# A Three-regime Theorem for flow-firing

Sarah Brauner, Galen Dorpalen-Barry, Selvi Kara, Caroline Klivans, Lisa
Schneider

###### Abstract.

Graphical chip-firing is a discrete dynamical system where chips are placed on
the vertices of a graph and exchanged via simple firing moves. Recent work has
sought to generalize chip-firing on graphs to higher dimensions, wherein
graphs are replaced by cellular complexes and chip firing becomes flow-
rerouting along the faces of the complex. Given such a system, it is natural
to ask (1) whether this firing process terminates and (2) if it terminates
uniquely (e.g. is confluent). In the graphical case, these questions were
definitively answered by Bjorner–Lovasz–Shor, who developed three regimes
which completely determine if a given system will terminate. Building on the
work of Duval–Klivans–Martin and Felzenszwalb-Klivans, we answer these
questions in a context called flow-firing, where the cellular complexes are
2-dimensional.

##  1\. Introduction

This paper concerns flow-firing, a higher-dimensional analogue of chip-firing.
In classical chip-firing chips are placed on the vertices of a graph and move
to other vertices via local moves dictated by the graph Laplacian [9]. Flow-
firing is a discrete model for cell complexes, where flow along one-
dimensional edges is diverted over two-dimensional faces as dictated by the
combinatorial Laplacian. Higher dimensional chip-firing was introduced in [2]
where the focus was on algebraic considerations. The study of the dynamical
properties of flow-firing was initiated in [3], and is continued here.
Consider the following two questions concerning the fundamental behavior of
termination and confluence for flow-firing:

$\bullet$ For which initial configurations does the flow-firing process
terminate?

$\bullet$ For which initial configurations does the flow-firing process
terminate uniquely?

In the classical case of chip-firing on a graph the answer to the first
question is provided by a theorem of Björner, Lovasz and Shor [1]. Their
result identifies three regimes for chip-firing behavior. Informally, the
theorem states that: (1) If the number of chips is small enough then the
process always terminates; (2) If the number of chips is large enough then the
process never terminates; and (3) If the number of chips is in a middle range,
then one can always find a configuration which terminates and one that does
not, see Theorem 2.5 for a precise statement.

In the graphical case, if a configuration terminates it always does so
uniquely, rendering the second question unnecessary. In flow-firing, however,
not all initial configurations terminate uniquely. A setting that terminates
uniquely was identified in [3]. Building from that work, our main result is a
generalization of the Three-Regime Theorem for uniqueness of termination for
flow-firing.

Denote by $\operatorname{{\sf K}}(n,r)$ the initial configuration known as the
_pulse_ of height $n$ and radius $r$ (see Section 2).

######  Theorem 1.1 (Three-Regime Theorem for Flow-Firing).

Let $r,n\in\operatorname{\mathbb{Z}}_{\geq 0}$.

  1. (1)

If $r\leq 1$, then firing from $\operatorname{{\sf K}}(n,r)$ terminates
uniquely in the Aztec diamond.

  2. (2)

If $2\leq r\leq\lceil\frac{n}{2}\rceil$, then firing from $\operatorname{{\sf
K}}(n,r)$ does not terminate uniquely, but can terminate in the Aztec diamond.

  3. (3)

If $r>\lceil\frac{n}{2}\rceil$, then $\operatorname{{\sf K}}(n,r)$ does not
terminate uniquely. Moreover, if $r\geq\lceil\frac{n}{\sqrt{3}}\rceil+1$, then
firing from $\operatorname{{\sf K}}(n,r)$ will never terminate in the Aztec
diamond.

Questions of termination and uniqueness have also been considered in other
variations of chip-firing. Two notable examples are labeled chip-firing [6, 7,
8] and root-system chip-firing [4, 5] where the emphasis is on local versus
global confluence properties. We prove confluence for subsystems of flow-
firing including certain path firings (Corollary 3.5).

The remainder of the paper proceeds as follows. Section 2 gives background on
flow-firing. We introduce the notion of firing along a path and study its
behavior in Section 3. In Sections 4, 5 and 6 we prove Theorem 1.1, addressing
each of the three regimes in turn. We then conclude in Section 6.3 by
discussing challenges to improving the bounds of Theorem 1.1.

### Acknowledgements

This project was initiated as part of the ICERM Research Community in
Algebraic Combinatorics in 2021/2022. We are thankful both to ICERM and the
organizers (Pamela E. Harris, Susanna Fishel, Rosa Orellana, and Stephanie van
Willigenburg) for their generous support. The first author was supported by an
NSF Graduate Research Fellowship.

##  2\. Background

Following [3], we consider flow-firing on the two dimensional grid complex. A
flow configuration is an assignment of integer values to the edges of the
complex. Visually, we depict flow as oriented, with positive values oriented
to the right and up, and negative values oriented to the left and down. For
example, the flow configuration on the left in Figure 1 has flow value zero on
all edges except the center edge which has a value of $2$ depicted as a
downward facing arrow and labeled with the magnitude $2$.

The degree of an edge is the number of two dimensional faces in which it is
contained. An edge can fire if it has at least as many units of flow as its
degree. In firing an edge, one unit of flow is diverted around each of the two
dimensional cells containing it; an example is shown in Figure 1. In our
visualizations, an edge with $1$ unit of flow will have a directed arrow but
we suppress the number $1$.

Figure 1. An example of flow-firing on the grid

The flow-firing process starts with a fixed initial flow configuration. From
all edges that can fire, one is selected and fired. This continues until no
firing moves are possible or continues forever. A flow configuration is stable
if no flow-firing moves are possible. Otherwise, a flow configuration is
called an intermediate configuration. From a fixed initial configuration, we
are interested in whether a stable configuration can be reached or not, and
whether that stable configuration is unique.

A flow configuration is conservative if for each vertex $v$, the flow into $v$
is equal to the flow out of $v$. Conservative flows are of particular interest
because of the following.

######  Theorem 2.1 ([3, Proposition 2, Theorem 4]).

Let $\operatorname{{\sf K}}$ be a flow configuration.

  * •

If there is a vertex $v$ with

$|{\sf inflow}(v)-{\sf outflow}(v)|>4,$

then flow-firing never terminates.

  * •

If $\operatorname{{\sf K}}$ is conservative, then flow-firing terminates after
a finite number of steps.

Moreover, conservative flow configurations are induced by face
representations. A face representation of a flow configuration is an
assignment of integer values to the $2$-cells of the complex. Visually, we
depict flow as oriented, with positive values oriented clockwise and negative
values oriented counter-clockwise, see below.

(a) Edge representation

(b) Face representation

Henceforth, we assume our configurations are conservative and can therefore be
analyzed using the face representation. Given such a configuration
$\operatorname{{\sf K}}$, we denote the weight of a face $f$ by
$\operatorname{{\sf K}}_{f}$. The support of a face configuration is the set
of faces $f$ whose weights are nonzero. In the grid, the distance between
faces $f$ and $g$, written $\operatorname{dist}(f,g)$ is the _Manhattan
distance_. For instance, $\operatorname{dist}(f,f)=0$, and if $f$ and $g$ are
neighbors, i.e. share an edge, then $\operatorname{dist}(f,g)=1$.

The flow-firing rules described earlier can be translated to firing rules on
faces. Let $\operatorname{{\sf K}}$ be a configuration and suppose $f,g$ are
faces of the grid with weights $w_{f}$ and $w_{g}$ respectively. If $w_{f}\geq
w_{g}+2$, then a face fire of $f$ towards $g$ results in a new configuration
$\operatorname{{\sf K}}^{\prime}$ with weights $w^{\prime}_{f}=w_{f}-1$ and
$w^{\prime}_{g}=w_{g}+1$.

######  Example 2.2.

Below are three face configurations. The second picture (b) is obtained from
the first picture (a) by firing the face with weight $2$ to the right. The
last picture (c) is obtained from (a) by by firing the face with weight $3$ to
the left.

$\ytableau{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>12\\\ \\\ \hskip
56.9055pt\ytableau{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>111\\\ \\\ \hskip
56.9055pt\ytableau{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>12\\\ \\\ $

(a) (b) (c)

We consider a modification of face-firing called face-firing with a marked
face. Fix a marked face $f_{0}$ and an integer $n$, the weight of $f_{0}$. The
marked face acts simultaneously as a source and a sink, while eschewing some
of the properties of both. We use the usual face-firing rules away from
$f_{0}$, but adapt the rules when firing into or out of $f_{0}$ in the
following way: if $f$ is adjacent to $f_{0}$ and has weight
$\operatorname{{\sf K}}_{f}<n$, we can fire from $f_{0}$ to $f$. If on the
other hand $\operatorname{{\sf K}}_{f}>n$, we can fire from $f$ to $f_{0}$. If
$\operatorname{{\sf K}}_{f}=n$, then we cannot fire between $f_{0}$ and $f$.
Notably, for any configuration $\operatorname{{\sf K}}$, the value
$\operatorname{{\sf K}}_{f_{0}}$ never changes as a result of firing. Example
2.3 illustrates face firing moves with a distinguished face.

######  Example 2.3.

Below are three configurations on the grid complex with a marked face
(shaded). To obtain the second configuration (b) from the first configuration
(a), we fire the marked face to the left. This firing is only allowed because
we are firing the marked face. In the second picture (b), the values of the
two faces involved are the same, so we cannot fire the marked face to the
left. The third picture (c) is obtained from the second configuration (b) by
firing the top middle box to the right.

$\ytableau{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>1*(teal!40)2\\\ \\\ \hskip
56.9055pt\ytableau{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>2*(teal!40)2\\\ \\\ \hskip
56.9055pt\ytableau{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>1*(teal!40)2\\\ \\\ $

(a) (b) (c)

The (total) weight of a configuration $\operatorname{{\sf K}}$ is the sum of
the non-zero face weights in $\operatorname{{\sf K}}$:

$\mathrm{weight}(\operatorname{{\sf K}}):=\sum_{f}\operatorname{{\sf K}}_{f}$

and the support radius of $\operatorname{{\sf K}}$ is
$\mathrm{len}(\operatorname{{\sf
K}}):=\max\\{\operatorname{dist}(f,f_{0}):\operatorname{{\sf K}}_{f}\neq
0\\}.$

There are two configurations that will play a significant role in our
analysis. The first configuration is the Aztec diamond, which is the
configuration whose faces $f$ have the following weights

$\operatorname{Az}(n)_{f}:=\text{max}\\{n-\operatorname{dist}(f_{0},f)+1,0\\}\quad\text{for
all faces }f\not=f_{0}\,,$

and $\operatorname{Az}(n)_{f_{0}}=n$.

The second configuration is the pulse of height $n$ and radius $r$, which we
denote by $\operatorname{{\sf K}}(n,r)$. In the edge representation,
$\operatorname{{\sf K}}(n,r)$ can be described as a simple closed curve of
radius $r$ around $f_{0}$, with $n$ units of flow oriented clockwise. In the
face representation, $\operatorname{{\sf K}}(n,r)$ is the configuration such
that $\operatorname{{\sf K}}(n,r)_{f}=n$ if $\operatorname{dist}(f_{0},f)\leq
r$ and $\operatorname{{\sf K}}(n,r)_{f}=0$, otherwise. Examples of
$\operatorname{{\sf K}}(4,2)$ and $\operatorname{Az}(2)$ are shown below. In
the edge representation, $\operatorname{{\sf K}}(4,2)$ is a closed curve with
$4$ units of flow along the curve and $\operatorname{Az}(2)$ has $1$ unit of
flow on every edge in the discrete $\ell_{1}$-ball centered at the
distinguished face.

. &
*(violet!40!white)4
*(violet!40!white) 4 *(violet!40!white) 4 *(violet!40!white) 4
*(violet!40!white)4 *(violet!40!white) 4 *(teal!40) 4 *(violet!40!white)4 *(violet!40!white)4
*(violet!40!white) 4 *(violet!40!white) 4 *(violet!40!white) 4
*(violet!40!white)4

. &
*(violet!40!white)1
*(violet!40!white) 1 *(violet!40!white) 2 *(violet!40!white) 1
*(violet!40!white)1 *(violet!40!white) 2 *(teal!40) 2 *(violet!40!white)2 *(violet!40!white)1
*(violet!40!white) 1 *(violet!40!white) 2 *(violet!40!white) 1
*(violet!40!white)1

Pulse $\operatorname{{\sf K}}(4,2)$ Aztec diamond $\operatorname{Az}(2)$

The significance of these two configurations originates with the following
theorem, which proved the first instance of confluent behavior in the context
of flow-firing.

######  Theorem 2.4 ([3, Theorem 9]).

Every sequence of flow-firing moves on $\operatorname{{\sf K}}(n,0)$
terminates in $\operatorname{Az}(n)$.

The three regimes theorem for flow-firing (Theorem 1.1) provides a fuller
description of the possible behavior of initial configurations around a marked
face, in particular showing that the pulse is essentially the only initial
configuration that gives a confluent system.

For context and comparison, we give the precise statements of the graphical
theorem before proceeding with the proof of the flow-firing theorem.

######  Theorem 2.5 ([1, Theorem 3.3]: Three Regimes for Graphical Chip-
Firing).

Let $G$ be a finite connected graph with $n$ vertices and $m$ edges. Let
$\bf{c}$ be a configuration with $N$ chips in total. Then

  1. (1)

If $N<m$ then the chip-firing process terminates after finitely-many firing.

