# Strong law of large numbers for the stochastic six vertex model

Hindy Drillick and Yier Lin  Department of Mathematics, Columbia University
<EMAIL_ADDRESS>Department of Statistics, University of Chicago
<EMAIL_ADDRESS>
###### Abstract.

We consider the inhomogeneous stochastic six vertex model with periodicity
starting from step initial data. We prove that it converges almost surely to a
deterministic limit shape. For the proof, we map the stochastic six vertex
model to a deformed version of the discrete Hammersley process [Sep97,
BEGG16]. Then we construct a colored version of the model and apply Liggett’s
superadditive ergodic theorem. The construction of the colored model includes
a new idea using a Boolean-type product, which generalizes and simplifies the
method used in [DL22].

##  1\. Introduction

###  1.1. The S6V model

The stochastic six vertex (S6V) model [GS92] is a classical model in two-
dimensional statistical physics. It is a stochastic version of the six vertex
model [Lie67, Bax16], which is a natural model for crystal lattices with
hydrogen bonds. We associate six possible configurations to each lattice point
on the first quadrant, as in Figure 1. We view the lines from the south and
the west as _inputs_ and the lines to the north and the east as _outputs_. The
model is _stochastic_ in the sense that when we fix the inputs, the weights of
possible configurations with those inputs sum up to 1. The configurations
chosen for neighboring vertices need to be compatible so that the output lines
of a vertex are the input lines of the vertices to its immediate north and
east.

By [BCG16], we know that the S6V model belongs to the Kardar-Parisi-Zhang
(KPZ) universality class—a class of models that exhibit universal statistical
behavior in their large time/large-scale limits, see [KPZ86, Qua11, Cor12] for
an overview of this topic. Some recent works on the S6V model include [Agg17,
Agg18, BG19, Lin19, ST19, Agg20, CGST20, Dim20, Kua21, Lin22] and references
therein.

In this paper, we view the S6V model as a stochastic path ensemble on the
first quadrant with the boundary condition that there is a line entering each
of the vertices in $\\{1\\}\times\mathbb{Z}_{\geq 1}$ from the left, and no
lines entering $\mathbb{Z}_{\geq 1}\times\\{1\\}$ from the bottom. This
boundary data is called step initial data. Fix a collection of parameters
$\\{b_{1}(i,j),b_{2}(i,j)\\}_{i,j}$ that take values in $[0,1]$. Starting from
$(1,1)$, we tile each vertex with one of the six vertex configurations in
Figure 1, where we only consider configurations whose input lines match the
input lines of the given vertex. We assign an allowed configuration with
probability given by the weight of the configuration. This tiling construction
then progresses sequentially in the linear order
$(2,1),(1,2),(3,1),(2,2),(1,3),\dots$ to the entire quadrant. Note that we can
view the stochastic path ensemble as a family of upright paths ordered in the
northwest direction, as depicted in the left panel of Figure 2.

Let $h$ be the height function of the S6V model defined on $\mathbb{Z}_{\geq
0}^{2}$ where $h(x,y)$ records the number of lines that pass through or to the
right of $(x,y)$, see the left panel of Figure 2.

Type | I | II | III | IV | V | VI
---|---|---|---|---|---|---
Configuration |  |  |  |  |  |
Weight | 1 | 1 | $b_{1}(i,j)$ | $1-b_{1}(i,j)$ | $b_{2}(i,j)$ | $1-b_{2}(i,j)$
Figure 1. Six types of configurations for the S6V model

######  Assumption 1.1.

There exist periodicities $I,J\in\mathbb{Z}_{\geq 1}$ such that
$b_{k}(x+I,y)=b_{k}(x,y)$ and $b_{k}(x,y+J)=b_{k}(x,y)$ for arbitrary $x,y$
and $k\in\\{1,2\\}$.

######  Theorem 1.2.

There exists a Lipschitz function $g$ such that with probability $1$,

$\lim_{n\to\infty}\frac{h(\lfloor nx\rfloor,\lfloor
ny\rfloor)}{n}=g(x,y),\qquad\forall x,y\in\mathbb{R}_{\geq 0}.$ (1.1)

######  Remark 1.3.

The S6V model considered here is inhomogeneous and not integrable in most
cases. The parameters $b_{1}$ and $b_{2}$ have periodicity $I$ in space and
periodicity $J$ in time. Hence, there are $IJ$ degrees of freedom. If $I,J=1$,
the S6V model is homogeneous and its limit shape has been explicitly derived
in [BCG16, Theorem 1.1] and [Agg20, Theorem 5.1] (also see [RS18] for a
derivation of the limit shape using a variational principle). Note that the
derivations of explicit limit shapes in these two papers rely respectively on
integrability and the existence of explicit stationary distributions for the
model. For general $I,J$, due to the lack of these properties, we don’t think
that there is an explicit formula for $g$.

######  Remark 1.4.

Take $I=J=1$ and consider the homogeneous S6V model where $b_{1}(i,j)$ and
$b_{2}(i,j)$ do not depend on $i,j$. By [BCG16, Theorem 1.1] or [Agg20,
Theorem 5.1], if $b_{1}\leq b_{2}$, we have

$g(x,y)=\begin{cases}\frac{\big{(}\sqrt{y(1-b_{1})}-\sqrt{x(1-b_{2})}\big{)}^{2}}{b_{2}-b_{1}}&\qquad\frac{1-b_{2}}{1-b_{1}}<\frac{x}{y}<\frac{1-b_{1}}{1-b_{2}},\\\
0&\qquad\frac{x}{y}\geq\frac{1-b_{1}}{1-b_{2}},\\\
y-x&\qquad\frac{x}{y}\leq\frac{1-b_{2}}{1-b_{1}}.\end{cases}$ (1.2)

If $b_{1}\geq b_{2}$, we have

$g(x,y)=\begin{cases}0&\qquad x\geq y,\\\ y-x&\qquad x\leq y.\end{cases}$

With Theorem 1.2, we strengthen the convergence in probability in [BCG16,
Agg20] to the almost sure level.

###  1.2. The complemented S6V model

To prove Theorem 1.2, it is convenient to study the S6V model after
_horizontal complementation_. If there is a horizontal line, we erase it; if
there is no horizontal line, we add it, see Figure 2 for an example. This
procedure converts an ensemble of upright paths to an ensemble of downright
paths. Note that this model has appeared in [ABW22] without a specific name,
and we call it the _complemented S6V (CS6V) model_.

Let $H$ be the height function of the CS6V model defined on $\mathbb{Z}_{\geq
0}^{2}$, where $H(x,y)$ records the number of lines that pass through or to
the left of $(x,y)$. It is straightforward that $H(x,y)=y-h(x,y)$, see the
right panel of Figure 2. Note that by interpreting the stochastic paths as a
collection of downright paths ordered in the northeast direction, these
downright paths are the level lines of the height function.

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Figure 2. Left panel: A sampling of the S6V model on the first quadrant, with
the blue numbers denoting the height function. Right panel: The CS6V model
obtained by horizontally complementing the S6V model.

There is another way to sample the CS6V model on the first quadrant that is
more similar to how we defined the S6V model above. Consider the six vertex
configurations in Figure 3. These configurations are obtained by horizontally
complementing the configurations in Figure 1. We consider _empty boundary
data_ —the boundary data obtained by horizontally complementing the step
initial data. In other words, no lines enter the quadrant from either the left
boundary $\\{1\\}\times\mathbb{Z}_{\geq 1}$ or the bottom boundary
$\mathbb{Z}_{\geq 1}\times\\{1\\}$. We sequentially tile the vertices
$(1,1),(2,1),(1,2),(3,1),(2,2),(1,3),\dots$ with one of the six vertex
configurations. We can view the tiling as a family of downright paths ordered
in the northeast direction. This path ensemble has the same law as the one
obtained by complementing the S6V model.

Type | I | II | III | IV | V | VI
---|---|---|---|---|---|---
Configuration at $(i,j)$ |  |  |  |  |  |
Weight | 1 | 1 | $b_{1}(i,j)$ | $1-b_{1}(i,j)$ | $b_{2}(i,j)$ | $1-b_{2}(i,j)$
Figure 3. Six types of configurations for the CS6V model

Due to the relationship $H(x,y)=y-h(x,y)$, proving Theorem 1.2 is equivalent
to proving the following theorem.

