimal sum of and by

where . See Figure 2 for an example. For a Young diagram , define its closure and its height function so that . Then the terminology comes from the fact that , where is the infimal convolution (see for example Section 5 of [Roc]).

The following lemma in particular establishes that the set of Young diagrams equipped with the operation is a commutative monoid. We omit the routine proof.

Figure 2: An example of the infimal sum of two Young diagrams.
Lemma 2.1.

Let be Young diagrams. The infimal sum has the following properties:

  1. is a Young diagram.

  2. .

  3. .

  4. .

  5. .

  6. If , then .

Assume and are Young diagrams. Let consist of all Young diagrams such that . The infimal difference of and is defined as

Lemma 2.2.

Let be Young diagrams. The infimal difference has the following properties:

  1. .

  2. .

Proof.

Property (1) holds since an intersection of Young diagrams is a Young diagram and by Lemma 2.1(5). Property (2) holds since implies . ∎

2.3 Enhanced growth

Given a pair of enhancements , we form a pair of Young diagrams such that the row counts of are given by and the column counts of are given by . Therefore we use the two pairs interchangeably to describe an enhanced dynamics. The following lemma explains why enhanced growth is simpler than regular growth.

Lemma 2.3.

Let be enhancements that span for a zero-set . The set of occupied sites satisfies the following for all :

  1. The set is a Young diagram.

  2. The concave corners of must grow: if , then for any .

  3. The sites with identical enhancements become occupied simultaneously: if and , then if and only if .

Proof.

We prove Property (1) by induction. When , . Now suppose is a Young diagram for some . Assume , , and . As is a Young diagram, and . As and , and is a Young diagram, implies . Therefore is also a Young diagram.

To prove Property (2), suppose . By Property (1), for all and . Thus for all and , which contradicts the fact that spans for .

Let and . Then if and only if . Therefore if and only if . This proves property (3). ∎

Lemma 2.4.

Fix a zero-set and let be enhancements that span for . Let be maximal intervals of equal column lengths of , and similarly let be maximal intervals of equal row lengths of . Then

for all .

Proof.

We use induction, beginning with the trivial base case . The inductive step follows from Lemma 2.3 (2) and (3). ∎

Corollary 2.5.

Let have nonzero row counts and have nonzero column counts. If span for , then .

The next lemma provides the key connection between the infimal sum and enhanced growth.

Lemma 2.6.

Fix a zero-set . The enhancements span for if and only if

Proof.

The pair of enhancements does not span if and only if there exists so that and are both in . For this to happen, we must have . But and , thus , and so .

Conversely, if , there exist and , so that . Then , , and so . Thus no point outside of becomes occupied, and consequently does not span. ∎

2.4 Perturbations of zero-sets

Let be the first time that the regular dynamics given by covers a line in . Define

We omit the simple proof of the following lemma.

Lemma 2.7.

For any zero-set ,

As already remarked, is not apparently monotone with respect to inclusion. We do however have a weaker form of monotonicity which is the subject of the next lemma.

Lemma 2.8.

For any zero-set ,

Proof.

To prove the first inequality, by symmetry we only need to show . Assume . Let . If and are large enough, then and . Therefore .

For the second inequality, we again only need to show . Assume , where and . Let

be row enhancements with finite support . For are large enough, , and . Thus . ∎

The next lemma gives a converse inequality.

Lemma 2.9.

For any zero-set ,

Proof.

Assume that and , with respective number of rows and columns and , span for the zero-set . Then and span for and do so in at most steps. The enhanced dynamics given by and agree on the rectangle by Lemma 2.3. Therefore, the dynamics given by covers this rectangle by time , and then needs at most two additional steps to fully occupy . The inequality follows. ∎

2.5 Thin sets

As we will see in the next lemma, it is advantageous to permute rows and columns to arrange sites in a thin set in a certain manner reminiscent of convexity (see Figure 3). We formalize this arrangement next.

Figure 3: A thin set in the standard arrangement with , , and .

Two sets are equivalent if there are permutations of rows and columns of that map to . It is clear that the spanning times of two equivalent sets are the same.