  2. (2)

If $m\leq N\leq 2m-n$ then there exists an initial configuration which
stabilizes and also one which does not.

  3. (3)

If $N>2m-n$ then the chip-firing process never stabilizes.

##  3\. Path-Firing

We will first restrict our attention to a local flow-firing process on paths,
called path-firing. After developing properties of path-firing, we will be
ready to prove Theorem 1.1 beginning in Section 4.

A path $\operatorname{{\mathcal{P}}}$ is a collection of successive faces
$\operatorname{{\mathcal{P}}}=(f_{1},\dots,f_{k})$ such that
$\operatorname{dist}(f_{i-1},f_{i})=1$ for $2\leq i\leq k$. A flow-firing
process involving only the faces of $\operatorname{{\mathcal{P}}}$ is called
path-firing. We say that one path-fires along a path
$\operatorname{{\mathcal{P}}}$. Given a path $\operatorname{{\mathcal{P}}}$, a
configuration $\operatorname{{\sf R}}$ is an assignments of weights to the
faces of $\operatorname{{\mathcal{P}}}$. We use $\operatorname{{\sf R}}_{i}$
to denote the weight of $f_{i}$ in $\operatorname{{\sf R}}$. A configuration
$\operatorname{{\sf R}}$ on $\operatorname{{\mathcal{P}}}$ is called a path
stable configuration if there are no possible path fires on
$\operatorname{{\sf R}}$. As in the case of configurations on the grid, we
will define the (total path) weight of $\operatorname{{\sf R}}$ to be

$\mathrm{weight}(\operatorname{{\sf
R}})=\sum_{f_{i}\in\operatorname{{\mathcal{P}}}}\operatorname{{\sf R}}_{i}.$

It will be clear from context whether we are interested in the weight of a
path or total configuration. Define the length of a path configuration
$\operatorname{{\sf R}}$ to be the

$\mathrm{len}(\operatorname{{\sf
R}}):=\max_{f_{i}\in\operatorname{{\mathcal{P}}}}\
\\{\operatorname{dist}(f_{1},f_{i}):\operatorname{{\sf R}}_{i}\neq 0\\}.$

A path is called standard if each $f_{i}$ has value
$\max\\{0,n-\operatorname{dist}(f_{0},f_{i})+1\\}$. In other words, a standard
path is one in which weight of $f$ matches $\operatorname{Az}(n)_{f}$.
Importantly, if no faces in $\operatorname{{\mathcal{P}}}$ violate
$\operatorname{Az}(n)$, then we will show in Lemma 4.1 there is always a way
to fire along $\operatorname{{\mathcal{P}}}$ to obtain a standard path.

In what follows, we will first constrain the type of configurations possible
when path firing along an arbitrary path $\operatorname{{\mathcal{P}}}$ in
§3.1; then in §3.2 we will analyze a particular class of paths and
configurations, which will be integral to the proof of the second and third
regimes in Theorem 1.1.

###  3.1. Constraints from firing rules

The following observation will play an important role in our analysis.

######  Observation 3.1.

Suppose one begins with a path configuration $\operatorname{{\sf R}}$ on
$\operatorname{{\mathcal{P}}}$ such that the face weights of
$\operatorname{{\sf R}}$ form a weakly decreasing sequence $\operatorname{{\sf
R}}_{1}\geq\dots\geq\operatorname{{\sf R}}_{k}$. Then the face weights of any
configuration $\operatorname{{\sf R}}^{\prime}$ obtained by path-firing on
$\operatorname{{\sf R}}$ must also form a weakly decreasing sequence.

All of the path configurations we analyze to prove Theorem 1.1 will have face
weights that form a weakly decreasing sequence. Note that by Observation 3.1,
this means that firing from face $f_{i}$ can only be directed towards face
$f_{i+1}$—in other words, firing can only happen in one direction along the
path. The next two lemmas put limitations on what types of weakly decreasing
sequences can appear.

######  Lemma 3.2.

Suppose the face weights of $\operatorname{{\sf R}}$ form a weakly decreasing
sequence. Then after any sequence of path fires on $\operatorname{{\sf R}}$
along $\operatorname{{\mathcal{P}}}$, it is impossible to obtain:

  1. (1)

A configuration $\operatorname{{\sf R}}^{\prime}$ of the form

$\ldots$$\operatorname{{\sf R}}^{\prime}_{i-1}$$\operatorname{{\sf
R}}^{\prime}_{i}=a$$\operatorname{{\sf
R}}^{\prime}_{i+1}=a$$\operatorname{{\sf
R}}^{\prime}_{i+2}=a$$\operatorname{{\sf R}}^{\prime}_{i+3}$$\ldots$

unless the faces $f_{i},f_{i+1}$ and $f_{i+2}$ never fire; or

  2. (2)

A configuration $\operatorname{{\sf R}}^{\prime}$ of the form

$\cdots$$\operatorname{{\sf R}}^{\prime}_{i-2}$$\operatorname{{\sf
R}}^{\prime}_{i-1}=a$$\operatorname{{\sf
R}}^{\prime}_{i}=a$$\operatorname{{\sf
R}}^{\prime}_{i+1}=a-1$$\cdots$$a-t-1$$\operatorname{{\sf
R}}^{\prime}_{i+t}=a-t$$\operatorname{{\sf
R}}^{\prime}_{i+t+1}=a-t$$\operatorname{{\sf R}}^{\prime}_{i+t+2}$$\cdots$

where the weights of faces between $\operatorname{{\sf R}}^{\prime}_{i}$ and
$\operatorname{{\sf R}}^{\prime}_{i+t}$ decrease by exactly $1$ from left to
right.

###### Proof of Lemma 3.2 (1).

Suppose there exists a configuration $\operatorname{{\sf R}}^{\prime}$
obtained from $\operatorname{{\sf R}}$ via path-firing on
$\operatorname{{\mathcal{P}}}$ of the form

$\cdots$$\operatorname{{\sf
R}}^{\prime}_{i-1}$$i-1$$a$$i$$a$$i+1$$a$$i+2$$\operatorname{{\sf
R}}^{\prime}_{i+3}$$i+3$$\cdots$

where $n\geq\operatorname{{\sf R}}^{\prime}_{i-1}>a>\operatorname{{\sf
R}}^{\prime}_{i+3}\geq 0.$

Assume without loss of generality that $\operatorname{{\sf R}}^{\prime}$ is
the first configuration where $\operatorname{{\sf
R}}^{\prime}_{i}=\operatorname{{\sf R}}^{\prime}_{i+1}=\operatorname{{\sf
R}}^{\prime}_{i+2}$, and let $\operatorname{{\sf R}}^{\prime\prime}$ be the
configuration immediately preceding $\operatorname{{\sf R}}^{\prime}$ in the
sequence of path fires. We will consider the firing moves on
$\operatorname{{\sf R}}^{\prime\prime}$ which would be necessary to create
this triple in $\operatorname{{\sf R}}^{\prime}$.

Note that in order to have $\operatorname{{\sf
R}}^{\prime}_{i}=\operatorname{{\sf R}}^{\prime}_{i+1}=\operatorname{{\sf
R}}^{\prime}_{i+2}$, either the faces $f_{i-1}$, $f_{i}$, $f_{i+1}$, or
$f_{i+2}$ would need to be fired in $\operatorname{{\sf R}}^{\prime\prime}$.

If firing $f_{i+2}$ had resulted in $\operatorname{{\sf R}}^{\prime},$ then
$\operatorname{{\sf R}}^{\prime\prime}_{i+1}=a$ and $\operatorname{{\sf
R}}^{\prime\prime}_{i+2}=a+1$, a contradiction. In particular, firing
$f_{i+2}$ on $\operatorname{{\sf R}}^{\prime\prime}$ would have given a
configuration of the following form:

$\cdots$$\operatorname{{\sf
R}}^{\prime}_{i-1}$$i-1$$a$$i$$a$$i+1$$a+1$$i+2$$c$$i+3$$\cdots$

Similarly, if firing $f_{i+1}$ resulted in $\operatorname{{\sf R}}^{\prime}$,
then we would have $\operatorname{{\sf R}}^{\prime\prime}_{i}=a$ and
$\operatorname{{\sf R}}^{\prime\prime}_{i+1}=a+1$, again a contradiction. We
obtain an analogous contradiction if $f_{i-1}$ or $f_{i}$ were the last faces
to fire. Hence the configuration $\operatorname{{\sf R}}^{\prime}$ is
impossible to obtain via path fires unless the faces $f_{i-1}$, $f_{i}$,
$f_{i+1}$, and $f_{i+2}$ were never fired. ∎

###### Proof of Lemma 3.2 (2).

The second part of Lemma 3.2 naturally extends the proof of $(1)$. Suppose we
obtain a configuration $\operatorname{{\sf R}}^{\prime}$ from
$\operatorname{{\sf R}}$ violating the statement of Lemma 3.2 (2). Again,
assume without loss of generality that $\operatorname{{\sf R}}^{\prime}$ is
the first such configuration. As before, we consider the last configuration
$\operatorname{{\sf R}}^{\prime\prime}$ that produced $\operatorname{{\sf
R}}^{\prime}$.

By an analogous argument to that made in the proof of Lemma 3.2 $(1)$, the
last firing move made to go from $\operatorname{{\sf R}}^{\prime\prime}$ to
$\operatorname{{\sf R}}^{\prime}$ could not have been from the faces $f_{i-2}$
and $f_{i+t-1}$. Likewise, faces $f_{i}$ and $f_{i+t+1}$ could not have fired
last. Thus the only options for the firing face are $f_{i-1},f_{i+t}$, and a
face between $f_{i}$ and $f_{i+t-1}$. To handle these settings, we proceed by
induction on $t$, which is the (minimum) distance between the repeated
weights.

Base Case: Let $t=1$. Then $\operatorname{{\sf R}}^{\prime}$ is of the form

$\cdots$$\operatorname{{\sf R}}^{\prime}_{i-1}=a$$i-1$$\operatorname{{\sf
R}}^{\prime}_{i}=a$$i$$\operatorname{{\sf
R}}^{\prime}_{i+1}=a-1$$i+1$$\operatorname{{\sf
R}}^{\prime}_{i+2}=a-1$$i+2$$\cdots$

From our previous discussion, the last face to fire is either $f_{i-1}$ or
$f_{i+1}$. In both cases, this would mean that $\operatorname{{\sf
R}}^{\prime\prime}$ had a sequence of three consecutive faces with the same
value, which is impossible by Lemma 3.2 (1).

Inductive Step: Now assume that the lemma holds for all intervals smaller than
$t$. Again, we consider the last face fired to move from $\operatorname{{\sf
R}}^{\prime\prime}$ to $\operatorname{{\sf R}}^{\prime}$. There are three
cases.

_Case I: The last face to fire was $f_{i-1}$._ Then $\operatorname{{\sf
R}}^{\prime\prime}$ would be of the form:

$\cdots$$\operatorname{{\sf
R}}^{\prime\prime}_{i-1}=a+1$$i-1$$\operatorname{{\sf
R}}^{\prime\prime}_{i}=a-1$$i$$\operatorname{{\sf
R}}^{\prime\prime}_{i+1}=a-1$ $i+1$$\cdots$$\cdots$$\operatorname{{\sf
R}}^{\prime\prime}_{i+t-1}=a-t+1$ $i+t-1$$\operatorname{{\sf
R}}^{\prime\prime}_{i+t}=a-t$ $i+t$$\operatorname{{\sf
R}}^{\prime\prime}_{i+t+1}=a-t$ $i+t+1$$\cdots$

This pattern violates the inductive hypothesis, since the distance between the
repeated weights $f_{i+1}$ and $f_{i+t}$ is strictly less than $t$ and values
between repeated weights of faces decrease exactly by 1.

_Case II: The last face to fire is $f_{i+t}$._ By a similar argument,
$\operatorname{{\sf R}}^{\prime\prime}$ would be of the form:

$\cdots$$\operatorname{{\sf
R}}^{\prime\prime}_{i-1}=a$$i-1$$\operatorname{{\sf
R}}^{\prime\prime}_{i}=a$$i$$\operatorname{{\sf R}}^{\prime\prime}_{i+1}=a-1$
$i+1$$\cdots$$\cdots$$\operatorname{{\sf R}}^{\prime\prime}_{i+t-1}=a-t+1$
$i+t-1$$\operatorname{{\sf R}}^{\prime\prime}_{i+t}=a-t+1$
$i+t$$\operatorname{{\sf R}}^{\prime\prime}_{i+t+1}=a-t$ $i+t+1$$\cdots$

Note that $i-(i-t+1)=t-1$, which means the distance between the repeated
weights are less than $t$, a contradiction.

_Case III: The last face to fire is $f_{i+m}$ for $1<m<t-1$._ In this case,
$\operatorname{{\sf R}}^{\prime\prime}$ would have had a repeated weight less
than $t$ away from the first pair of repeated weights $f_{i-1}$ and $f_{i}$,
shown below. This violates our inductive hypothesis.