######  Theorem 1.5.

There exists a Lipschitz function $g$ such that with probability $1$,

$\lim_{n\to\infty}\frac{H(\lfloor nx\rfloor,\lfloor
ny\rfloor)}{n}=y-g(x,y),\qquad\forall x,y\in\mathbb{R}_{\geq 0}.$

###  1.3. The CS6V model and discrete Hammersley process

[Sep97] studied Ulam’s problem on the planar lattice. This model is also
referred to as the discrete Hammersley process in [BEGG16, CG19]. Consider the
integer lattice on $\mathbb{Z}_{\geq 0}^{2}$ and a random set $\xi$ of integer
points chosen independently with probability $p$. Define the partial order
$\prec$ such that $(x_{1},y_{1})\prec(x_{2},y_{2})$ if and only if
$x_{1}<x_{2}$ and $y_{1}<y_{2}$.

Let $H^{\mathsf{d}}$ denote the height function defined on $\mathbb{Z}_{\geq
0}^{2}$:

$H^{\mathsf{d}}(x,y)=\max\Big{\\{}L:\text{ there exist integer points
}(x_{1},y_{1})\prec\dots\prec(x_{L},y_{L})\in\xi\cap[1,x]\times[1,y]\Big{\\}}.$

As noticed in [BEGG16], we can construct a collection of downright paths which
play the role of the level lines of $H^{\mathsf{d}}$. We first use a downright
path to connect the minimal points of $\xi$ under the order $\prec$. We remove
these points from $\xi$ and connect the new minima to obtain the second line
and so on. The height function starts from $H^{\mathsf{d}}(0,0)=0$ and
whenever we cross a downright path from southwest to northeast, the height
function increases by $1$, see Figure 4.

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Figure 4. Illustration of the discrete Hammersley process. The $\times$’s are
chosen independently on the lattice with probability $p$.

The following theorem shows that the CS6V model is a deformed version of the
discrete Hammersley process.

######  Theorem 1.6.

Take $b_{1}=0$ and $b_{2}=1-p$. The CS6V model degenerates to the discrete
Hammersley process.

###### Proof.

It suffices to show that the collection of stochastic lines in Figure 4 has
the same distribution as those in the CS6V model when $b_{1}=0$ and
$b_{2}=1-p$. By the definition of the discrete Hammersley process, we can also
sample the stochastic lines in Figure 4 by sequentially sampling the
configurations of the vertices in the linear order
$(1,1),(2,1),(1,2),(3,1),(2,2),(1,3),\dots$. If there is only a single line
entering a vertex from either the left or bottom, then the line continues. If
there are lines entering from both the left and bottom, then they annihilate
each other and form a corner. If no lines enter from the left and bottom, then
with probability $1-b_{2}=p$ the vertex is chosen to be a part of $\xi$, which
corresponds to choosing a configuration of type VI from Figure 3. Otherwise,
we choose a configuration of type V. This coincides with the sequential
sampling rule for the CS6V model. ∎

Using this, we recover the almost sure limit shape of the discrete Hammersley
process established in [Sep97, Theorem 1] and [BEGG16, Theorem 1].

######  Corollary 1.7.

With probability $1$, for all $x,y\geq 0$,

$\lim_{t\to\infty}\frac{H^{\mathsf{d}}(tx,ty)}{t}=\begin{cases}(1-p)^{-1}\big{(}2\sqrt{pxy}-(x+y)p\big{)}&\qquad\text{
if }p<\min(\tfrac{x}{y},\tfrac{y}{x}),\\\
\min(x,y)&\qquad\text{else}.\end{cases}$

###### Proof.

One can readily check that the right-hand side above equals $y-g(x,y)$ when
$b_{1}=0$ and $b_{2}=1-p$, using the formula of $g$ in Remark 1.4. The
corollary then follows from a combination of Theorems 1.5 and 1.6. ∎

###  1.4. Connection to the $t$-PNG model

The $t$-PNG model introduced in [ABW22] is a deformation of the polynuclear
growth (PNG) model [PS00]. We fix a parameter $t\in[0,1]$ and place a Poisson
point process with intensity $1$ on the upper-right quadrant representing
_nucleations_. We draw lines emanating from each of these nucleations in both
the upward and rightward directions until they collide with one another. We
call these collision points _intersection points_. Given the Poisson
nucleations, we sample the outcomes of the intersection points (lines will
either cross or annihilate each other) starting with the intersection point
which has the smallest sum of $x$\- and $y$\- coordinates and moving
sequentially outward. At an intersection point, the two lines will cross each
other with probability $t$ and will annihilate each other with probability
$1-t$, forming a corner. Note that when lines cross, they might generate new
intersection points. We refer to Figure 5 for a sampling of the $t$-PNG model.

$\times$$\times$$\times$$\times$$\times$ Figure 5. A sampling of the $t$-PNG
model, where “$\times$” are the Poisson nucleations.

Consider the CS6V model where $b_{1}$ and $b_{2}$ do not depend on $i,j$. As
noticed by [ABW22], if we take $b_{1}=t$ and $b_{2}=1-\epsilon^{2}$ and
simultaneously scale the $x$\- and $y$-axis by $\epsilon^{-1}$, the CS6V model
degenerates to the $t$-PNG model as $\epsilon\to 0$.

Using integrable methods, [ABW22] proved a weak law of large numbers and a
fluctuation theorem for the height function of the $t$-PNG model. In [DL22],
we proved a strong law of large numbers of the $t$-PNG model by constructing a
colored $t$-PNG model and using Liggett’s superadditive ergodic theorem (one
can easily go from the usual subadditive ergodic theorem to the superadditive
version by placing negative signs as needed).

###  1.5. Proof ideas

For the discrete Hammersley process defined in Section 1.3, the strong law of
large numbers follows as a direct consequence of the superadditive ergodic
theorem (see Theorem 3.1 for the full statement). To apply the superadditive
ergodic theorem, one needs to construct a family of superadditive random
variables $\\{X_{m,n}:0\leq m\leq n\\}$ where $\\{X_{0,n},n\geq 0\\}$ records
the height function. A subfamily of the random variables also needs to be
ergodic. For the discrete Hammersley process, one can simply define

$X_{m,n}:=\max\Big{\\{}L:\text{ there exist integer points
}(x_{1},y_{1})\prec\dots\prec(x_{L},y_{L})\in\xi\cap[m+1,n]\times[m+1,n]\Big{\\}}.$
(1.3)

It is straightforward to check that $\\{X_{m,n}:0\leq m\leq n\\}$ satisfies
the condition required by the superadditive ergodic theorem.

For the CS6V model, one can also define a family of random variables
$\\{X_{m,n}:0\leq m\leq n\\}$ by slightly modifying the above definition and
setting

$X_{m,n}:=\max\Big{\\{}L:\text{ there exist integer points
}(x_{1},y_{1})\preceq\dots\preceq(x_{L},y_{L})\in\eta\cap[m+1,n]\times[m+1,n]\Big{\\}},$

where $(x_{1},y_{1})\preceq(x_{2},y_{2})$ if $x_{1}\leq x_{2}$ and $y_{1}\leq
y_{2}$, and $\eta$ represents the set of vertices of type III and VI in Figure
3. It is easy to verify that the definition of $X_{m,n}$ above coincides with
that in (1.3) when $b_{1}=0$ and $b_{2}=1-p$.

We check whether the conditions in the superadditive ergodic theorem are still
satisfied for this modified family of random variables. Indeed, the random
variables are still superadditive, and $\\{X_{0,n}:n\geq 0\\}$ records the
height function as desired. However, the ergodic (and even stationary)
property no longer holds, since the random variable $X_{m,n}$ depends on the
lines entering the box $[m+1,n]\times[m+1,n]$ from the left and bottom
boundaries, and the distribution of these lines varies with $m,n$.

To overcome this difficulty, we apply a similar idea as in [DL22] and
construct a colored CS6V model. The colored CS6V model is a generalization of
the CS6V model, where each line is assigned a color. We denote the different
colors by integers $i\in\mathbb{N}$, and we say that the color $i$ has higher
priority than the color $j$ if $i<j$. We allow multiple (but only finitely
many) lines with different colors to travel together, with the restriction
that lines traveling together must have different colors.