An equivalence class of thin sets is given by finite (possibly empty) weakly decreasing vectors and with integer entries of at least , which specify the number of occupied sites in the rows and columns that contain at least two sites, and a number of isolated occupied sites. We now identify a specific representative of this equivalence class.

We say that a thin set is in the standard arrangement (see Figure 3) if the following hold:

  • The row counts , , and column counts , , are weakly decreasing.

  • If and , then and are on the same line.

To achieve the standard arrangement, let (resp., ) be the number of entries of (resp., ), and consider the rectangle . The sites in comprise, in order, the following diagonally adjacent intervals connecting the top left corner of this rectangle with its bottom right corner: vertical intervals of sites, , followed by single sites, and followed by horizontal intervals of sites, (see Figure 3). It is straightforward to see that the standard arrangement is unique.

Lemma 2.10.

Let be a thin set in the standard arrangement. Assume is a Young diagram. Let and . Then

  1. If , then for all , and .

  2. If and , then .

  3. The set is the union of a Young diagram and .

Moreover, is the union of a Young diagram and for all .

Proof.

To prove (1), it is, by symmetry, enough to prove the inequality for the row counts. Let and , where and . Let , , , and . As is a Young diagram, . As is in the standard arrangement, . Therefore,

This establishes (1), which immediately implies (2). Then (3) follows, as , where

is a Young diagram. Finally, the last claim follows by induction. ∎

3 Special cases

In this section we prove results on for two special cases of neighborhood growth. First is line growth, where is a single rectangle. The second is L-growth, where is a union of two rectangles: , where such that and . It turns out that the bounds for L-growth do not depend on the larger numbers and .

Lemma 3.1.

Let and let . If , then there exists at least one site such that . Similarly, if , then there exists at least one site such that .

Proof.

By symmetry we may assume that . Then and . There exists at least one site such that . Since , we must have that . This implies . ∎

Proposition 3.2.

If , then with , then

Proof.

We only address the case when , as the case is similar. Let be a spanning set for . By Lemma 3.1, there are at least covered rows and covered columns at time . At , every column containing at least one site that lies outside of the spanned rows and spanned columns becomes covered. As spans, there must be at least covered columns at . Therefore .

For the lower bound let be a thin set in the standard arrangement given by , , and . It is straightforward to check that spans in steps. ∎

Lemma 3.3.

Let and let . In every two time steps, at least one line is covered.

Proof.

Suppose . There exists such that . Suppose . Then either , or and .

If , then column is covered in . Otherwise, and , and . There exists such that . Then which implies that . Thus row is covered in . ∎

Proposition 3.4.

If , then

Proof.

Without loss of generality assume . By Lemma 3.3 at least one row or column is spanned in every two steps, and Lemma 2.7 gives us

and the upper bound follows by induction.

For the lower bound, we consider two cases. If , then by Lemma 2.8, . Otherwise . Let . Let be a thin set in the standard arrangement given by , , and , where . One can check that , if , and , otherwise. Therefore by Lemma 2.8, . ∎

4 Enhanced growth vs. growth from thin sets

In this section we prove Theorem 1.2.

Lemma 4.1.

For any zero-set ,

Proof.

Let be some enhancements that span for , given respectively by infinite vectors and of respective finite supports and . Form infinite vectors and .

Observe that the enhancements span for . In fact, the enhanced dynamics with zero-set and enhancements and the enhanced dynamics with zero-set and enhancements have the same occupied set at any time . It is important to note that when covers the rectangle , it in fact fully occupies .

Choose large enough so that the square satisfies and . For let and , and for let . Let be the set of occupied sites at time under the enhanced dynamics given by . By Lemma 2.3, for all and therefore the enhancements span for .

Define a thin set in the standard arrangement given by the row vector , the column vector , and . Observe that . Let be the set of occupied sites at time starting from under the regular dynamics given by .

We claim that for all . We use induction to prove this claim, which clearly holds at . Assume that the claim holds for some . Let . By Lemma 2.10, and again by Lemma 2.3, . Therefore if and only if which proves the induction step.

When the enhanced dynamics given by covers , the enhanced dynamics given by also covers , and thus fully occupies . Therefore, by the claim in the previous paragraph, , and then