$\cdots$$\operatorname{{\sf
R}}^{\prime\prime}_{i-1}=a$$i-1$$\operatorname{{\sf
R}}^{\prime\prime}_{i}=a$$i$$\operatorname{{\sf R}}^{\prime\prime}_{i+1}=a-1$
$i+1$$\cdots$$\cdots$$\operatorname{{\sf R}}^{\prime\prime}_{i+m-1}=a-m+1$
$i+m-1$$\operatorname{{\sf R}}^{\prime\prime}_{i+m}=a-m+1$
$i+m$$\operatorname{{\sf
R}}^{\prime\prime}_{i+m+1}=a-m-2$$i+m+1$$\cdots$$\cdots$

∎

###  3.2. Constraints from counting

Lemma 3.2 significantly limits the types of configurations that can arise via
firing along any path configuration $\operatorname{{\sf R}}$ whose weights are
weakly decreasing. We will now use counting arguments to further narrow the
types of configurations that arise when firing along certain path
configurations.

Our ultimate goal is to understand final configurations obtained from flow-
firing on $\operatorname{{\sf K}}(n,r)$. To do so, we will first analyze path-
firing on a single row of $\operatorname{{\sf K}}(n,r)$. For any
$\ell\in\operatorname{\mathbb{Z}}_{\geq 0}$, define

$\operatorname{{\mathcal{P}}}(\ell):=(f_{1}\ldots,f_{n-\lceil\frac{n}{2}\rceil+\ell}),$

and a configuration $\operatorname{{\sf R}}(\ell)$ on
$\operatorname{{\mathcal{P}}}(\ell)$:

$\operatorname{{\sf
R}}(\ell):=\underbrace{(n,n,\cdots,n}_{\ell~{}\textrm{times}},\underbrace{0,0,\cdots,0)}_{n-\lceil\frac{n}{2}\rceil~{}\textrm{times}}.$

The remainder of this section is devoted to considering how one can path fire
from $\operatorname{{\sf R}}(\ell)$ along
$\operatorname{{\mathcal{P}}}(\ell)$.

Since $\operatorname{{\mathcal{P}}}(\ell)$ does _not_ include the marked face
$f_{0}$, the number of chips will stay constant while firing along
$\operatorname{{\mathcal{P}}}(\ell)$. Thus,
$\mathrm{weight}(\operatorname{{\sf
R}}(\ell))=\mathrm{weight}(\operatorname{{\sf R}})$ where $\operatorname{{\sf
R}}$ is any configuration we obtain by path firing on $\operatorname{{\sf
R}}(\ell)$.

######  Observation 3.3.

Let $\operatorname{{\sf R}}^{\prime}$ be a path stable configuration obtained
from $\operatorname{{\sf R}}(\ell)$. Let $S_{\ell}$ be the subset of all faces
$\\{f_{1},\ldots,f_{n-\lceil\frac{n}{2}\rceil+\ell}\\}$ that have been fired
in $\operatorname{{\mathcal{P}}}(\ell)$ to reach $\operatorname{{\sf
R}}^{\prime}$ from $\operatorname{{\sf R}}(\ell)$. It follows from Observation
3.1 and Lemma 3.2 that the weights of faces $f\in S_{\ell}$ in
$\operatorname{{\sf R}}^{\prime}$ are weakly decreasing along
$\operatorname{{\mathcal{P}}}(\ell)$ and the weights of adjacent faces in $S$
differ by exactly one in all but at most one position.

Note that the weight of face $f_{i}$ is $n$ if and only if $f_{i}$ never
fires. When considering face weights with repeated values, it will be simpler
to ignore the faces that have never fired. We will say that a configuration
$\operatorname{{\sf R}}$ has a repeated weight if there is a pair of faces
$f_{i},f_{i+1}\in S_{\ell}$ that have the same face weights. (In other words,
we will not count the faces that have never fired as having repeated weights.)

Next, we use Lemma 3.2 and Observation 3.3 to determine which faces of
$\operatorname{{\mathcal{P}}}(\ell)$ must fire in order to obtain a path
stable configuration from $\operatorname{{\sf R}}(\ell)$.

######  Lemma 3.4.

Write $\ell=\lfloor\frac{n}{2}\rfloor+k$ for $k\in\operatorname{\mathbb{Z}}$
and $\ell\in\operatorname{\mathbb{Z}}_{>0}$. Let $\operatorname{{\sf
R}}^{\prime}$ be a path stable configuration obtained from $\operatorname{{\sf
R}}(\ell)$, and $S_{\ell}$ be the set of faces in
$\operatorname{{\mathcal{P}}}(\ell)$ that fire to obtain $\operatorname{{\sf
R}}^{\prime}$. Let $r=\mathrm{len}(\operatorname{{\sf R}}^{\prime})$. We have
the following description of $S_{\ell}$:

  * •

If $k\leq 0$, then $S_{\ell}=\\{f_{1},\cdots,f_{r-1}\\}$.

  * •

If $k>0$ then $S_{\ell}=\\{f_{k+1},f_{k+2},\cdots,f_{r-1}\\}$.

###### Proof.

Let $\operatorname{{\sf R}}^{\prime}$ be a path stable configuration obtained
from path-firing on $\operatorname{{\sf R}}(\ell)$ hence

$\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})=\mathrm{weight}(\operatorname{{\sf R}}(\ell))=n\cdot\ell.$

Let $0\leq t<\ell$ be the largest index such that $f_{t}\not\in S_{\ell}$. If
$t=0$, then $S_{\ell}=\\{f_{1},\cdots,f_{r-1}\\}$. If $t>0,$ then
$\operatorname{{\sf R}}^{\prime}_{t}=n$ and $\operatorname{{\sf
R}}^{\prime}_{t+1}=\operatorname{{\sf R}}^{\prime}_{t}-1=n-1$.

We will first consider $k>0$. To see why the statement holds, note that

$\mathrm{weight}(\operatorname{{\sf R}}^{\prime})=n\cdot\ell=n\cdot\lfloor
n/2\rfloor+k\cdot n.$

By Lemma 3.2 and Observation 3.3, the faces in $S_{\ell}$ will have weights in
$\operatorname{{\sf R}}^{\prime}$ of the form $n-m,n-m-1,n-m-2,\cdots,1$ for
$1\leq m<n$, with at most one weight occurring twice, which we shall write as
$0\leq n-p<n$. If there is no repeated face weight, set $p=n$; this is a
slight abuse of notation because $f\not\in S_{\ell}$ if $f$ has weight $0$,
but it will not change the results and streamlines the two cases (e.g.
presence or absence of a repeated weight).

Thus the total weight of the faces in $S_{\ell}$ is

$\frac{(n-m)(n-m+1)}{2}+n-p.$

Since $k>0$, for any choices of $0\leq(n-p)<n$ and $1\leq m<n$, we have

$\mathrm{weight}(\operatorname{{\sf R}}^{\prime})=n\cdot\lfloor
n/2\rfloor+k\cdot n>\frac{(n-m)(n-m+1)}{2}+n-p,$

and thus we cannot have $S_{\ell}=\\{f_{1},\cdots,f_{r}\\}$. This implies that
there is at least one face $f_{t}\not\in S_{\ell}$ with face value $n$, and so
$f_{t+1}=n-1$, and we can write $\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})$ as

(1) $\mathrm{weight}(\operatorname{{\sf R}}^{\prime})=n\cdot\ell=n\cdot
t+\frac{(n-1)n}{2}+n-p,$

where $0\leq n-p<n$ is the repeated weight. Using the fact that
$t,\ell,p\in\operatorname{\mathbb{Z}}_{\geq 0}$, and that
$\ell=\lfloor\frac{n}{2}\rfloor+k$, we solve for $t$ and obtain $t=k$ when
$k>0$.

When $k\leq 0$, we proceed in a similar manner, except that now it suffices to
show $t=0$. The faces in $S_{\ell}$ must have weights $n-1,n-2,\cdots,1$ and

$\mathrm{weight}(\operatorname{{\sf R}}^{\prime})=t\cdot
n+\frac{(n-1)n}{2}+(n-p)=n\cdot\lfloor n/2\rfloor+kn.$

Solving for $t$ shows that it must be 0. ∎

In general flow-firing processes are highly non-confluent, i.e. order of
firing moves matter and effects the terminal configurations. One surprising
outcome of our analysis is that path firing does lead to confluent behavior.
Using Lemma 3.4 and the fact that path firing on $\operatorname{{\sf
R}}(\ell)$ along $\operatorname{{\mathcal{P}}}(\ell)$ does not change the
weight of the configuration, we obtain a setting where flow-firing is
confluent.

######  Theorem 3.5.

For any $\ell\geq 0$, path firing on $\operatorname{{\sf R}}(\ell)$ terminates
uniquely.

###### Proof.

Let $\operatorname{{\sf R}}^{\prime}$ be a the path-stable configuration
obtained from $\operatorname{{\sf R}}(\ell)$. We claim that by Lemma 3.2,
$\operatorname{{\sf R}}^{\prime}$ is uniquely determined by the set $S_{\ell}$
given in Lemma 3.4 and the fact that $\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})=n\cdot\ell$. First, if $f_{i}\not\in S_{\ell}$, then
$\operatorname{{\sf R}}^{\prime}_{i}=n$; otherwise, for $f_{j}\in S_{\ell}$,
we have $\operatorname{{\sf R}}^{\prime}_{j}<n$. Moreover, adjacent faces in
$S_{\ell}$ have weights that decrease by exactly one except at the repeated
weight face pair (if it exists). Thus by Lemma 3.4,
$\mathrm{weight}(\operatorname{{\sf R}}^{\prime})$ can be written uniquely as

$\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})=n\cdot\ell=(\ell-|S_{\ell}|)\cdot n+\frac{(n-m)(n-m+1)}{2}+n-p,$

where $n-p$ is the repeated face weight (again, taking $n=p$ if there is no
repeat), and $m=1$ if $\ell-|S_{\ell}|>0$. For fixed $n,\ell$, these values in
turn uniquely determine the face weights of $\operatorname{{\sf R}}^{\prime}$.
∎

Having established that path-firing $\operatorname{{\sf R}}(\ell)$ yields a
unique path-stable configuration $\operatorname{{\sf R}}^{\prime}$, we next
bound the support $\mathrm{len}(\operatorname{{\sf R}}^{\prime})$ of
$\operatorname{{\sf R}}^{\prime}$.

######  Proposition 3.6.

Let $\ell\leq\lceil\frac{n}{2}\rceil$ and $\operatorname{{\sf R}}^{\prime}$ be
the unique path-stable configuration obtained from path-firing on
$\operatorname{{\sf R}}(\ell)$.

  * •

If $\operatorname{{\sf R}}^{\prime}$ has no repeated weights, then
$\mathrm{len}(\operatorname{{\sf R}}^{\prime})=n-m$ for
$m\geq\lceil\frac{n}{2}\rceil-\ell$.

  * •

If $\operatorname{{\sf R}}^{\prime}$ has a single repeated weight, then
$\mathrm{len}(\operatorname{{\sf R}}^{\prime})=n-m+1$ for
$m\geq\lceil\frac{n}{2}\rceil-\ell+1$.

###### Proof.

As in the previous proof, the main idea is to examine the equality
$\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})=\mathrm{weight}(\operatorname{{\sf R}}(\ell))=n\cdot\ell$ for
each case.

  1. (1)

Case 1: $n$ is odd and $\ell=\frac{n+1}{2}$. In this case, by Lemma 3.4,
$f_{1}\not\in S_{\ell}$ (e.g. it never fires), and so we have

$\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})=n\cdot\frac{n+1}{2}=n+\frac{(n-1)n}{2}.$

Hence $\operatorname{{\sf R}}^{\prime}$ has no repeated face and
$\mathrm{len}(\operatorname{{\sf R}}^{\prime})=n$.

  2. (2)

Case 2: $\ell\leq\lfloor n/2\rfloor$ and $\operatorname{{\sf R}}^{\prime}$ has
no repeated weight.

$\mathrm{weight}(R^{\prime})=n\cdot\ell=\frac{(n-m)(n-m+1)}{2}$

for some $m\leq n$. Hence $\mathrm{len}(\operatorname{{\sf R}}^{\prime})=n-m$.
We will show that $m\geq\lceil\frac{n}{2}\rceil-\ell$.

Since $m\geq 0$,

$m^{2}-m=2n(m+\ell)-n^{2}-n\geq 0.$

As a result,

$2n(m+\ell)\geq n(n+1)\implies
m+\ell\geq\frac{n+1}{2}\geq\Big{\lceil}\frac{n}{2}\Big{\rceil}.$

  3. (3)

Case 3: $\ell\leq\lfloor n/2\rfloor$ and $\operatorname{{\sf R}}^{\prime}$ has
a repeated weight $0<n-p\leq n-m$. Thus $\mathrm{len}(\operatorname{{\sf
R}}^{\prime})=n-m+1$. As before,

$\mathrm{weight}(\operatorname{{\sf
R}}^{\prime})=n\cdot\ell=\frac{(n-m)(n-m+1)}{2}+n-p$

and by a similar argument, we solve for $m$ to show that
$m\geq\lceil\frac{n}{2}\rceil-\ell+1$. ∎

##  4\. The first regime

We are now ready to prove Theorem 1.1.

######  Theorem 1.1.

Let $r,n\in\operatorname{\mathbb{Z}}_{\geq 0}$.