The colored CS6V model is defined by specifying a sampling rule for when
horizontal lines and vertical lines meet. The sampling rule is given by a
family of stochastic matrices $\\{\mathsf{L}^{n},n\in\mathbb{N}\\}$ that are
consistent. More concretely, the matrix $\mathsf{L}^{n}$ has both rows and
columns indexed by $\\{0,1\\}^{n}\times\\{0,1\\}^{n}$. The matrix elements are
given by $\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l})$, where
the four vectors $\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}\in\\{0,1\\}^{n}$
specify the number of lines (either zero or one) of each color in
$\\{1,\dots,n\\}$ on the bottom, left, top, and right of an intersection,
respectively (see Figure 6). The stochastic matrices give a probability
measure on the output lines $\mathbf{k},\mathbf{l}$ from an intersection point
given the input lines $\mathbf{i},\mathbf{j}$. In Section 2, we will
explicitly define the matrices $\\{\mathsf{L}^{n}\\}_{n\geq 1}$ and show that
they satisfy three properties. We will now give a brief overview of these
properties, and the precise statements will be given in Propositions 2.5, 2.6,
and 2.7.

$\mathbf{i}$$\mathbf{j}$$\mathbf{k}$$\mathbf{l}$$\mathbf{i}$$\mathbf{j}$$\mathbf{k}$$\mathbf{l}$
Figure 6. Left panel: Fix $n\in\mathbb{N}$. At an intersection point, we have
lines with colors in $\\{1,\dots,n\\}$ in each direction, with at most one
line per color in each direction. Let
$\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}\in\\{0,1\\}^{n}$ denote the
number of lines on the bottom, left, top and right directions, respectively,
where the $m$-th coordinate of each vector records the number of lines with
color $m$. Right panel: Take $n=3$. Let red, blue, and orange denote the
colors $1$, $2$, and $3$. We illustrate an example of the configuration with
$\mathbf{i}=(1,0,0)$, $\mathbf{j}=(0,1,1)$, $\mathbf{k}=(1,1,1)$, and
$\mathbf{l}=(0,0,0)$.

Property 1 (Color Ignorance): Lines with higher priority colors ignore those
with lower priority colors. For instance, the lines with colors that belong to
$\\{1,\dots,m\\}$ ignore the behavior of lines with colors greater than $m$.
This means that if we sample the $n$-colored model and ignore the lines with
colors greater than $m$, the remaining lines will behave as the $m$-colored
model. Because of this, we can define the random variables $\\{X_{m,n},0\leq
m\leq n\\}$ in a way that maintains ergodicity, as we will see in Section 3.

Property 2 (Mod 2 Erasure): Fix arbitrary integers $1\leq
r_{1}<\dots<r_{m}\leq n$. We can project the matrix $\mathsf{L}^{n}$ to
$\mathsf{L}^{m}$ if we replace the colors in $\\{r_{k-1}+1,\dots,r_{k}\\}$
with color $k$ for each $k\in\\{1,\dots,m\\}$ and then erase every pair of
lines with the same color traveling together. Because of this property, we can
project the colored CS6V model down to the single-colored CS6V model. This
will ensure that the random variables $\\{X_{0,n},n\geq 0\\}$ will record the
height function of the CS6V model, which is the quantity that we are
interested in studying.

Property 3 (Monotonicity of the Height Function): Suppose we have a sampling
of the two-colored CS6V configuration on a rectangle $[1,x]\times[1,y]$ where
we assign the vertex configurations in the rectangle using the matrix
$\mathsf{L}^{2}$, and we allow only second color lines to enter from the left
and bottom boundary. We can naturally extend the definition of $H(x,y)$ to
apply to the case of non-empty boundary data by defining $H(x,y)$ as the sum
of the number of lines entering the box $[1,x]\times[1,y]$ from the left
boundary and the number of lines exiting the box $[1,x]\times[1,y]$ from the
top boundary. Let $H^{1}(x,y)$ be the height function when we only consider
lines of the first color. Let $H^{2}(x,y)$ be the height function of the
projection of both colors to the single-colored model after we applied the
procedure of mod $2$ erasure. Then

$H^{1}(x,y)\leq H^{2}(x,y).$

In other words, adding a second color to the model does not decrease the
height function. This property will be important for proving the
superadditivity of the random variables $\\{X_{m,n},0\leq m\leq n\\}$.

####  1.5.1. Novelty of the paper

In [DL22], we studied the $t$-PNG model, which is a one-parameter model. The
construction of the $\mathsf{L}$-matrices therein comes from a sophisticated
guess. The proof of stochasticity of $\\{\mathsf{L}^{n}\\}_{n\geq 1}$ and
Properties 1–3 for the $t$-PNG model is then based on a case by case check.
The CS6V model is a two-parameter model, so defining the colored CS6V model
via guessing is more challenging. One novelty of the paper is that we use a
Boolean-type product to define the $\mathsf{L}$-matrices, which significantly
simplifies both the definition and the proof of the properties. This Boolean-
type product construction can also be applied to give a simpler definition of
the colored $t$-PNG model, see Appendix A.

Other novelties come from working in a discrete setting and the inhomogeneity
in the weights. For the $t$-PNG model, due to a scaling property, we only need
to prove the strong law of large numbers in the diagonal direction. For the
CS6V model, there is no similar scaling property, so we need to prove the
strong law of large numbers in every direction. To this end, for each rational
direction, we construct a colored model on the first quadrant and prove the
strong law of large numbers in that direction. The convergence in all
directions then follows from a density argument.

####  1.5.2. Another possible approach

We want to remark that it is also possible to prove Theorem 1.2 without
complementing the S6V model. Indeed, [BF87, AFS04] prove a law of large
numbers for a family of translation invariant exclusion processes. Their proof
also relies on the superadditive ergodic theorem, but in a different way. An
important ingredient is to couple the exclusion processes so that we can run
them under the same random environment, starting at different times and
initial data. The S6V model is also an exclusion-type interacting particle
system, and to apply their idea, we need to find such a coupling for the S6V
model. Another interesting question is to study the almost sure convergence of
the S6V model to the limit shape under general initial data. Note that this
result has been proved for a family of finite-range exclusion processes in
[BGRS10]. We leave both of these questions to future work.

### Outline

In Section 2, we define a family of $\mathsf{L}$-matrices using a Boolean-type
product. Then we prove that the $\mathsf{L}$-matrices are stochastic and
satisfy Properties 1–3. In Section 3, we construct a colored CS6V model on the
first quadrant using the $\mathsf{L}$-matrices. We apply Liggett’s
superadditive ergodic theorem to the model and prove Theorem 1.5.

### Acknowledgments

We thank Ivan Corwin for suggesting the question and for helpful comments on
the paper. We thank Pablo Ferrari for helpful discussion. HD was supported by
the National Science Foundation Graduate Research Fellowship under Grant No.
DGE-1644869, as well as the W.M. Keck Foundation Science and Engineering grant
on “Extreme Diffusion” and the Fernholz Foundation’s Minerva summer fellows
program.

##  2\. Definition of the $\mathsf{L}$-matrices

In this section, we first define the $\mathsf{L}^{1}$-matrix, which encodes
the weights of the usual single-colored CS6V model. Then, we introduce the
Boolean-type product and use it to define the $\mathsf{L}^{n}$-matrices. In
the following, we take generic parameters $b_{1},b_{2}\in[0,1]$.

######  Definition 2.1.

The matrix $\mathsf{L}^{1}$ is indexed by a 4-tuple $i,j,k,l\in\\{0,1\\}$,
where $i,j,k,l$ denote the number of lines (either zero or one) on the bottom,
left, top, and right of a vertex. We define

$\displaystyle\mathsf{L}^{1}(1,0;1,0)=1,\qquad\mathsf{L}^{1}(0,1;0,1)=1,\qquad\mathsf{L}^{1}(0,0;0,0)=b_{2},$
$\displaystyle\mathsf{L}^{1}(0,0;1,1)=1-b_{2},\qquad\mathsf{L}^{1}(1,1;1,1)=b_{1},\qquad\mathsf{L}^{1}(1,1;0,0)=1-b_{1}.$

For all other $i,j,k,l\in\\{0,1\\}$, we set $\mathsf{L}^{1}(i,j;k,l)=0$. For
fixed input lines $i,j\in\\{0,1\\}$, $\mathsf{L}^{1}(i,j;\cdot,\cdot)$ is a
probability measure on the output lines.