  1. (1)

If $r\leq 1$, then firing from $\operatorname{{\sf K}}(n,r)$ terminates
uniquely in the Aztec diamond $\operatorname{Az}(n)$.

  2. (2)

If $2\leq r\leq\lceil\frac{n}{2}\rceil$, then firing from $\operatorname{{\sf
K}}(n,r)$ does not terminate uniquely, but can terminate in the Aztec diamond
$\operatorname{Az}(n)$.

  3. (3)

If $r>\lceil\frac{n}{2}\rceil$, then $\operatorname{{\sf K}}(n,r)$ does
terminate uniquely. Moreover, if $r\geq\lceil\frac{n}{\sqrt{3}}\rceil+1$, then
firing from $\operatorname{{\sf K}}(n,r)$ will never terminate in the Aztec
diamond $\operatorname{Az}(n)$.

Our approach to showing a configuration _can_ terminate in
$\operatorname{Az}(n)$ (e.g. Regimes 1 and 2) relies on finding an
intermediate configuration of $\operatorname{{\sf K}}(n,r)$ that does not
“violate” the Aztec diamond. A configuration $\operatorname{{\sf K}}$ violates
the Aztec diamond $\operatorname{Az}(n)$ if some face $f$ has
$\operatorname{{\sf K}}_{f}>\operatorname{Az}(n)_{f}$. After finding such a
configuration, Lemma 4.1 allows us to conclude that the initial configuration
$\operatorname{{\sf K}}(n,r)$ _can_ terminate in $\operatorname{Az}(n)$.

######  Lemma 4.1.

Let $\operatorname{{\sf K}}$ be a configuration with $\operatorname{{\sf
K}}_{f_{0}}=n$. If $\operatorname{{\sf K}}_{f}\leq\operatorname{Az}(n)_{f}$
for all faces $f$, then there is a sequence of firing moves sending
$\operatorname{{\sf K}}$ to $\operatorname{Az}(n)$.

###### Proof.

The core of this proof comes from two observations.

  1. (1)

Let $f,g$ be adjacent faces with $g$ strictly further from $f_{0}$ than $g$,
so that $\operatorname{dist}(f_{0},f)+1=\operatorname{dist}(f_{0},g)$. If
$K_{f}=\operatorname{Az}(n)_{f}$ and $K_{g}<\operatorname{Az}(n)_{g}$, then
$K_{g}\leq K_{f}-2$ and hence we can fire from $f$ to $g$.

  2. (2)

If $g$ is adjacent to $f_{0}$ and $K_{g}<\operatorname{Az}(n)_{g}$, then
$K_{g}\leq n-1$ and we can fire from $f_{0}$ to $g$.

Let $g$ be a face such that $K_{g}<\operatorname{Az}(n)_{g}$ but all faces
strictly closer to $f_{0}$ satisfy $K_{f}=\operatorname{Az}(n)_{f}$.

If $g$ is adjacent to $f_{0}$, then by (2) above, we can fire from $f_{0}$ to
$g$. If $g$ is not adjacent to $f_{0}$, then there is some face $f$ adjacent
to $g$ and strictly closer to $f_{0}$; by (1) we can fire from $f$ to $g$.
Repeat this process until $\operatorname{{\sf
K}}_{g}\leq\operatorname{Az}(n)_{g}$ for all faces $g$. ∎

We use this, along with arguments made in [3], to prove part (1) of Theorem
1.1.

######  Proposition 4.2.

If $r\leq 1$, then $\operatorname{{\sf K}}(n,r)$ terminates uniquely in the
Aztec diamond.

###### Proof.

The $r=0$ case was proven in [3, Theorem 9]. If $r=1$, then
$\operatorname{{\sf K}}(n,1)$ is an intermediate configuration obtained from
$\operatorname{{\sf K}}(n,0)$ and it terminates uniquely at
$\operatorname{Az}(n)$ by [3, Theorem 9]. ∎

######  Example 4.3.

Consider $\operatorname{{\sf K}}(3,r)$. If $r=0$, then $\operatorname{{\sf
K}}(n,0)$ is the pulse and uniquely terminates at $\operatorname{Az}(n)$. If
$r=1$, observe that $\operatorname{{\sf K}}(3,1)$ terminates uniquely at
$\operatorname{Az}(3)$ (shown on the right below).

. & *(black!20!white)
*(black!20!white) *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(violet!40!white)3 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(violet!40!white)3 *(teal!40) 3 *(violet!40!white)3 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(violet!40!white)3 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(black!20!white)
*(black!20!white)
. & *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(teal!40) 3 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1

We will see in the subsequent section that although there are many more
configurations that _can_ terminate in the Aztec diamond, in general this
termination will not be unique.

##  5\. The Second Regime

We now turn to the second regime, which requires the most analysis. The main
goal of this section is to prove part (2) of Theorem 1.1, namely the following
theorem.

######  Theorem 5.1.

If $2\leq r\leq\lceil\frac{n}{2}\rceil$, flow firing on $\operatorname{{\sf
K}}(n,r)$ does not terminate uniquely. In particular, there exists a sequence
of firings that terminate in $\operatorname{Az}(n)$ and a sequence of firings
that terminate in a configuration that is not $\operatorname{Az}(n)$.

We illustrate the subtlety of this regime below in Example 5.2.

######  Example 5.2.

Below draw three configurations. The first (a) is $\operatorname{{\sf
K}}(3,2)$. The second two are different stable configurations that can be
obtained from (a) by different sets of firing moves. The second configuration
(b) does not violate the Aztec diamond $\operatorname{Az}(3)$, but the third
configuration (c) does violate $\operatorname{Az}(3)$.

. &
3
3 3 3
3 3 *(teal!40) 3 3 3
3 3 3
3

. & *(black!20!white) 1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(teal!40) 3 *(black!20!white)3 *(black!20!white) 2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white) 2 *(black!20!white)1
*(black!20!white)1
. & 1 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(teal!40) 3 *(black!20!white)3 *(black!20!white) 2 *(black!20!white)1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1
*(black!20!white)1 *(black!20!white) 2 *(black!20!white)1
*(black!20!white) 1

(a) (b) (c)

Theorem 1.1 will be proved in two parts. In Section 5.1 (Proposition 5.6), we
show that firing on $\operatorname{{\sf K}}(n,r)$ can terminate in a
configuration that violates the Aztec diamond; this proves that
$\operatorname{{\sf K}}(n,r)$ need not terminate in $\operatorname{Az}(n)$.
Then in Section 5.2 (Proposition 5.10), we construct an explicit sequence of
face fires on $\operatorname{{\sf K}}(n,r)$ which terminate in the Aztec
diamond, thereby showing that $\operatorname{{\sf K}}(n,r)$ _can_ terminate in
$\operatorname{Az}(n)$.

######  Remark 5.3.

In what follows, we state our results for Regime 2 in terms of
$\operatorname{{\sf K}}(n,r)$. However, Theorems 5.6 and 5.10 hold more
generally for a closed curve with weight $n$ strictly containing
$\operatorname{{\sf K}}(n,1)$ and contained within $\operatorname{{\sf
K}}(n,r)$ for $2\leq r\leq\lceil\frac{n}{2}\rceil$.

###  5.1. Second regime: termination violating the Aztec diamond.

Our goal in this section is to show that the configuration $\operatorname{{\sf
K}}(n,r)$ need not terminate in the Aztec diamond when $2\leq
r\leq\lceil\frac{n}{2}\rceil$. The main tool is the _Flooding Lemma_ (Lemma
5.5), which in turn depends on the following lemma:

######  Lemma 5.4.

Let $\operatorname{{\sf K}}$ be any configuration and let $f$ and $g$ be
adjacent faces with

$\operatorname{dist}(f_{0},f)<\operatorname{dist}(f_{0},g)<n\,.$

If $f$ violates the Aztec diamond but $g$ does not, then one can fire from $f$
to $g$. In the resulting configuration $\operatorname{{\sf K}}^{\prime}$ ,
$\operatorname{{\sf K}}^{\prime}_{f}$ will violate the Aztec diamond if
$\operatorname{{\sf K}}_{f}>\operatorname{Az}(n)+1$, and $\operatorname{{\sf
K}}^{\prime}_{g}$ will violate the Aztec diamond if $\operatorname{{\sf
K}}_{g}=\operatorname{Az}(n)_{g}$.

###### Proof.

To see the first part of the lemma, we need to show that $\operatorname{{\sf
K}}_{f}-\operatorname{{\sf K}}_{g}\geq 2$. Recall that

$\operatorname{Az}(n)=\max\\{n-\operatorname{dist}(f_{0},f)+1,0\\}.$

Since $\operatorname{dist}(f_{0},f)<\operatorname{dist}(f_{0},g)<n$, we have
that $\operatorname{Az}(n)_{g}<\operatorname{Az}(n)_{f}$. Since they are
adjacent,

$\operatorname{Az}(n)_{f}-\operatorname{Az}(n)_{g}=1.$

Thus

$\operatorname{{\sf
K}}_{f}\geq\operatorname{Az}(n)_{f}+1=\operatorname{Az}(n)_{g}+2.$

On the other hand, since $\operatorname{{\sf K}}_{g}$ does not violate the
Aztec diamond, $\operatorname{{\sf K}}_{g}\leq\operatorname{Az}(n)_{g}$. Thus

$\operatorname{{\sf K}}_{f}\geq\operatorname{{\sf K}}_{g}+2$

as desired. The second part of the claim is clear from inspection. The
conditions stated are the only possible cases because we have assumed that $f$
violates the Aztec diamond while $g$ does not. ∎

In what follows, we prove Lemma 5.5 (the Flooding Lemma). The intuition is
that one can always “push” a violation of the Aztec diamond further from
$f_{0}$. We will use this idea to create an intermediate configuration of
$\operatorname{{\sf K}}(n,r)$ that can never terminate in the Aztec diamond.
In the proof of the Flooding Lemma, we will be interested in paths whose first
step is always $f_{0}$, and will use the convention
$\operatorname{{\mathcal{P}}}=(f_{0},f_{1},\dots,f_{\ell})$. In general,
however, paths do not need to start at $f_{0}$.

######  Lemma 5.5 (Flooding lemma).

Let $f$ be a face such that $\operatorname{{\sf
K}}_{f}>\operatorname{Az}(n)_{f}>0$. Then there is configuration
$\operatorname{{\sf K}}^{\prime}$, reachable from $\operatorname{{\sf K}}$ by
flow firing moves, and a face $g$ such that

  * •

$\operatorname{dist}(f_{0},g)=1+\operatorname{dist}(f_{0},f)$ and

  * •

$\operatorname{{\sf K}}^{\prime}_{g}>\operatorname{Az}(n)_{g}$.

###### Proof.

We explicitly construct $\operatorname{{\sf K}}^{\prime}$ as in the statement
of the theorem. Let $V$ be the set of faces that violate the Aztec diamond in
the configuration $\operatorname{{\sf K}}$. Take $f\in V$ to be a face with
maximal distance from $f_{0}$. By the maximality of $f$, every neighbor $g$ of
$f$ with $\operatorname{dist}(f_{0},g)=1+\operatorname{dist}(f_{0},f)$ does
_not_ violate the Aztec diamond. Choose such a neighbor $g$ and a path
$\operatorname{{\mathcal{P}}}=(f_{0},f_{1},\cdots,f_{\ell}=g)$ that does not
pass through $f$.

We claim that one can fire along $\operatorname{{\mathcal{P}}}$ until $g$
violates the Aztec diamond. There are two cases:

  * •

_Case I: For $1\leq i\leq\ell$, no $f_{i}$ in $\operatorname{{\mathcal{P}}}$
violates the Aztec diamond._
First fire along $\operatorname{{\mathcal{P}}}$ to create a standard path.
This is always possible because $f_{i}$ never violates the Aztec diamond. Note
that after this process, $g$ has value $\operatorname{Az}(n)_{g}$. By Lemma
5.4, we can fire from $f$ into $g$, which means that $g$ will then violate the
Aztec diamond.

  * •

_Case II: There is some $1\leq i\leq\ell$ where $f_{i}$ violates the Aztec
diamond._
Pick the violating face $f_{i}$ that is furthest from $f_{0}$. Thus $f_{i+1}$
does not violate $\operatorname{Az}(n)$, and so by Lemma 5.4 we can fire from
$f_{i}$ to $f_{i+1}$. Repeat this process; it necessarily will terminate
because we are in the conservative flow setting. After termination, either we
are in _Case I_ or the violation has been pushed to
$\operatorname{{\mathcal{P}}}_{\ell}=g$.

Thus, the resulting configuration $\operatorname{{\sf K}}^{\prime}$ has
$\operatorname{{\sf K}}^{\prime}_{g}>\operatorname{Az}(n)_{g}$ since $g$
violates the Aztec diamond. ∎

The Flooding Lemma implies that for a configuration $\operatorname{{\sf
K}}(n,r)$ with $r$ large enough, one can construct a terminal configuration
that violates $\operatorname{Az}(n)$.

######  Theorem 5.6.

Given a configuration $\operatorname{{\sf K}}(n,r)$ with $2\leq
r\leq\lceil\frac{n}{2}\rceil$, there is a stable configuration
$\operatorname{{\sf K}}^{*}$ such that $\operatorname{{\sf
K}}^{*}\neq\operatorname{Az}(n)$.

###### Proof.

Since $2\leq r\leq\lceil\frac{n}{2}\rceil$, there exists at least one face
violating $\operatorname{Az}(n)$. Let $V$ be the set of all such faces, and
choose $f\in V$ of maximal distance from $f_{0}$. Since $f$ is maximal, every
face $g$ with $\operatorname{dist}(g,f_{0})>\operatorname{dist}(f,f_{0})$ has
$\operatorname{{\sf K}}_{g}\leq\operatorname{Az}(n)_{g}$. Fix such a $g$ with
$\operatorname{dist}(g,f_{0})=\operatorname{dist}(f,f_{0})+1$. Applying the
Flooding Lemma (Lemma 5.5) to $f$ and $g$ gives a new configuration
$\operatorname{{\sf K}}^{\prime}$ such that some face $g$ with
$\operatorname{dist}(g,f_{0})>\operatorname{dist}(f,f_{0})$ has
$\operatorname{{\sf K}}_{g}>\operatorname{Az}(n)_{g}$.

Repeat this process until the violating face $h$ has
$\operatorname{dist}(f_{0},h)>n$, so that $\operatorname{Az}(n)_{h}=0$; note
that this is always possible by Lemma 5.4. Call this configuration
$\operatorname{{\sf K}}^{\prime}$. Since $\operatorname{{\sf
K}}^{\prime}_{h}>0$, no firing moves on $\operatorname{{\sf K}}^{\prime}$ can
ever decrease the weight of face of $h$, and so firing on $\operatorname{{\sf
K}}^{\prime}$ will result in a terminal configuration $\operatorname{{\sf
K}}^{*}$ with $\operatorname{{\sf K}}^{*}_{h}>0$. Hence $\operatorname{{\sf
K}}^{*}\neq\operatorname{Az}(n)$. ∎

Example 5.7 shows how iterative applications of the Flooding Lemma can be used
to obtain such a final configuration $\operatorname{{\sf K}}^{\ast}$.

######  Example 5.7.

Below we draw four configurations. The first (a) is $\operatorname{{\sf
K}}(3,2)$, and the other three are obtained from the first by successive
firing moves. Using the notation of Lemma 5.5, for each of the configurations
$\operatorname{{\sf K}}$ below, pick $f$ to be the northern-most nonzero face
that can fire in $\operatorname{{\sf K}}$ (shaded, in pink), and let $g$ be
the neighboring face (shaded, in yellow). Using these choices of $f$ and $g$,
we show a sequence of three iterations of the Flooding Lemma.

. &
*(yellow!40!white) *(magenta!20!white)3
3 3 3
3 3 *(teal!40) 3 3 3
3 3 3
3
. &
*(yellow!40!white) 1 2
*(magenta!20!white) 3 3 3
3 3 *(teal!40) 3 3
3 3 3
3
. & *(yellow!40!white)
*(magenta!20!white) 2 2
2 3 3
3 3 *(teal!40) 3 3 3
3 3 3
3
. & *(magenta!20!white) 1
1 2
2 3 3
3 3 *(teal!40) 3 3 3
3 3 3
3

(a) (b) (c) (d)

###  5.2. Second regime: termination in the Aztec diamond

Having constructed a stable configuration from $\operatorname{{\sf K}}(n,r)$
which is not equal to $\operatorname{Az}(n)$, we now wish to show that
$\operatorname{Az}(n)$ _can_ appear as a final configuration for
$\operatorname{{\sf K}}(n,r)$ when $2\leq r\leq\lceil\frac{n}{2}\rceil$. Our
goal is to show:

######  Theorem 5.8.

Given a configuration $\operatorname{{\sf K}}(n,r)$ with $2\leq
r\leq\lceil\frac{n}{2}\rceil$, there is a stable configuration
$\operatorname{{\sf K}}^{*}$ obtained from firing on $\operatorname{{\sf
K}}(n,r)$ such that $\operatorname{{\sf K}}^{*}=\operatorname{Az}(n)$.

To conclude Theorem 5.10, we will divide $\operatorname{{\sf K}}(n,r)$ and
$\operatorname{Az}(n)$ into four quadrants, as shown below in the case of
$\operatorname{{\sf K}}(4,2)$. Each quadrant is given in a different color,
and $\operatorname{Az}(4)$ is shaded in gray in the background. Note that the
marked face $f_{0}$ is not included in any of the quadrants.

. & *(black!20!white)
*(black!20!white) *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(violet!40!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(yellow!20!white) 4 *(violet!40!white) 4 *(violet!40!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(yellow!20!white) 4 *(yellow!20!white) 4 *(teal!85!white) 4 *(red!30!white) 4 *(red!30!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(green!20!white)4 *(green!20!white)4 *(red!30!white)4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(green!20!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white)*(black!20!white)
*(black!20!white)

Without loss of generality, we can study a single quadrant and then appeal to
symmetry to describe the remaining quadrants. We consider the rows in the
first quadrant of $\operatorname{{\sf K}}(n,r)$ (colored violet in the figure
above) and count them from top to bottom. Let
$\operatorname{{\mathcal{P}}}(\ell)$ be the path that is obtained from
extending the $\ell^{\text{th}}$ row of this quadrant to the boundary of
$\operatorname{Az}(n)$. The relevant configuration on
$\operatorname{{\mathcal{P}}}(\ell)$ is then

$\operatorname{{\sf
R}}(\ell):=\underbrace{(n,n,\cdots,n}_{\ell~{}\textrm{times}},\underbrace{0,0,\cdots,0)}_{n-\lceil\frac{n}{2}\rceil~{}\textrm{times}}.$

whose behavior was studied in Section 3.

######  Example 5.9.

Below we draw two configurations restricted to the first quadrant. In order to
illustrate how we use path firing, we shade the first and second rows
$\operatorname{{\mathcal{P}}}(1)$ (in teal) and
$\operatorname{{\mathcal{P}}}(2)$ (in violet). In the first figure (a), the
path configuration of $\operatorname{{\mathcal{P}}}(1)$ is $\operatorname{{\sf
R}}(1)=(4,0,0)$ and the path configuration of
$\operatorname{{\mathcal{P}}}(2)$ is $\operatorname{{\sf R}}(2)=(4,4,0,0)$.
Path stable configurations obtained from $\operatorname{{\sf R}}(1)$ and
$\operatorname{{\sf R}}(2)$ are $(2,1,1)$ and $(3,2,2,1)$, respectively, as
shown in the second figure (b).

. &

*(teal!40!white) 4 *(teal!40!white) *(teal!40!white)
*(violet!40!white)4 *(violet!40!white) 4 *(violet!40!white) *(violet!40!white)

. &

*(teal!40!white)2 *(teal!40!white)1 *(teal!40!white)1
*(violet!40!white)3 *(violet!40!white)2 *(violet!40!white)2 *(violet!40!white)1

(a) (b)

We are now ready to present the main theorem of this section.

######  Theorem 5.10.

It is possible for $\operatorname{{\sf K}}(n,r)$ to terminate in
$\operatorname{Az}(n)$ when $2\leq r\leq\lceil\frac{n}{2}\rceil$.

###### Proof.

We will show that path firing on $\operatorname{{\sf R}}(\ell)$ yields a path-
stable configuration that does not violate $\operatorname{Az}(n)$.

Note that $\operatorname{dist}(f_{0},f_{i})=r-\ell+i$ for each face $f_{i}$ of
$\operatorname{{\mathcal{P}}}(\ell)$ and

$\operatorname{Az}(n)_{f_{i}}=n-\operatorname{dist}(f_{0},f_{i})+1=n-r+\ell-i+1.$

Let $\operatorname{{\sf R}}^{\prime}$ be the unique (by Theorem 3.5) stable
configuration obtained from path-firing on $\operatorname{{\sf R}}(\ell)$.
Recall by Proposition 3.6, there are only two possible forms
$\operatorname{{\sf R}}^{\prime}$ can take. If $\operatorname{{\sf
R}}^{\prime}$ has no repeated weights, then $\mathrm{len}(\operatorname{{\sf
R}}^{\prime})=n-m$ for $m\geq\lceil\frac{n}{2}\rceil-\ell$. In this case,

$\operatorname{Az}(n)_{f_{i}}\geq n-m-i+1=\operatorname{{\sf R}}^{\prime}_{i}$

because $m\geq\lceil\frac{n}{2}\rceil-\ell$.

If $\operatorname{{\sf R}}^{\prime}$ has a single repeated weight,
$\mathrm{len}(\operatorname{{\sf R}}^{\prime})=n-m+1$ for
$m\geq\lceil\frac{n}{2}\rceil-\ell+1$. Thus

$\operatorname{Az}(n)_{f_{i}}\geq n-m-i+2\geq\operatorname{{\sf
R}}^{\prime}_{i}$

since $m\geq\lceil\frac{n}{2}\rceil-\ell+1$.

Thus, the value of $\operatorname{{\sf R}}^{\prime}_{i}$ cannot exceed
$\operatorname{Az}(n)_{f_{i}}$ along $\operatorname{{\mathcal{P}}}(\ell)$.
Firing $\operatorname{{\sf K}}(n,r)$ along all rows in the analogous way thus
yields a configuration which does not violate $\operatorname{Az}(n)$. Thus by
Lemma 4.1, there is a sequence of firing moves that can then be implemented to
obtain $\operatorname{Az}(n)$ as a stable configuration. ∎

We illustrate this process with Example 5.11 below.

######  Example 5.11.

In Example 5.9, we looked at two path configurations restricted to the first
quadrant. The full configurations are depicted below, and the labels
correspond to the labels of the restrictions from Example 5.9. The difference
is that now we perform analogous path firing moves in _all four quadrants_ to
obtain the second configuration (b) from the first (a). Note that the second
configuration (b) does not violate the Aztec diamond $\operatorname{Az}(4)$.
This means that we can apply Lemma 4.1 to obtain $\operatorname{Az}(4)$ from
(b) by firing moves.

. & *(black!20!white)
*(black!20!white) *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(violet!40!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(yellow!20!white) 4 *(violet!40!white) 4 *(violet!40!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(yellow!20!white) 4 *(yellow!20!white) 4 *(teal!85!white) 4 *(red!30!white) 4 *(red!30!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(green!20!white)4 *(green!20!white)4 *(red!30!white)4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(green!20!white) 4 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white)*(black!20!white)
*(black!20!white)

. & *(black!20!white)
*(black!20!white) *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white) *(teal!40!white)2 *(teal!40!white)1 *(teal!40!white)1
*(black!20!white)1 *(black!20!white)1 *(black!20!white)2 *(violet!40!white)3 *(violet!40!white)2 *(violet!40!white)2 *(violet!40!white)1
*(black!20!white)1 *(black!20!white) 2 *(black!20!white)2 *(black!20!white)3 *(teal!85!white) 4 *(black!20!white)3 *(black!20!white)2 *(black!20!white) 2 *(black!20!white) 1
*(black!20!white)1 *(black!20!white)2 *(black!20!white)2 *(black!20!white)3 *(black!20!white)2 *(black!20!white)1 *(black!20!white) 1
*(black!20!white) 1 *(black!20!white) 1 *(black!20!white)2 *(black!20!white) *(black!20!white)
*(black!20!white) *(black!20!white)*(black!20!white)
*(black!20!white)

(a) (b)

##  6\. The Third Regime

Finally, we turn to the third regime. We first show that when $r>\lceil
n/2\rceil$, $\operatorname{{\sf K}}(n,r)$ cannot terminate uniquely in Section
6.1. Then, we show that for $r>\lceil\frac{n}{\sqrt{3}}\rceil+1$, it is
impossible for $\operatorname{{\sf K}}(n,r)$ to terminate in the Aztec diamond
in Section 6.2.

###  6.1. Non-unique termination

We will show that $\operatorname{{\sf K}}(n,r)$ does not terminate uniquely
for $r>\lceil n/2\rceil$. Our proof relies on decomposing the grid in two
distinct ways, shown below:

Figure 3. Two distinct ways to decompose the grid into quadrants.

For $i=1,2$, we refer to $D_{i}$ as the decomposition of the grid. Our
argument introduces an algorithm that fires identically within each quadrant,
but produces two different final cofigurations using the two decompositions in
Figure 3.

######  Theorem 6.1.

If $r>\lceil n/2\rceil$ then $\operatorname{{\sf K}}(n,r)$ does not terminate
uniquely.

###### Proof.

Consider the following flow-firing algorithm on $\operatorname{{\sf K}}(n,r)$:

  1. (1)

Divide $\operatorname{{\sf K}}(n,r)$ into four quadrants as in either $D_{i}$
for $i=1,2$ as in Figure 3;

  2. (2)

While possible, path-fire within each quadrant along all horizontal paths
(e.g. rows) away from $f_{0}$;

  3. (3)

While possible, path-fire within each quadrant along all vertical paths (e.g.
columns) away from $f_{0}$;

  4. (4)

Return to step (2).

We claim that this process results in a stable configuration. By construction,
within each quadrant the resulting configuration is stable. We claim that any
adjacent faces between different quadrants must differ by at most one. This
follows from the fact that the quadrants are reflections of one another, with
face values shifted exactly one unit up (down) and right (left).