We proceed to give a closed form to the matrices $\\{\mathsf{L}^{n}\\}_{n\geq
1}$. Let

$W^{\prime}:=\\{0,1,b_{1},b_{2},1-b_{1},1-b_{2}\\}$

denote the set of possible weights for the CS6V model. Note that we treat the
elements in $W^{\prime}$ as indeterminates, so the size of $W^{\prime}$ does
not depend on the values of $b_{1}$ and $b_{2}$. We define a product $*$ on
the extended space

$W:=\\{0,1,b_{1},b_{2},(1-b_{1}),(1-b_{2}),b_{1}b_{2},b_{1}(1-b_{2}),(1-b_{1})b_{2},(1-b_{1})(1-b_{2})\\}.$
(2.1)

For $x,y\in W$, we define $[x]$ to be the set of factors of $x$. More
precisely, $[x]=\\{x\\}$ if $x\in W^{\prime}$, $[x]=\\{b_{1},b_{2}\\}$ if
$x=b_{1}b_{2}$ and similarly for other values. We define

$x*y:=\begin{cases}0&\qquad\text{ if
}\\{b_{1},1-b_{1}\\}\subseteq[x]\cup[y]\text{ or
}\\{b_{2},1-b_{2}\\}\subseteq[x]\cup[y],\\\
\prod_{z\in[x]\cup[y]}z&\qquad\text{ otherwise.}\end{cases}$

It is straightforward to check that $W$ is closed under $*$. Moreover, we
readily obtain the following.

######  Proposition 2.2.

The product $*$ satisfies the following three properties.

  1. (1)

(commutativity) $a*b=b*a$, for $a,b\in W$.

  2. (2)

(associativity) $a*(b*c)=(a*b)*c$, for $a,b,c\in W$.

  3. (3)

(distributivity) Suppose that $a_{1},...,a_{n}\in W$ such that
$\sum_{i=1}^{n}a_{i}\in W$. For any $c\in W$,

$\sum_{i=1}^{n}c*a_{i}=c*\sum_{i=1}^{n}a_{i}.$

Alternatively, we can also define $*$ first on $W^{\prime}$ and extend it to
$W$. For $x,y\in W^{\prime}$, we define $x*x=x$, and $x*y=0$ if
$\\{x,y\\}=\\{b_{1},1-b_{1}\\}$ or $\\{b_{2},1-b_{2}\\}$. This is why we call
$*$ a Boolean-type product. For other choices of $x,y\in W^{\prime}$, we
define $x*y=xy$. We then extend the product $*$ from $W^{\prime}$ to $W$ using
the commutative and associative properties.

We proceed to define the $\mathsf{L}^{n}$-matrices, using $W$ as the set of
possible weights. For $\mathbf{x}=(x_{1},\dots,x_{n})\in\\{0,1\\}^{n}$ and
$r\in\\{1,\dots,n\\}$, we define the $r$-fold projection

$\mathfrak{s}_{r}(\mathbf{x})=\Big{(}\sum_{m=1}^{r}x_{m}\Big{)}\mod 2.$

As an example, consider the vector $\mathbf{x}=(1,0,1,1).$ We have

$\displaystyle\mathfrak{s}_{1}(\mathbf{x})=1,\ \
\mathfrak{s}_{2}(\mathbf{x})=1,\ \ \mathfrak{s}_{3}(\mathbf{x})=0,\ \
\mathfrak{s}_{4}(\mathbf{x})=1.$

######  Definition 2.3.

Fix arbitrary $n\in\mathbb{Z}_{\geq 1}$. The matrix $\mathsf{L}^{n}$ is
indexed by a 4-tuple
$\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}\in\\{0,1\\}^{n}$, where
$\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}$ denote the number of lines
(either zero or one) of each color in $\\{1,\dots,n\\}$ on the bottom, left,
top, and right of an intersection, respectively. We define the matrix
$\mathsf{L}^{n}$ via

$\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l}):=\sideset{}{{}^{*}}{\prod}_{r\in\\{1,\dots,n\\}}\mathsf{L}^{1}(\mathfrak{s}_{r}(\mathbf{i}),\mathfrak{s}_{r}(\mathbf{j});\mathfrak{s}_{r}(\mathbf{k}),\mathfrak{s}_{r}(\mathbf{l})),$
(2.2)

where $\prod^{*}$ denotes the Boolean-type product of multiple terms under
$*$.

###  2.1. Properties of $\mathsf{L}^{n}$

We proceed to state and prove that the matrices $\mathsf{L}^{n}$ are
stochastic and that they satisfy the properties mentioned above. The key to
the proofs of stochasticity, color ignorance, and mod 2 erasure is the
distributivity of the Boolean-type product, which allows us to manipulate the
sums and products as we normally would.

######  Proposition 2.4 (Stochasticity).

$\mathsf{L}^{n}$ is a stochastic matrix, i.e., the entries of $\mathsf{L}^{n}$
are non-negative, and for any $\mathbf{i},\mathbf{j}\in\\{0,1\\}^{n}$,

$\sum_{\mathbf{k},\mathbf{l}\in\\{0,1\\}^{n}}\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l})=1.$

###### Proof.

We prove this by induction. The case $n=1$ follows from examination. Suppose
that $\mathsf{L}^{n-1}$ is stochastic. Using the definition for
$\mathsf{L}^{n}$, we can factor it as follows:

$\mathsf{L}^{n}(\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l})=\mathsf{L}^{n-1}(\mathbf{i}_{[1,n-1]},\mathbf{j}_{[1,n-1]},\mathbf{k}_{[1,n-1]},\mathbf{l}_{[1,n-1]})*\mathsf{L}^{1}(\mathfrak{s}_{n}(\mathbf{i}),\mathfrak{s}_{n}(\mathbf{j});\mathfrak{s}_{n}(\mathbf{k}),\mathfrak{s}_{n}(\mathbf{l})).$
(2.3)

Note that $(\mathfrak{s}_{n}(\mathbf{k}),\mathfrak{s}_{n}(\mathbf{l}))$ equals
each element of $\\{0,1\\}^{2}$ exactly once when we vary
$\mathbf{k},\mathbf{l}$ under the restriction
$\mathbf{k}_{[1,n-1]}=\mathfrak{k}$ and $\mathbf{l}_{[1,n-1]}=\mathfrak{l}$.
Applying this and Proposition 2.2 to the right-hand side of (2.3) yields

$\displaystyle\sum_{\mathbf{k},\mathbf{l}\in\\{0,1\\}^{n}}\mathsf{L}^{n}(\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l})=\Big{(}\sum_{\mathfrak{k},\mathfrak{l}\in\\{0,1\\}^{n-1}}\mathsf{L}^{n-1}(\mathbf{i}_{[1,n-1]},\mathbf{j}_{[1,n-1]};\mathfrak{k},\mathfrak{l})\Big{)}*\sum_{k,l\in\\{0,1\\}}\mathsf{L}^{1}(\mathfrak{s}_{n}(\mathbf{i}),\mathfrak{s}_{n}(\mathbf{j});k,l)=1.$

The last equality follows from the stochasticity of both $\mathsf{L}^{n-1}$
and $\mathsf{L}^{1}$. ∎

######  Proposition 2.5 (Color Ignorance).

Fix $m\in\\{1,\dots,n\\}$ and
$\mathfrak{i},\mathfrak{j},\mathfrak{k},\mathfrak{l}\in\\{0,1\\}^{m}$. For all
$\mathbf{i},\mathbf{j}\in\\{0,1\\}^{n}$ such that
$\mathbf{i}_{[1,m]}=\mathfrak{i}$ and $\mathbf{j}_{[1,m]}=\mathfrak{j}$, we
have

$\sum_{\begin{subarray}{c}\mathbf{k}_{[1,m]}=\mathfrak{k},\\\
\mathbf{l}_{[1,m]}=\mathfrak{l}\end{subarray}}\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l})=\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l}).$
(2.4)

###### Proof.

By Definition 2.3, we have

$\displaystyle\sum_{\begin{subarray}{c}\mathbf{k}_{[1,m]}=\mathfrak{k},\\\
\mathbf{l}_{[1,m]}=\mathfrak{l}\end{subarray}}\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l})=\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l})\sum_{\begin{subarray}{c}\mathbf{k}_{[1,m]}=\mathfrak{k},\\\
\mathbf{l}_{[1,m]}=\mathfrak{l}\end{subarray}}\
\sideset{}{{}^{*}}{\prod}_{i=m+1}^{n}\mathsf{L}^{1}(\mathfrak{s}_{i}(\mathbf{i}),\mathfrak{s}_{i}(\mathbf{j});\mathfrak{s}_{i}(\mathbf{k}),\mathfrak{s}_{i}(\mathbf{l})).$