Let $\operatorname{{\sf K}}_{i}$ be the stable configuration obtained from the
above process, with the choice of quadrants given by $D_{i}$ for $i=1,2$. Our
goal is to show that $\operatorname{{\sf K}}_{1}$ and $\operatorname{{\sf
K}}_{2}$ are two different stable configurations on $\operatorname{{\sf
K}}(n,r)$. If the supports of $\operatorname{{\sf K}}_{1}$ and
$\operatorname{{\sf K}}_{2}$ are different, it is immediate that
$\operatorname{{\sf K}}_{1}\neq\operatorname{{\sf K}}_{2}$. If the supports
are the same but there exists a face $f$ such that $(\operatorname{{\sf
K}}_{1})_{f}\neq(\operatorname{{\sf K}}_{2})_{f}$, then again
$\operatorname{{\sf K}}_{1}\neq\operatorname{{\sf K}}_{2}$. We claim that
these are the only possible situations. For the remainder of the proof,
suppose both the supports and face weights of $\operatorname{{\sf K}}_{1}$ and
$\operatorname{{\sf K}}_{2}$ are the same. This will lead to a contradiction.

We will label the face weights of $\operatorname{{\sf K}}_{1}$ and
$\operatorname{{\sf K}}_{2}$ as follows: first, label the faces of the grid by
$(j,k)\in\operatorname{\mathbb{Z}}^{2}$ so that
$f_{j,k}\in\operatorname{\mathbb{Z}}$ is the face weight at face $(j,k)$ for
$i=1,2$. Fix the indexing so that the distinguished face is $f_{0,0}$, and
$(j,k)$ increase to the right and up. We will focus on the first quadrant of
$\operatorname{{\sf K}}_{i}$, denoted by $Q_{i}$. Label the face weights in
both $Q_{i}$ as $a_{j,k}^{(i)}\in\operatorname{\mathbb{Z}}^{2}_{>0}$, where
the bottom-left face is $a_{1,1}^{(i)}$ for $i=1,2$ and again indices increase
right and up.

Figure 4. The face weights of the grid, $Q_{1}$, and $Q_{2}$ (respectively)

We make three observations:

  * •

because $Q_{1}$ and $Q_{2}$ were fired via the same process,
$a_{j,k}^{(1)}=a_{j,k}^{(2)}$ for all
$j,k\in\operatorname{\mathbb{Z}}^{2}_{>0}$. In other words, $Q_{2}$ is
obtained by shifting $Q_{1}$ up and left by 1;

  * •

by the way that the quadrants were chosen in $\operatorname{{\sf K}}_{1}$ and
$\operatorname{{\sf K}}_{2}$, we have $a_{1,1}^{(1)}=f_{1,0}^{(1)}$, while
$a_{1,1}^{(2)}=f_{0,1}^{(2)}$;

  * •

by the assumption that $\operatorname{{\sf K}}_{1}=\operatorname{{\sf
K}}_{2}$, we have $a_{j,k}^{(1)}=a_{j+1,k-1}^{(2)}$ for all $j,k$. Therefore
$a_{j,k}^{(1)}=a_{j-1,k+1}^{(1)}$ for all $j,k$.

Label the columns of $Q_{i}$ by $C_{1},\cdots C_{k}$, and let the length of
$C_{j}$ be $n_{j}$. Then by the last observation, we must have that
$n_{j+1}=n_{j}-1$ for $j=1,\ldots,k-1$. Thus, without loss of generality, we
may describe $Q_{1}$ as follows:

Figure 5. The face weights of $Q_{1}$ when the support and face weights of
$K_{1}$ and $K_{2}$ are the same

Here, we make use of the firing process described at the beginning of the
proof, Lemma 3.2, and Lemma 3.4, along with the bound $r>\lceil n/2\rceil$ to
obtain a contradiction. The details are omitted, but the general idea is to
use similar counting arguments as in Section 3.2 to obtain a contradiction. ∎

######  Example 6.2.

In the case of $\operatorname{{\sf K}}(3,3)$, the figures below show two
distinct final configurations. We decompose the grid as in Figure 3, and
perform firings as in the proof of Theorem 6.1.

. & *(violet!40!white) 1
*(yellow!20!white) 1 *(violet!40!white)2 *(violet!40!white) 1
*(yellow!20!white)1 *(yellow!20!white)2 *(violet!40!white)2 *(violet!40!white)2 *(violet!40!white)1
*(yellow!20!white) 1 *(yellow!20!white)2 *(yellow!20!white)2 *(violet!40!white)3 *(violet!40!white)3 *(violet!40!white)2 *(violet!40!white)1
*(yellow!20!white) 1 *(yellow!20!white) 2 *(yellow!20!white)3 *(yellow!20!white)3 *(teal!85!white) 3 *(red!30!white)3 *(red!30!white)3 *(red!30!white)2 *(red!30!white)1
*(green!20!white)1 *(green!20!white)2 *(green!20!white)3 *(green!20!white)3 *(red!30!white)2 *(red!30!white)2 *(red!30!white)1
*(green!20!white) 1 *(green!20!white) 2 *(green!20!white)2 *(red!30!white)2 *(red!30!white)1
*(green!20!white) 1 *(green!20!white)2*(red!30!white) 1
*(green!20!white) 1
. & *(yellow!20!white) 1
*(yellow!20!white) 1 *(yellow!20!white) 2 *(violet!40!white) 1
*(yellow!20!white)1 *(yellow!20!white)2 *(yellow!20!white)2 *(violet!40!white)2 *(violet!40!white)1
*(yellow!20!white) 1 *(yellow!20!white)2 *(yellow!20!white)3 *(yellow!20!white)3 *(violet!40!white)2 *(violet!40!white)2 *(violet!40!white)1
*(green!20!white) 1 *(green!20!white) 2 *(green!20!white)3 *(green!20!white)3 *(teal!85!white) 3 *(violet!40!white) 3 *(violet!40!white) 3 *(violet!40!white) 2 *(violet!40!white) 1
*(green!20!white)1 *(green!20!white)2 *(green!20!white)2 *(red!30!white)3 *(red!30!white)3 *(red!30!white)2 *(red!30!white)1
*(green!20!white) 1 *(green!20!white) 2 *(red!30!white)2 *(red!30!white)2 *(red!30!white)1
*(green!20!white) 1 *(red!30!white)2*(red!30!white) 1
*(red!30!white) 1

###  6.2. Regime 3: Termination outside of the Aztec diamond

Henceforth, let $N=\lceil\frac{n}{\sqrt{3}}\rceil+1$.

######  Theorem 6.3.

If $r\geq N$, then $\operatorname{{\sf K}}(n,r)$ cannot terminate in the Aztec
diamond $\operatorname{Az}(n)$.

######  Remark 6.4.

In fact, Theorem 6.3 holds for any closed curve of weight $n$ containing
$\operatorname{{\sf K}}(N,r)$.

###### Proof.

It is sufficient to prove that $\mathrm{weight}(\operatorname{{\sf
K}}(n,r))>\mathrm{weight}(\operatorname{Az}(n))$.

Let $\operatorname{{\sf K}}$ denote a single quadrant of the Aztec diamond.
The weight of $\operatorname{{\sf K}}$ is

$\displaystyle\mathrm{weight}(\operatorname{{\sf
K}})=(n+n-1+\cdots+1)+(n-1+n-2+\cdots+1)+\cdots+1$
$\displaystyle=\sum_{i=1}^{n}n\cdot i-\sum_{i=1}^{n}i^{2}+\sum_{i=1}^{n}i$
$\displaystyle=\frac{n(n+1)(n+2)}{6}.$

Thus summing over all four quadrants _and_ $f_{0}$, we obtain

$\mathrm{weight}(\operatorname{Az}(n))=n+4\left(\frac{n(n+1)(n+2)}{6}\right)=n+\frac{2}{3}(n)(n+1)(n+2)\,.$

Let $\operatorname{{\sf K}}^{\prime}$ denote a single quadrant of
$\operatorname{{\sf K}}(n,r)$. The weight of $\operatorname{{\sf K}}^{\prime}$
is

$\mathrm{weight}(\operatorname{{\sf K}}^{\prime})=nr+n(r-1)+\cdots
2n+n=n\cdot\frac{r(r+1)}{2}$

where the first expression comes from adding the total number of chips in each
row of the quadrant. Thus by similar logic, summing over all quadrants and
$f_{0}$ gives

$\mathrm{weight}(\operatorname{{\sf K}}(n,r))=n+4n\cdot\frac{r(r+1)}{2}.$

Suppose $r\geq\lceil\frac{n}{\sqrt{3}}\rceil+1$, and note that for $n$ a
positive integer,

$\lceil n/\sqrt{3}\rceil>n/\sqrt{3}.$

Thus

$\displaystyle\mathrm{weight}(\operatorname{{\sf K}}(n,r))>$ $\displaystyle
4n\left(\frac{\left(\frac{n}{\sqrt{3}}+1\right)\left(\frac{n}{\sqrt{3}}+2\right)}{2}\right)+n$
$\displaystyle=2n\left(\frac{n}{\sqrt{3}}+1\right)\left(\frac{n}{\sqrt{3}}+2\right)+n$
$\displaystyle=\frac{2}{3}\left(n^{3}+3\sqrt{3}n^{2}+6n\right)+n$
$\displaystyle>\frac{2}{3}\left(n^{3}+3n^{2}+2n\right)+n=\mathrm{weight}(\operatorname{Az}(n)).\qed$

###  6.3. Bound improvement

Proposition 6.3 naturally raises the question of whether it is possible for
$\operatorname{{\sf K}}(n,r)$ to terminate in the Aztec diamond when
$\lceil\frac{n}{\sqrt{3}}\rceil+1>r>\lceil n/2\rceil$. In the table below, we
present the values of $r=\lceil\frac{n}{\sqrt{3}}\rceil+1$ and $\lceil
n/2\rceil$ for $3\leq n\leq 24$, along with the minimum value of $r$ such that
$\mathrm{weight}(\operatorname{{\sf
K}}(n,r))>\mathrm{weight}(\operatorname{Az}(n))$.

n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathbf{\lceil\frac{n}{2}\rceil}$ | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 9 | 10 | 10 | 11 | 11 | 12 | 12
Minimum r | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12 | 12 | 13 | 14 | 14 | 15
$\mathbf{\lceil\frac{n}{\sqrt{3}}\rceil+1}$ | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 | 11 | 11 | 12 | 12 | 13 | 14 | 14 | 15 | 15

One can see that the minimum value of $r$ for which
$\mathrm{weight}(\operatorname{{\sf
K}}(n,r))>\mathrm{weight}(\operatorname{Az}(n))$ is precisely
$\lceil\frac{n}{\sqrt{3}}\rceil+1$ approximately one third of the time. Thus,
in these cases, the bound for Theorem 6.3 cannot be improved unless a
completely different approach is taken. On the other hand, for $n>8$, there is
a gap between $\lceil\frac{n}{2}\rceil$ and the minimum value of $r$. This
suggests finding a better upper bound for Regime 2 may be more viable. We
present below examples that show improving the bounds for Regime 2 and Regime
3 may be quite subtle.

  1. (1)

It is possible to find configurations $\operatorname{{\sf K}}(n,r)$ with
$\lceil\frac{n}{2}\rceil<r<\lceil\frac{n}{\sqrt{3}}\rceil+1$ that can
terminate in $\operatorname{Az}(n)$. Consider the case $n=8$; the only ball
contained in the region between Regimes 2 and 3 is $\operatorname{{\sf
K}}(8,5)$. First observe that $\mathrm{weight}(\operatorname{{\sf
K}}(8,5))=\mathrm{weight}(\operatorname{Az}(8))$. For this initial
configuration, there is a sequence of firing moves that terminate at
$\operatorname{Az}(8)$.

  2. (2)

It is possible to find configurations $\operatorname{{\sf K}}(n,r)$ with
$\lceil\frac{n}{2}\rceil<r<\lceil\frac{n}{\sqrt{3}}\rceil+1$ that cannot
terminate in $\operatorname{Az}(n)$. Consider the case $n=4$; the only ball
contained in the region between Regimes 2 and 3 is $\operatorname{{\sf
K}}(4,3)$. As one can see from the table above,
$\mathrm{weight}(\operatorname{{\sf
K}}(4,3))>\mathrm{weight}(\operatorname{Az}(4))$. So, it is not possible for
$\operatorname{{\sf K}}(4,3)$ to terminate at $\operatorname{Az}(4)$.

  3. (3)

It is possible to find configurations within $\operatorname{{\sf K}}(n,r)$
that can terminate in $\operatorname{Az}(n)$, even if $\operatorname{{\sf
K}}(n,r)$ cannot. Suppose that $n=4$. Consider the configuration
$\operatorname{{\sf K}}$ on the left in the figure on the following page,
which is contained in $\operatorname{{\sf K}}(4,3)$:

. & *(blue!40!white) 4
*(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4
*(black!20!white) *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(black!20!white)
*(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(teal!50!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4
*(black!20!white) *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(black!20!white)
*(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4
*(blue!40!white) 4

. & *(blue!40!white) 4
*(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4
*(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4
*(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(teal!50!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4
*(black!20!white) *(blue!40!white) 4 *(blue!40!white) 4 *(blue!40!white) 4 *(black!20!white)
*(black!20!white) *(blue!40!white) 4 *(black!20!white)
*(blue!40!white) 4

Note that $\mathrm{weight}(\operatorname{{\sf
K}})=\mathrm{weight}(\operatorname{Az}(4))$. For this initial configuration
$\operatorname{{\sf K}}$, one can find a sequence of firing moves that
terminate at the Aztec diamond.

  4. (4)

It is possible to find a configuration $\operatorname{{\sf K}}$ with
$\mathrm{weight}(\operatorname{{\sf
K}})=\mathrm{weight}(\operatorname{Az}(n))$ that cannot terminate in
$\operatorname{Az}(n)$. Again, take $n=4$ and consider the configuration on
the right in the figure above. This configuration _cannot_ terminate in
$\operatorname{Az}(4)$.

## References

  * [1] Anders Björner, László Lovász, and Peter W. Shor.  Chip-firing games on graphs.  European J. Combin., 12(4):283–291, 1991.
  * [2] Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin.  Critical groups of simplicial complexes.  Ann. Comb., 17(1):53–70, 2013.
  * [3] Pedro Felzenszwalb and Caroline Klivans.  Flow-firing processes.  Journal of Combinatorial Theory, Series A, 177:105308, 2021.
  * [4] Pavel Galashin, Sam Hopkins, Thomas McConville, and Alexander Postnikov.  Root system chip-firing i: interval-firing.  Mathematische Zeitschrift, 292(3):1337–1385, 2019.
  * [5] Pavel Galashin, Sam Hopkins, Thomas McConville, and Alexander Postnikov.  Root System Chip-Firing II: Central-Firing.  International Mathematics Research Notices, 2021(13):10037–10072, 06 2019.
  * [6] Sam Hopkins, Thomas McConville, and James Propp.  Sorting via chip-firing.  Electron. J. Combin., 24(3):Paper No. 3.13, 20, 2017.
  * [7] Caroline Klivans and Patrick Liscio.  Confluence in labeled chip-firing.  Journal of Combinatorial Theory, Series A, 186:105542, 2022.
  * [8] Caroline Klivans and Patrick Liscio.  Move and configuration posets.  Arxiv preprint, 2022.
  * [9] Caroline J. Klivans.  The mathematics of chip-firing.  Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2019.


# Global spin alignment of vector mesons and strong force fields in heavy-ion
collisions

Jinhui Chen ID Key Laboratory of Nuclear Physics and Ion-beam Application
(MoE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
Zuo-Tang Liang ID Key Laboratory of Particle Physics and Particle Irradiation
(MOE), Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao, Shandong 266237, China  Yu-Gang Ma ID Key Laboratory of
Nuclear Physics and Ion-beam Application (MoE), Institute of Modern Physics,
Fudan University, Shanghai 200433, China  Qun Wang ID Peng Huanwu Center for
Fundamental Theory and Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China

## Introduction.

Particles and fields are two fundamental forms of matter in our natural world.
Particles are excitations or quanta of their corresponding fields, which are
characterized by quantum numbers such as mass, charge, spin, and parity.
Photons, for example, are quanta of electromagnetic fields that mediate
electromagnetic forces among charged particles. In collisions at high energy
scales, gluons appear as quanta of color fields that mediate interaction among
quarks as elementary particles of strong interaction. At low energy scales,
strong interaction is often characterized by mesons as effective degrees of
freedom of quark and gluons, whose existence was proposed by Yukawa in 1935 in
analogy with electromagnetic fields (Yukawa:1935xg, ). We now know nuclear
forces as strong interaction at low energies have many components of meson
fields (Sawada:1962, ; Manohar:1983md, ) $\sigma$, $\pi$, $\rho$, $\omega$,
etc.. As the energy scale increases in nuclear reactions other meson fields
carrying strangeness quantum number may come into play, such as $K$, $\phi$,
etc.. As quanta of strong force fields, all these mesons with their specified
quantum numbers were discovered before 1960s. Due to their short Compton
wavelengths, experimentally it is much easier to detect particles than wave-
like fields that are elusive and strongly fluctuate.

Recently, for the first time, the print of one kind of strong force field
could be detected through the global spin alignment of vector mesons in heavy-
ion collisions in the STAR experiment (STAR:2022fan, ). This is another
breakthrough after the one in the measurement of the global spin polarization
of $\Lambda$ in heavy-ion collisions (STAR:2017ckg, ). The global spin
polarization arises from the initial orbital angular momentum (OAM) in nuclear
collisions that is partially converted to the local OAM or vorticity leading
to the polarization of hadrons through spin-orbit coupling in the interaction
(Liang:2004ph, ; Liang:2004xn, ; Wang, ). The spin alignment of vector mesons
in heavy-ion collisions along the OAM direction was first proposed by Liang
and Wang some years ago (Liang:2004xn, ). The observable is the 00 element
$\rho_{00}$ of the spin density matrix for vector mesons which can be measured
by the angular distribution of their strong decay daughters in their rest
frame (Abelev:2008ag, ). The STAR data for $\rho_{00}$ show a surprisingly
large positive deviation from 1/3, which is orders of magnitude larger than
predictions by conventional mechanisms (STAR:2022fan, ). In Ref.
(Sheng:2019kmk, ), Sheng, Oliva and Wang proposed that the local correlation
or fluctuation of the $\phi$ meson field can produce a large positive
deviation for $\rho_{00}$ from 1/3 and thus provided a possible explanation.
Such $\phi$ field may originate from non-perturbative strong interaction
coupled to $s$ and $\overline{s}$ and is connected with vacuum properties of
quantum chromodynamics (QCD)(Shuryak:1981ff, ; Shifman:1978bx, ).

## Vector meson spin alignment

The spin state of a system of spin-$S$ particles can be described by the spin
density matrix $\hat{\rho}$, which is a $(2S+1)\times(2S+1)$ complex and
Hermitian matrix for spin-$S$ particles with unit trace. The number of
independent real variables in $\hat{\rho}$ is $4S(S+1)$. For example, for
spin-1/2 particles, there are 3 independent real variables corresponding to a
polarization vector $\mathbf{P}$ as in
$\rho=(1/2)(1+\boldsymbol{\sigma}\cdot\mathbf{P})$ where $\boldsymbol{\sigma}$
are Pauli matrices. For spin-1 particles such as vector mesons, there are 8
independent real variables corresponding to a polarization vector $\mathbf{P}$
(3 variables) and a symmetric traceless tensor $T_{ij}$ (also called the
tensor polarization, 5 variables). As a spin-1/2 particle, the $\Lambda$
hyperon’s spin polarization can be measured through its parity-violating weak
decay $\Lambda\rightarrow\mathrm{p}\pi^{-}$, since the preferred direction of
the daughter proton’s momentum is along its spin in the rest frame. However,
this is not the case for vector mesons since they mainly decay through strong
interaction in which the parity is conserved. So the elements of the spin
density matrix that can be measured are its tensor components $T_{ij}$ and
$\rho_{00}$ is associated with them. For the $\phi$ meson’s strong decay
$\phi\rightarrow K^{+}K^{-}$, the daughter particle’s polar angular
distribution is given by

$\frac{dN}{d\mathrm{cos}\theta}=\frac{3}{4}\left[1-\rho_{00}+(3\rho_{00}-1)\mathrm{cos}^{2}\theta\right].$
(1)

We see that $\rho_{00}$ appears in the coefficient of $\mathrm{cos}^{2}\theta$
which can be determined by measuring the polar angle distribution. If
$\rho_{00}=1/3$, the distribution is a constant indicating that the daughter
particles are emitted isotropically. If $\rho_{00}\neq 1/3$, the probabilities
of the vector meson in three spin states are not equal and then the emission
of daughter particles is anisotropic. For $\rho_{00}\gtrless 1/3$, the polar
angle distribution is in a cigar/discus shape (as shown in Fig. 1).
Correspondingly, if $\rho_{00}>1/3$, the vector meson has a larger probability
to be in the spin state $\lambda=0$ so that its average polarization vector
(not the spin) is aligned to the spin quantization axis. If $\rho_{00}<1/3$,
the vector meson has a larger probability to be in the spin states
$\lambda=\pm 1$ and its average polarization vector is aligned in the plane
perpendicular to the spin quantization axis. The quantity $\rho_{00}$ is thus
referred to the spin alignment of the vector meson.

Figure 1: (Color online) The shape of $dN/d\mathrm{cos}\theta$ for the
daughter particle in the vector meson’s decay with $\rho_{00}=1/2>1/3$ (left
panel) and $\rho_{00}=1/6<1/3$ (right panel).

## Experimental results.

In Ref. STAR:2022fan , $\phi$ and $K^{*0}$ mesons are observed by paring of
their decay daughters $K^{\pm}$ and $K\pi$, respectively, with subtraction of
the combinatorial background and application of the so-called event mixing and
rotation methods. Detail studies show that both techniques can effectively
break the correlation between pairs in real events, and the yields of $\phi$
and $K^{*0}$ from two techniques are consistent STAR:2022fan . The spin
quantization direction is chosen to be the normal direction of the second-
order event plane constructed from charge particles collected by the STAR Time
Projection Chamber (TPC). Then the polar angle distribution of Eq. (1) is
analyzed, and $\rho_{00}$ is extracted after correction for detection
efficiency and acceptance. The spin quantization direction can be constructed
by different detectors such as the shower maximum detector and the beam-beam
counter, and they all give consistent results on the global spin alignment of
$\phi$ and $K^{*0}$ STAR:2022fan . In the following, we only discuss results
with respect to the TPC event plane.

Figure 2:  (Color online) Measurements of $\rho_{00}$ with respect to the
second-order event plane in high energy heavy-ion collisions. Stars represent
the data for $\phi$ mesons STAR:2022fan ; Acharya:2019vpe , circles represent
the data for $K^{*0}$ mesons STAR:2022fan ; Acharya:2019vpe . The solid red
line is a fit to data by the model based on strong force fields Sheng:2019kmk
.

Figure 2 shows the $\rho_{00}$ data for $\phi$ and $K^{*0}$ mesons in Au+Au
collisions at $\mathrm{\sqrt{s_{NN}}}$= 11.5 to 200 GeV measured by the STAR
collaboration STAR:2022fan  from a dedicated Beam Energy Scan program and
multiple years of high statistics event collection of Au+Au collisions at 200
GeV. The centrality categorizes events according to the observed number of
tracks in each collision. Here the 0% centrality corresponds to exactly head-
on collisions, which produce the most tracks, while the 100% centrality
corresponds to barely glancing collisions, which produce the fewest tracks.
The STAR measurements presented in Fig. 2 are for the centrality interval of
20% to 60%, where one expects the largest OAM among collisions and better
signal to noise ratio in the experimental analysis. A complete set of results
for centrality and transverse momentum dependence can be found in Ref.
STAR:2022fan . The STAR’s data give

$\displaystyle\rho_{00}^{\phi}$ $\displaystyle=$ $\displaystyle 0.3512\pm
0.0017(\mathrm{stat.})\pm 0.0017(\mathrm{syst.}),$
$\displaystyle\rho_{00}^{K^{*0}}$ $\displaystyle=$ $\displaystyle 0.3356\pm
0.0034(\mathrm{stat.})\pm 0.0043(\mathrm{syst.}),$

which are obtained by averaging over results at energies from 11.5 to 62.4 GeV
for $\phi$, and from 11.5 to 54.4 GeV for $K^{*0}$ STAR:2022fan . Taking the
total uncertainties as the sum in quadrature of statistical and systematic
uncertainties, the results suggest that $\rho_{00}^{\phi}$ is above 1/3 with a
significance of 7.4 $\sigma$, indicating a significant global spin alignment
for the $\phi$ meson. The values of $\rho_{00}^{K^{*0}}$, however, are
consistent with 1/3. Measurements in Pb+Pb collisions at
$\mathrm{\sqrt{s_{NN}}}$= 2.76 TeV Acharya:2019vpe  in the $p_{T}$ bin nearest
to the mean $p_{T}$ for STAR data in the range from 1.0 to 5.0 GeV/c are also
shown for comparison. They are consistent with the STAR data where magnitude
of the spin alignment for both $\phi$ and $K^{*0}$ lie at 1/3 at the highest
beam energies within large uncertainties.