The last product can be rewritten as $\mathsf{L}^{n-m}(i,j,k,\ell)$ where
$i=(\mathfrak{s}_{m+1}(\mathbf{i}),\cdots,\mathfrak{s}_{n}(\mathbf{i}))$, and
where $j,k$, and $\ell$ are defined similarly. Summing
$\mathsf{L}^{n-m}(i,j,k,\ell)$ over $\mathbf{k}_{[1,m]}=\mathfrak{k}$ and
$\mathbf{l}_{[1,m]}=\mathfrak{l}$ reduces to summing the same object over
$k,\ell\in\\{0,1\\}^{n-m}$, which equals $1$ due to stochasticity. This
concludes the result. ∎

We call $\pi$ a partition of $\\{1,\dots,n\\}$ if it takes the form of

$\pi=\big{\\{}\\{1,\dots,r_{1}\\},\\{r_{1}+1,\dots,r_{2}\\},\dots,\\{r_{m-1}+1,\dots,r_{m}\\}\big{\\}}$

for some $m\leq n$ and $1=r_{1}<r_{2}<\dots<r_{m}=n$. We define a projection
map $g_{\pi}:\\{0,1\\}^{n}\to\\{0,1\\}^{m}$ such that

$g_{\pi}(x_{1},\dots,x_{n})=\bigg{(}\Big{(}\sum_{i=r_{k-1}+1}^{r_{k}}x_{i}\Big{)}\text{
mod }2\bigg{)}_{k=1}^{m}.$

We define $\ell(\pi)=m$ to be the length of the partition $\pi$.

######  Proposition 2.6 (Mod 2 Erasure).

Fix a partition $\pi$ of $\\{1,\dots,n\\}$ such that $\ell(\pi)=m$. Fix
$\mathfrak{i},\mathfrak{j},\mathfrak{k},\mathfrak{l}\in\\{0,1\\}^{m}$. For all
$\mathbf{i},\mathbf{j}\in\\{0,1\\}^{n}$ satisfying
$g_{\pi}(\mathbf{i})=\mathfrak{i}$ and $g_{\pi}(\mathbf{j})=\mathfrak{j}$, we
have

$\sum_{\begin{subarray}{c}g_{\pi}(\mathbf{k})=\mathfrak{k}\\\
g_{\pi}(\mathbf{l})=\mathfrak{l}\end{subarray}}\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l})=\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l}).$

###### Proof.

Define $A=\\{r_{1},...,r_{m}\\}$. We define $i,j$ to be the unique $n-m$
dimensional vectors such that

$\displaystyle(\mathfrak{s}_{1}(i),\cdots,\mathfrak{s}_{n-m}(i))$
$\displaystyle=(\mathfrak{s}_{1}(\mathbf{i}),\cdots,\widehat{\mathfrak{s}}_{r_{1}}(\mathbf{i}),\cdots,\widehat{\mathfrak{s}}_{r_{m}}(\mathbf{i}),\cdots,\mathfrak{s}_{n}(\mathbf{i})),$
$\displaystyle(\mathfrak{s}_{1}(j),\cdots,\mathfrak{s}_{n-m}(j))$
$\displaystyle=(\mathfrak{s}_{1}(\mathbf{j}),\cdots,\widehat{\mathfrak{s}}_{r_{1}}(\mathbf{j}),\cdots,\widehat{\mathfrak{s}}_{r_{m}}(\mathbf{j}),\cdots,\mathfrak{s}_{n}(\mathbf{j})),$

where $\widehat{x}$ denotes the removal of $x$ in the vector. We have

$\sum_{\begin{subarray}{c}g_{\pi}(\mathbf{k})=\mathfrak{k}\\\
g_{\pi}(\mathbf{l})=\mathfrak{l}\end{subarray}}\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l})=\sum_{\begin{subarray}{c}g_{\pi}(\mathbf{k})=\mathfrak{k}\\\
g_{\pi}(\mathbf{l})=\mathfrak{l}\end{subarray}}\sideset{}{{}^{*}}{\prod}_{i\in
A}\mathsf{L}^{1}(\mathfrak{s}_{i}(\mathbf{i}),\mathfrak{s}_{i}(\mathbf{j});\mathfrak{s}_{i}(\mathbf{k}),\mathfrak{s}_{i}(\mathbf{l}))\sideset{}{{}^{*}}{\prod}_{i\notin
A,i\leq
n}\mathsf{L}^{1}(\mathfrak{s}_{i}(\mathbf{i}),\mathfrak{s}_{i}(\mathbf{j});\mathfrak{s}_{i}(\mathbf{k}),\mathfrak{s}_{i}(\mathbf{l})).$
(2.5)

Since $g_{\pi}(\mathbf{k})=\mathfrak{k}$ and
$g_{\pi}(\mathbf{l})=\mathfrak{l}$, we have $\sideset{}{{}^{*}}{\prod}_{i\in
A}\mathsf{L}^{1}(\mathfrak{s}_{i}(\mathbf{i}),\mathfrak{s}_{i}(\mathbf{j});\mathfrak{s}_{i}(\mathbf{k}),\mathfrak{s}_{i}(\mathbf{l}))=\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l})$.
Using this and Proposition 2.2, the right-hand side of (2.5) can be rewritten
as

$\displaystyle\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l})\sum_{\begin{subarray}{c}g_{\pi}(\mathbf{k})=\mathfrak{k}\\\
g_{\pi}(\mathbf{l})=\mathfrak{l}\end{subarray}}\,\sideset{}{{}^{*}}{\prod}_{i\notin
A,i\leq
n}\mathsf{L}^{1}(\mathfrak{s}_{i}(\mathbf{i}),\mathfrak{s}_{i}(\mathbf{j});\mathfrak{s}_{i}(\mathbf{k}),\mathfrak{s}_{i}(\mathbf{l}))=\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l})\sum_{k,l\in\\{0,1\\}^{n-m}}\mathsf{L}^{n-m}(i,j,k,l)=\mathsf{L}^{m}(\mathfrak{i},\mathfrak{j};\mathfrak{k},\mathfrak{l}).$

The first equality is due to (2.2). The last equality follows from the
stochasticity of $\mathsf{L}^{n-m}.$ ∎

The next proposition implies that if we start with a CS6V model on
$[1,x]\times[1,y]$ with empty boundary data and wish to add some boundary data
(i.e., lines entering the box from the bottom and the left), then we can use
the colored model to sample the CS6V model with the boundary data in a way so
that the height function at $(x,y)$ cannot decrease with the addition of the
boundary data.

######  Proposition 2.7 (Monotonicity of the height function).

Suppose we have a two-colored CS6V model on $[1,x]\times[1,y]$, where the
vertex configurations are assigned using the matrix $\mathsf{L}^{2}$, and the
only lines entering from the bottom and left boundaries are lines of the
second color. Let $H^{1}(x,y)$ denote the height function when we only
consider lines of the first color (i.e., the lines of the $1$-fold
projection). Let $H^{2}(x,y)$ denote the height function of the $2$-fold
projection of both colors to the single-colored model. Then

$H^{1}(x,y)\leq H^{2}(x,y).$

###### Proof.

Consider the rectangle $[1,x]\times[1,y]$. Let $A_{1}$ (resp. $B_{1}$) be the
number of first-color lines leaving the top (resp. right) boundary. Let
$A_{2}$ (resp. $B_{2}$) be the number of second-color lines that leave the
same boundary and are accompanied by a first-color line. Let $A_{3}$ (resp.
$B_{3}$) be the number of second-color lines that leave the same boundary
without accompanying first-color lines. Let $A_{4}$ (resp. $B_{4}$) be the
number of second-color lines entering the left (resp. bottom) boundary. We
have

$\displaystyle H^{1}(x,y)=A_{1}=B_{1},\qquad
H^{2}(x,y)=A_{1}-A_{2}+A_{3}+A_{4}=B_{1}-B_{2}+B_{3}+B_{4}.$

By examining all possible two-color configurations in Figure 10, we know that
for a vertex configuration, the number of second-color lines entering from the
left and bottom is no smaller than the number of lines exiting above and to
the right. This implies that the number of second-color lines entering the
rectangle from the left and bottom boundaries is no smaller than the number of
second-color lines exiting the top and right boundaries. Hence,
$A_{4}+B_{4}\geq A_{2}+A_{3}+B_{2}+B_{3}$. Hence, we have either $A_{4}\geq
A_{2}+A_{3}$ or $B_{4}\geq B_{2}+B_{3}$. Assume without loss of generality
that the first inequality holds. Then we have

$H^{1}(x,y)=A_{1}\leq A_{1}-A_{2}+A_{3}+A_{4}=H^{2}(x,y).\qed$

######  Remark 2.8.

It is possible to define a colored model in the same way as in Definition 2.3,
using ordinary multiplication instead of the Boolean-type product. This will
still yield a model with the properties of color ignorance and mod 2 erasure.
However, with this definition, the colored model will no longer satisfy the
monotonicity of the height function. To see why this is, note that with the
Boolean definition, we cannot have the configurations in Figure 7. This is
because these configurations have weights $b_{1}*(1-b_{1})=0$ and
$b_{2}*(1-b_{2})=0$. However, with the ordinary product, these configurations
would have non-zero weights. As demonstrated in Figure 8, these configurations
will yield samplings that fail the monotonicity of the height function.

Figure 7. The weights of the above configurations equal
$b_{1}*(1-b_{1})=b_{2}*(1-b_{2})=0$ using the Boolean-type product, and equal
$b_{1}(1-b_{1})$, $b_{2}(1-b_{2})$ using ordinary multiplication. $1$-fold
projection${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2}$$2$-fold
projection${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}$
Figure 8. A sampling that violates the monotonicity of the height function
since the height function in the top-right corner of its $2$-fold projection
is smaller than the height function in the top-right corner of its $1$-fold
projection.

##  3\. The colored model and Proof of Theorem 1.5

With the help of the matrices $\\{\mathsf{L}^{n}\\}_{n\in\mathbb{Z}_{\geq
1}}$, we apply the superadditive ergodic theorem to the colored CS6V model and
prove the existence of the limit shape. The argument is similar to [DL22,
Section 3] with certain adaptations.

We are going to construct $\\{X_{m,n}:0\leq m\leq n\\}$ as discussed in the
introduction, using the colored CS6V model. Before doing that, let us recall
Liggett’s superadditive ergodic theorem. For our purposes, we formulate it in
the superadditive setting by placing negative signs where necessary.

######  Theorem 3.1 ([Lig12, page 277]).

Suppose $\\{X_{m,n}\\}$ is a collection of random variables that is indexed by
integers $0\leq m\leq n$ and satisfies:

  1. (i)

Almost surely $X_{0,0}=0$ and $X_{0,n}\geq X_{0,m}+X_{m,n}$ for $0\leq m\leq
n$.

  2. (ii)

For each $k\geq 1$, $\\{X_{(n-1)k,nk}:n\geq 1\\}$ is a stationary and ergodic
process.

  3. (iii)

$\\{X_{m,m+k}:k\geq 0\\}\overset{d}{=}\\{X_{m+1,m+k+1}:k\geq 0\\}$ for each
$m\geq 0$.

  4. (iv)

$\mathbb{E}[X_{0,1}^{-}]<\infty$ where $x^{-}=\max(-x,0)$.

Then there exists a constant $\gamma=\sup_{n\geq
1}\frac{\mathbb{E}[X_{0,n}]}{n}\in(-\infty,\infty]$ satisfying

$\displaystyle\gamma=\lim_{n\to\infty}\frac{X_{0,n}}{n}\text{ a.s.}$

We proceed to define a colored CS6V model on the first quadrant. Recall that
$I,J$ are the periodicities of the weights of the CS6V model, as in Assumption
1.1. Fix a direction $(x,y)\in\mathbb{Q}_{>0}^{2}$, and define

$N:=\text{ the smallest positive integer such that
}\frac{Nx}{I},\frac{Ny}{J}\in\mathbb{N}.$ (3.1)

We have omitted the dependence of $N$ on $x,y$. Assign color $-k$ to the
vertex $(a,b)$ if either $a\in[N(k-1)x+1,Nkx]$ or $b\in[N(k-1)y+1,Nky]$, see
the left panel of Figure 9. We use negative numbers to label the colors so
that we can have infinitely many colors of increasing priority. Given the
input lines of a vertex with color $-k$, we sample the output lines according
to stochastic matrices $\mathsf{L}^{k}$ with parameters $b_{1}=b_{1}(i,j)$ and
$b_{2}=b_{2}(i,j)$. The colored CS6V model is then sampled sequentially at the
vertices $(1,1),(2,1),(1,2),(3,1),(2,2),(1,3),\dots$ to the entire quadrant,
using the $\mathsf{L}$-matrices assigned to different vertices. By definition,
$\mathsf{L}^{k}$ is defined for the positive colors $1,\ldots,k$, so we just
map the colors $-k,\ldots,-1$ to $1,\ldots,k$, preserving their order. One can
check that if a vertex of color $-k$ has no input lines, then the only
possible output lines emanating from that vertex can be of color $-k$. See the
right panel of Figure 9 for a sampling of the colored CS6V model.

We proceed to define the random variables $\\{X_{m,n},m,n\in\mathbb{Z}_{\geq
0},m\leq n\\}$ that satisfy the condition of Theorem 3.1. Let $v_{z,Nny}$
denote the number of vertical lines exiting the vertex $(z,Nny)$ with colors
that belong to $\\{-n,\dots,-m-1\\}$. We define

$X_{m,n}:=\sum_{z=Nmx+1}^{Nnx}\big{(}v_{z,Nny}\text{ mod }2\big{)}.$ (3.2)

The next proposition encodes the height function of the CS6V model in
$\\{X_{m,n}:0\leq m\leq n\\}$.

Figure 9. Left panel: We illustrate how we color the vertices on the first
quadrant when $I=3,J=2,x=2$, and $y=1$. In this case, $N=6$. Let blue, red,
and olive represent the colors $-1,-2$, and $-3$. Each rectangle contains
$12\times 6$ vertices with the same color. Right panel: A sampling of the
colored CS6V model. The lines emanating from a vertex inherit the color of
that vertex. Due to the property of color ignorance, the red lines can be
sampled ignoring the blue lines, and the olive lines can be sampled ignoring
both the blue and red lines. Note that lines of different colors can travel
together across the same vertices.

######  Proposition 3.2.

We have $\\{X_{0,k},k\in\mathbb{Z}_{\geq
0}\\}=\\{H(Nkx,Nky),k\in\mathbb{Z}_{\geq 0}\\}$.

###### Proof.

It suffices to show that for each $n\in\mathbb{N}$,
$\\{X_{0,k},k=0,\dots,n\\}\overset{d}{=}\\{H(Nkx,Nky),k=0,\dots,n\\}$. The
lines in the square
$[\frac{1}{2},Nnx+\frac{1}{2}]\times[\frac{1}{2},Nny+\frac{1}{2}]$ have colors
that belong to $\\{-n,\dots,-1\\}$. Replace these colors with a single color.
Then, by the property of mod 2 erasure, the resulting model reduces to the
single-colored CS6V model. Note that $X_{0,k}$ in (3.2) is equal to the number
of vertical lines that cross the segment
$[\frac{1}{2},Nkx+\frac{1}{2}]\times\\{Nky+\frac{1}{2}\\}$, which is
$H(Nkx,Nky)$. This implies that
$\\{X_{0,k},k=0,\dots,n\\}\overset{d}{=}\\{H(Nkx,Nky),k=0,\dots,n\\}$. ∎

######  Proposition 3.3.

The stochastic process $\\{X_{m,n}:m,n\in\mathbb{Z}_{\geq 0},m\leq n\\}$
satisfies conditions (ii)–(iv) of Theorem 3.1.