## Quark coalescence model.

The quark coalescence model (QCM) is the convenient theoretical tool to
describe the hadron production or hadronization in heavy-ion collisions.
Therefore one can calculate the spin density matrix for vector mesons in terms
of those of the quarks and antiquarks in QCM (Liang:2004xn, ; Sheng:2019kmk,
). A non-relativistic QCM is the most simple and transparent one to this end
(Sheng:2019kmk, ; Sheng:2023cpb, ). In QCM, the meson’s spin density operator
can be constructed as
$\hat{\rho}_{M}=\hat{\rho}_{q}\otimes\hat{\rho}_{\overline{q}}$, where
$\hat{\rho}_{q}$ and $\hat{\rho}_{\overline{q}}$ are spin density operators of
the quark and antiquark in spin and momentum space respectively. The elements
$\rho_{\lambda_{1}\lambda_{2}}^{\mathrm{M}}(\mathbf{x},\mathbf{p})$ can be
obtained by projecting $\rho_{M}$ onto two mesons’ spin and momentum states
and then taking a Fourier transformation with respect to the relative momentum
of two states. The $\rho_{00}$ for the $\phi$ meson in phase space reads
(Sheng:2019kmk, )

$\displaystyle\rho_{00}^{\phi}(\mathbf{x},\mathbf{p})$ $\displaystyle\approx$
$\displaystyle\frac{1}{3}-\frac{2}{3}\left\langle
P_{q}^{y}\left(\mathbf{x}_{1},\mathbf{p}_{1}\right)P_{\bar{q}}^{y}\left(\mathbf{x}_{2},\mathbf{p}_{2}\right)\right\rangle$
(2)
$\displaystyle+\frac{2}{9}\left\langle\mathbf{P}_{q}\left(\mathbf{x}_{1},\mathbf{p}_{1}\right)\cdot\mathbf{P}_{\bar{q}}\left(\mathbf{x}_{2},\mathbf{p}_{2}\right)\right\rangle,$

where $\mathbf{x}_{1}=\mathbf{x}+\Delta\mathbf{x}/2$,
$\mathbf{x}_{2}=\mathbf{x}-\Delta\mathbf{x}/2$,
$\mathbf{p}_{1}=\mathbf{p}/2+\Delta\mathbf{p}$,
$\mathbf{p}_{2}=\mathbf{p}/2-\Delta\mathbf{p}$, the average is taken over
$\Delta\mathbf{x}$ and $\Delta\mathbf{p}$ weighted by the $\phi$ meson’s wave
function, $\mathbf{P}_{q}(\mathbf{x}_{1},\mathbf{p}_{1})$ and
$\mathbf{P}_{\overline{q}}(\mathbf{x}_{2},\mathbf{p}_{2})$ are spin
polarization vectors of the quark and antiquark respectively, and the spin
quantization direction is taken along the ${y}$ direction.

Equation (2) clearly shows that the vector meson’s spin alignment is
determined by the local correlation of quark’s and antiquark’s polarization
functions $\mathbf{P}_{q}(\mathbf{x}_{1},\mathbf{p}_{1})$ and
$\mathbf{P}_{\overline{q}}(\mathbf{x}_{2},\mathbf{p}_{2})$ inside the phase
space limited by the meson’s wave function.

## Local correlation of $\phi$ fields.

It is well known that fermions such as quarks at rest have magnetic moments
proportional to their spins that are polarized along the direction of the
magnetic field. For moving fermions, the electric field comes into play in the
form of $\mathbf{p}\times\mathbf{E}$ (the spin-orbit coupling or spin-Hall
effect). Similarly vorticity fields and vector meson fields can also polarize
quarks and antiquarks. For $s$ and $\overline{s}$ that make the $\phi$ meson,
the spin polarization vector is

$\displaystyle\mathbf{P}_{q/\overline{q}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{\omega}+\frac{1}{2m_{s}}\boldsymbol{\varepsilon}\times{\bf
p}$ (3) $\displaystyle\pm\frac{g_{\phi}}{2m_{s}T}{\bf
B}_{\phi}\pm\frac{g_{\phi}}{2m_{q}E_{p}T}{\bf E}_{\phi}\times{\bf p},$

where $T$ is the effective temperature of the quark matter when $s$ and
$\overline{s}$ combine into the $\phi$ meson, $\boldsymbol{\varepsilon}$ and
$\boldsymbol{\omega}$ are the electric and magnetic part of the thermal
vorticity tensor, and $\mathbf{E}_{\phi}$ and $\mathbf{B}_{\phi}$ are the
electric and magnetic part of the $\phi$ field, respectively. Here we have
neglected the effects from electromagnetic fields since they die away quickly
in the late stage of the matter evolution in heavy-ion collisions. The effects
from vorticity fields can also be neglected because their magnitudes from the
measured $\Lambda$ polarization are too small to account for the observed
$\rho_{00}^{\phi}$, but their terms are kept in Eq. (3) just as a contrast to
those of $\phi$ fields. The main difference between terms of vorticity fields
and $\phi$ fields in Eq. (3) is the sign for antiquarks: vorticity fields do
not distinguish quarks from antiquarks while $\phi$ fields do (same as
electromagnetic fields). We see in the following that this is essential to
obtained a large vector meson spin alignment $\rho_{00}$ in the coalescence of
$s$ and $\bar{s}$ into the $\phi$ meson.

Substituting Eq. (3) into Eq. (2), one obtains that $\rho_{00}^{\phi}$ depends
on the local correlation of $\phi$ fields inside the $\phi$ meson’s wave
function. A more rigorous approach based on the relativistic QCM has been
formulated using the closed-time-path (Schwinger-Keldysh) method in quantum
field theory (Sheng:2022wsy, ; Sheng:2022ffb, ). The result obtained for
$\rho_{00}^{\phi}$ reads (Sheng:2022wsy, ; Sheng:2022ffb, )

$\displaystyle\rho_{00}^{\phi}(t,\mathbf{x},\mathbf{p})$
$\displaystyle\approx$
$\displaystyle\frac{1}{3}+C_{1}\left[\frac{1}{3}\boldsymbol{\omega}^{\prime}\cdot\boldsymbol{\omega}^{\prime}-(\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\omega}^{\prime})^{2}\right]$
(4)
$\displaystyle+C_{2}\left[\frac{1}{3}\boldsymbol{\varepsilon}^{\prime}\cdot\boldsymbol{\varepsilon}^{\prime}-(\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\varepsilon}^{\prime})^{2}\right]$
$\displaystyle-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T^{2}}\left\\{C_{1}\left[\frac{1}{3}{\bf
B}_{\phi}^{\prime}\cdot{\bf
B}_{\phi}^{\prime}-(\boldsymbol{\epsilon}_{0}\cdot{\bf
B}_{\phi}^{\prime})^{2}\right]\right.$
$\displaystyle+\left.C_{2}\left[\frac{1}{3}{\bf E}_{\phi}^{\prime}\cdot{\bf
E}_{\phi}^{\prime}-(\boldsymbol{\epsilon}_{0}\cdot{\bf
E}_{\phi}^{\prime})^{2}\right]\right\\},$

where the fields with primes are in the $\phi$ meson’s rest frame, and $C_{1}$
and $C_{2}$ are functions of $m_{s}$ (strange quark mass) and $m_{\phi}$
($\phi$ meson mass). We see that all terms appear in fields squared. The
momentum dependence of $\rho_{00}^{\phi}$ can be obtained by rewriting
$\rho_{00}^{\phi}$ in terms of fields in the lab frame using Lorentz
transformation. By taking averages over space-time on the hadronization hyper-
surface of the $\phi$ meson, one obtains $\rho_{00}^{\phi}$ as functions of
momentum that can be compared with STAR’s data (STAR:2022fan, ). The
parameters are in the form of $\left\langle
g_{\phi}^{2}\mathbf{E}_{\phi}^{2}/T^{2}\right\rangle$ and $\left\langle
g_{\phi}^{2}\mathbf{B}_{\phi}^{2}/T^{2}\right\rangle$ and reflect local
fluctuations of $\phi$ fields (Sheng:2022wsy, ). If we assume that local field
fluctuations are different in the transverse (labeled as $i$) and longitudinal
direction (labeled as $z$) with respect to the beam direction $z$ in heavy-ion
collisions, then $\rho_{00}^{\phi}$ depends on two parameters
$F_{T}^{2}\equiv\langle g_{\phi}^{2}B_{\phi,i}^{2}/T^{2}\rangle=\langle
g_{\phi}^{2}E_{\phi,i}^{2}/T^{2}\rangle$ and $F_{z}^{2}\equiv\langle
g_{\phi}^{2}B_{\phi,z}^{2}/T^{2}\rangle=\langle
g_{\phi}^{2}E_{\phi,z}^{2}/T^{2}\rangle$. The values of two parameters are
determined by fitting STAR’s data on $\rho_{00}$ as functions of collision
energies in Fig. 2. With fitted values of two parameters one can predict the
transverse momentum spectra of $\rho_{00}^{\phi}$ at all available collision
energies which are in good agreement with STAR’s data (STAR:2022fan, ).

## Summary and outlook

An unexpected large global spin alignment of $\phi$ mesons has been observed
by the STAR Collaboration (STAR:2022fan, ) in relativistic heavy-ion
collisions. By using the quark coalescence model for hadron production, Refs.
Sheng:2019kmk ; Sheng:2022wsy ; Sheng:2022ffb  provide a good interpretation
of the data (STAR:2022fan, ). According to this interpretation, such a large
global spin alignment of $\phi$ mesons is induced by the local correlation or
fluctuation of strong force fields on the hadronization hyper-surface. The
average values $\langle\mathbf{E}_{\phi}^{2}\rangle$ and
$\langle\mathbf{B}_{\phi}^{2}\rangle$ reflect the local fluctuation of $\phi$
fields and are expected to be calculable using lattice QCD. These studies open
a new avenue to investigate properties of the strongly interacting quark
matter as well as non-perturbative properties of strong interaction. In
addition, relativistic heavy-ion collisions are usually called “small bang” in
contrast to the Big Bang of the universe. In such an analog, hadronization on
the freeze-out hyper-surface corresponds to the stage of the early universe in
which particles are decoupled from the interaction during the Big Bang. The
vector meson’s spin alignment is similar to polarization modes of cosmic
microwave background radiation.

We note that such an explanation is still subject to debate and further
verification. More systematical studies in both experiments and theories need
to be done to clarify the deep physics behind the phenomena.

## Conflict of interest.

The authors declare that they have no conflict of interest.

## Acknowledgements.

The authors thank A.H. Tang for insightful discussion. This work was supported
in part by the National Key Research and Development Program of China under
Contract No. 2022YFA1604900, by the National Natural Science Foundation of
China (NSFC) under Contract Nos. 11890710, 11890713, 11890714, 12025501,
12135011 and 12147101, and by the Strategic Priority Research Program of the
Chinese Academy of Sciences (CAS) under Grant No. XDB34030102.

## References

  * [1] Hideki Yukawa.  On the Interaction of Elementary Particles I.  Proc. Phys. Math. Soc. Jap., 17:48–57, 1935.
  * [2] Shoji Sawada, Tamotsu Ueda, Wataro Watari, and Minoru Yonezawa.  One Boson Exchange Model in Proton-Proton Scattering: An Approach to the Strong Interaction.  Prog. Theor. Phys., 28:991–1025, 1962.
  * [3] Aneesh Manohar and Howard Georgi.  Chiral Quarks and the Nonrelativistic Quark Model.  Nucl. Phys. B, 234:189–212, 1984.
  * [4] M. S. Abdallah et al.  Pattern of global spin alignment of $\phi$ and K∗0 mesons in heavy-ion collisions.  Nature, 614(7947):244–248, 2023.
  * [5] L. Adamczyk et al.  Global $\Lambda$ hyperon polarization in nuclear collisions: evidence for the most vortical fluid.  Nature, 548:62–65, 2017.
  * [6] Zuo-Tang Liang and Xin-Nian Wang.  Globally polarized quark-gluon plasma in non-central A+A collisions.  Phys. Rev. Lett., 94:102301, 2005.  [Erratum: Phys. Rev. Lett.96,039901(2006)].
  * [7] Zuo-Tang Liang and Xin-Nian Wang.  Spin alignment of vector mesons in non-central A+A collisions.  Phys. Lett., B629:20–26, 2005.
  * [8] Xin-Nian Wang.  Vector meson spin alignment by the strong force field.  Nucl. Sci. Tech., 34:15, 2023.
  * [9] B. I. Abelev et al.  Spin alignment measurements of the $K^{*0}$(892) and $\phi$ (1020) vector mesons in heavy ion collisions at $\sqrt{s_{NN}}$ = 200 GeV.  Phys. Rev., C77:061902, 2008.
  * [10] Xin-Li Sheng, Lucia Oliva, and Qun Wang.  What can we learn from the global spin alignment of $\phi$ mesons in heavy-ion collisions?  Phys. Rev. D, 101(9):096005, 2020.  [Erratum: Phys.Rev.D 105, 099903 (2022)].
  * [11] Edward V. Shuryak.  The Role of Instantons in Quantum Chromodynamics. 1. Physical Vacuum.  Nucl. Phys. B, 203:93, 1982.
  * [12] Mikhail A. Shifman, A. I. Vainshtein, and Valentin I. Zakharov.  QCD and Resonance Physics. Theoretical Foundations.  Nucl. Phys. B, 147:385–447, 1979.
  * [13] Shreyasi Acharya et al.  Evidence of Spin-Orbital Angular Momentum Interactions in Relativistic Heavy-Ion Collisions.  Phys. Rev. Lett., 125:012301, 2020.
  * [14] Xin-Li Sheng, Zuo-Tang Liang, and Qun Wang.  Global spin alignments of vector meson in heavy ion collisions.  Acta. Phys. Sin. (in Chinese), 72:072502, 2023.
  * [15] Xin-Li Sheng, Lucia Oliva, Zuo-Tang Liang, Qun Wang, and Xin-Nian Wang.  Spin alignment of vector mesons in heavy-ion collisions.  5 2022.
  * [16] Xin-Li Sheng, Lucia Oliva, Zuo-Tang Liang, Qun Wang, and Xin-Nian Wang.  Relativistic spin dynamics for vector mesons.  6 2022.