###### Proof.

We first prove (ii). Consider the square
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$.
There are lines flowing inside through the left boundary
$\\{Nmx+\frac{1}{2}\\}\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$ and the bottom
boundary $[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times\\{Nmy+\frac{1}{2}\\}$. These
lines have colors belonging to $\\{-m,\dots,-1\\}$. The vertices in
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$
have colors in $\\{-n,\dots,-m-1\\}$. Note that the color $i$ takes priority
over $j$ if $i<j$, so the lines that emanate from the vertices in
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$
have higher priority than the lines entering through the left and bottom
boundaries. Therefore, by Proposition 2.5, the behavior of the lines with
colors $\\{-n,\dots,-m-1\\}$ in the rectangle
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$
does not depend on the lower priority lines entering from the left and bottom.
Hence, the distribution of $X_{m,n}$ is independent of the number and location
of the lines entering the bottom and left boundaries of
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$.
This implies that for each $k\geq 1$, the random variables
$\\{X_{(n-1)k,nk},n\geq 1\\}$ are independent. It is straightforward to see
that $X_{(n-1)k,nk}$ has the same distribution as $H(Nkx,Nky)$ for all $n\geq
1$, therefore this sequence is i.i.d, and hence it is stationary and ergodic.

We proceed to prove (iii). It suffices to show that for arbitrary
$m\in\mathbb{Z}_{\geq 0}$,

$\\{X_{m,m+k},k\geq 0\\}\overset{d}{=}\\{X_{0,k},k\geq 0\\}.$ (3.3)

We look at the colored CS6V model restricted to
$[mNx+\frac{1}{2},\infty)\times[mNy+\frac{1}{2},\infty)$. Note that there are
lines with colors in $\\{-m,\dots,-1\\}$ entering from the left and bottom
boundaries of $[mNx+\frac{1}{2},\infty)\times[mNy+\frac{1}{2},\infty)$. By
Proposition 2.5, the behavior of lines in
$[mNx+\frac{1}{2},\infty)\times[mNy+\frac{1}{2},\infty)$ with colors less than
$-m$ is not affected by the lower priority lines entering from the boundary.
Since $b_{1}(mNx+\cdot,mNy+\cdot)=b_{1}(\cdot,\cdot)$ and
$b_{2}(mNx+\cdot,mNy+\cdot)=b_{2}(\cdot,\cdot)$, this implies that after a
shift by $(mNx,mNy)$, the lines with colors
$i_{1},\dots,i_{k}\in\mathbb{Z}_{\leq-m-1}$ in
$[mNx+\frac{1}{2},\infty)\times[mNy+\frac{1}{2},\infty)$ behave the same (in
distribution) as the lines with colors $i_{1}+m,\dots,i_{k}+m$ in
$[\frac{1}{2},\infty)\times[\frac{1}{2},\infty)$. Hence, we conclude (3.3).

Finally, we have $X_{0,1}^{-}=0$ because $X_{0,1}$ is non-negative. Hence,
(iv) holds. ∎

Let us proceed to prove that $\\{X_{m,n},m,n\in\mathbb{Z}_{\geq 0},m\leq n\\}$
also satisfies the superadditive condition (i) in Theorem 3.1. We begin with
some preparation. In the square
$[\frac{1}{2},Nnx+\frac{1}{2}]\times[\frac{1}{2},Nny+\frac{1}{2}]$, we replace
the colors $\\{-m,\dots,-1\\}$ with the color $-1$ and replace the colors
$\\{-n,\dots,-m-1\\}$ with the color $-2$. After that, as long as there are
two lines with the same color that travel together, we erase them. By
Proposition 2.6, the resulting model is a two-colored CS6V model. In
particular, the vertices have color $-1$ in the L-shaped area
$[\frac{1}{2},Nmx+\frac{1}{2}]\times[\frac{1}{2},Nny+\frac{1}{2}]\cup[\frac{1}{2},Nnx+\frac{1}{2}]\times[\frac{1}{2},Nmy+\frac{1}{2}]$.
The vertices have color $-2$ in the square
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$.

For the resulting two-colored CS6V model, let $Q_{1}$ be the number of
vertical lines with color $-1$ that cross the segment
$[\frac{1}{2},Nmx+\frac{1}{2}]\times\\{Nmy+\frac{1}{2}\\}$, let $Q_{2}$ be the
number of vertical lines with color $-2$ that cross the segment
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times\\{Nny+\frac{1}{2}\\}$, and let
$P_{1}$ be the number of horizontal lines with color $-1$ that cross
$\\{Nmx+\frac{1}{2}\\}\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$. Finally, let
$R$ be the number of single vertical lines of either color that cross
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times\\{Nny+\frac{1}{2}\\}$ (i.e., lines
that do not travel in a pair).

######  Lemma 3.4.

The following result holds:

$\displaystyle X_{0,m}=Q_{1},$ (3.4) $\displaystyle X_{m,n}=Q_{2},$ (3.5)
$\displaystyle X_{0,n}=Q_{1}+P_{1}+R.$ (3.6)

###### Proof.

Recall that we obtain the two-colored CS6V model in
$[\frac{1}{2},Nnx+\frac{1}{2}]\times[\frac{1}{2},Nny+\frac{1}{2}]$ by
replacing the colors $\\{-n,\dots,-m-1\\}$ with the color $-2$, replacing the
colors $\\{-m,\dots,-1\\}$ with the color $-1$, and erasing pairs of lines
with the same color. The erasure corresponds to the mod 2 erasure procedure in
(3.2). Hence, equations (3.4) and (3.5) directly follow from (3.2).

We proceed to prove (3.6). Note that $X_{0,n}$ is the number of single
vertical lines that cross the segment
$[\frac{1}{2},Nnx+\frac{1}{2}]\times\\{Nny+\frac{1}{2}\\}$ in the two-colored
CS6V model. We decompose

$X_{0,n}=Y_{1}+R,$ (3.7)

where $Y_{1}$ equals the number of vertical lines with the color $-1$ that
leave $[\frac{1}{2},Nmx+\frac{1}{2}]\times\\{Nny+\frac{1}{2}\\}$. We turn to
the rectangle
$[\frac{1}{2},Nmx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$, in
which there are only lines with color $-1$. The CS6V model is conservative in
the sense that the number of lines on the right and bottom of a vertex equals
the number of lines on the top and left of that vertex. Therefore, the number
of lines that cross the bottom and right boundaries of the rectangle
$[\frac{1}{2},Nmx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$ is
equal to the number of lines that cross the top and left boundaries, hence,

$Y_{1}=Q_{1}+P_{1}.$ (3.8)

Using this together with (3.7) and (3.8), we conclude (3.6). ∎

######  Proposition 3.5.

We have $X_{0,0}=0$ and $X_{0,n}\geq X_{0,m}+X_{m,n}$ for $0\leq m\leq n$.
Hence, $\\{X_{m,n},m\leq n\in\mathbb{Z}_{\geq 0}\\}$ satisfies (i) of Theorem
3.1.

###### Proof.

By definition, we have $X_{0,0}=0$. We proceed to show that $X_{0,n}\geq
X_{0,m}+X_{m,n}$. By Lemma 3.4, this is equivalent to $R+P_{1}\geq Q_{2}$.
This follows from applying Proposition 2.7 to the rectangle
$[Nmx+\frac{1}{2},Nnx+\frac{1}{2}]\times[Nmy+\frac{1}{2},Nny+\frac{1}{2}]$,
where only lower priority lines enter from the bottom-left boundary. ∎

######  Proposition 3.6.

Fix arbitrary $x,y\in\mathbb{Q}_{\geq 0}$, there exists a constant
$\widetilde{g}(x,y)\in\mathbb{R}_{\geq 0}$ such that almost surely,

$\lim_{n\to\infty}\frac{H(\lfloor nx\rfloor,\lfloor
ny\rfloor)}{n}=\widetilde{g}(x,y).$

###### Proof.

If $x=0$ or $y=0$, we know that $H(\lfloor nx\rfloor,\lfloor ny\rfloor)=0$ for
all $n$, thus $\widetilde{g}(x,y)=0$. Now we can assume that
$x,y\in\mathbb{Q}_{>0}$. Recall the definition of $X_{m,n}$ from (3.2). By
Propositions 3.3 and 3.5, $\\{X_{m,n},m\leq n\in\mathbb{Z}_{\geq 0}\\}$
satisfy the conditions of the superadditive ergodic theorem. Therefore, there
exists a constant $\gamma(x,y)$ such that almost surely,
$\lim_{n\to\infty}\frac{X_{0,n}}{n}=\gamma(x,y)$. By Proposition 3.2, we know
that almost surely, $\lim_{n\to\infty}\frac{H(nNx,nNy)}{n}=\gamma(x,y)$.
Recall that $N$ is the positive integer defined in (3.1). Since the height
function $h:\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq
0}$ is a Lipschitz function, we conclude that almost surely,

$\lim_{n\to\infty}\frac{H(\lfloor nx\rfloor,\lfloor
ny\rfloor)}{n}=\gamma(x,y)/N:=\widetilde{g}(x,y).\qed$

###### Proof of Theorem 1.5.

Using the fact that $\mathbb{Q}_{\geq 0}^{2}$ is countable and Proposition
3.6, almost surely,

$\lim_{n\to\infty}\frac{H(\lfloor nx\rfloor,\lfloor
ny\rfloor)}{n}=\widetilde{g}(x,y)$

for all $(x,y)\in\mathbb{Q}_{>0}^{2}$. By definition, $H$ is a Lipschitz
function, hence $\widetilde{g}:\mathbb{Q}_{\geq 0}^{2}\to\mathbb{R}$ is also a
Lipschitz function. We can thus extend $\widetilde{g}$ to be a Lipschitz
function on $\mathbb{R}_{\geq 0}^{2}$. By the density of $\mathbb{Q}_{\geq
0}^{2}$ in $\mathbb{R}_{\geq 0}^{2}$ and the Lipschitz property, we have that
almost surely,

$\lim_{n\to\infty}\frac{H(\lfloor nx\rfloor,\lfloor
ny\rfloor)}{n}=\widetilde{g}(x,y)$

for all $(x,y)\in\mathbb{R}_{\geq 0}^{2}$. Taking
$g(x,y)=y-\widetilde{g}(x,y)$, we conclude the proof. ∎

##  Appendix A The Boolean-type product for the $t$-PNG model

Taking $b_{1}=t$ and $b_{2}=1$, the set $W$ in (2.1) reduces to
$\\{0,t,1-t,1\\}$. The Boolean-type product reduces to the following: for
$a,b\in\\{0,t,1-t,1\\}$,

$a*b:=\begin{cases}0&\qquad\text{ if }\\{a,b\\}=\\{t,1-t\\},\\\ a&\qquad\text{
if }a=b,\\\ ab&\qquad\text{ else.}\end{cases}$

Recall from [DL22, Definition 1.7] that the $\mathsf{L}^{n}$-matrix of the
$t$-PNG model is defined as

$\mathsf{L}^{n}(\mathbf{i},\mathbf{j};\mathbf{k},\mathbf{l}):=\mathfrak{min}_{r\in\\{1,\dots,n\\}}\Big{(}\mathsf{L}^{1}(\mathfrak{s}_{r}(\mathbf{i}),\mathfrak{s}_{r}(\mathbf{j});\mathfrak{s}_{r}(\mathbf{k}),\mathfrak{s}_{r}(\mathbf{l}))\Big{)},$

where $\mathfrak{min}$ is a modified version of the minimum. For
$x_{1},\dots,x_{n}\in\\{0,t,1-t,1\\}$,

$\mathfrak{min}\big{(}x_{1},\dots,x_{n}\big{)}:=\begin{cases}0&\qquad\text{if
$x_{i}=t$ and $x_{j}=1-t$ for some $i,j\in\\{1,\dots,n\\}$,}\\\
\min\big{(}x_{1},\dots,x_{n}\big{)}&\qquad\text{else.}\\\ \end{cases}$

It is straightforward to verify that $\mathfrak{min}(a,b)=a*b$ for
$a,b\in\\{0,t,1-t,1\\}$, and thus
$\mathfrak{min}(x_{1},\dots,x_{n})=x_{1}*\cdots*x_{n}.$ Hence, the definition
[DL22, Definition 1.7] is equivalent to Definition 2.3 when $b_{1}=t$ and
$b_{2}=1$.

##  Appendix B Two-colored CS6V configurations

In this section, we list the configurations of the two-colored CS6V model.
Note that the weights of the vertex configurations are given by the matrix
$\mathsf{L}^{2}$.

$1$$1$$b_{1}$$b_{2}$$1-b_{1}$$1-b_{2}$$1-b_{1}$$b_{1}$$1$$b_{1}$$1-b_{1}$$b_{1}$$1-b_{1}$$1$$b_{1}$$1-b_{1}$$b_{2}$$1-b_{2}$$b_{2}$$1-b_{2}$$b_{2}$$1-b_{2}$$b_{2}$$1-b_{2}$$b_{1}b_{2}$$(1-b_{1})(1-b_{2})$$b_{1}(1-b_{2})$$(1-b_{1})b_{2}$$b_{1}b_{2}$$(1-b_{1})b_{2}$$(1-b_{2})b_{1}$$(1-b_{1})(1-b_{2})$
Figure 10. The two-colored CS6V configurations with non-zero weight given by
$\mathsf{L}^{2}$. Red has a higher priority than blue.

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# An Autonomous Probing System
for Collecting Measurements at Depth
from Small Surface Vehicles

Yuying Huang∗,1, Yiming Yao∗,1, Johanna Hansen∗,1, Jeremy Mallette1, Sandeep
Manjanna1,2, Gregory Dudek1, and David Meger1 1Mobile Robotics Lab, McGill
University, Montreal, Canada 2Computer and Information Science, University of
Pennsylvania, Philadelphia, USA ∗indicates equal contribution

###### Abstract

This paper presents the portable autonomous probing system (APS), a low-cost
robotic design for collecting water quality measurements at targeted depths
from an autonomous surface vehicle (ASV). This system fills an important, but
often overlooked niche in marine sampling by enabling mobile sensor
observations throughout the near-surface water column without the need for
advanced underwater equipment. We present a probe delivery mechanism built
with commercially available components and describe the corresponding open-
source simulator and winch controller. Finally, we demonstrate the system in a
field deployment and discuss design trade-offs and areas for future
improvement. Project details are available on  our website

###### Index Terms:

marine robotics, sensing, in situ sampling

##  I Introduction

Active monitoring of water resources such as estuaries, lakes, and bays is of
importance for improving our understanding of how water quality impacts
natural ecosystems and human populations. Our knowledge of these systems is
often limited by the expense of gathering frequent and diverse samples. While
there has been wide adoption of autonomous boats[1] for capturing in situ data
from the surface, underwater observations are mostly out of reach.

Our work is motivated by the desire to increase the availability of vertical
profiles in the water column in small bodies of water. The need for in situ
data from throughout the water column has been well-established in the
oceanographic [2] and limnological [3] communities. The demand for this water-
column data is perhaps best illustrated by the incredible success and adoption
of data from the ARGO profiling float program [4]. In lakes, the need for
portable depth profiling systems is clearly motivated. Lofton _et al._ [5]
employ water-column lake observations of nutrients, temperature, and light to
investigate phytoplankton distribution in over $51$ northern lakes.

(a) ASV

(b) Probe

Figure 1: This figure captures our autonomous probing system (APS) deploying a
water quality sensor called a sonde to a specified depth in the water column.

This paper presents the design and demonstration of a low-cost system to
capture samples to valuable near-surface water column data (up to $20~{}m$).
Our platform, consisting of a modular frame, off-the-shelf winch, and sensor
probe (depicted in Fig. 2) mounts on a fully controllable autonomous surface
vehicle (ASV). In the following sections, we describe the mechanical design
(Section III), present a new open-source simulator (Section IV), and finally,
we present results from field trials (Section V) and discuss design
limitations and room for improvement (Section VI).

(a) CAD Diagram

(b) Winch

(c) Slip Ring

(d) EXO Sensor

Figure 2: Fig. 2(a) shows the CAD diagram of the entire system we used in our
experiments and in our FEA analysis. Fig. 2(b) shows the commercially
available winch that we utilized for probe deployment and retrieval. This view
of the winch and tether is from the _winchcam_ , a camera mounted directly
above the winch to provide remote monitoring. We add an electronic slip ring
(Fig. 2(c)) to the winch and an underwater tether to enable continuous
communication with the probe. In our experiments, we employ the EXO Sonde as
our probe, depicted in Fig. 2(d).

There have been consistent efforts to reduce the cost (both human and
monetary) of acquiring marine data. Shallow ($0-20~{}m$) marine phenomena are
traditionally measured either by passive sensors, such as fixed moorings or
drifting nodes such as the aforementioned profiling floats [6, 4]. While these
are powerful sources of data at depth, both are limited in where they can
sample. Moorings are typically deployed to a static location and can be
expensive to relocate. Drifting sensors cannot be explicitly utilized for
actively gathering information from scientifically important positions as they
are not spatially controllable.

Underwater vehicles, such as [7, 8] enable highly specific sampling throughout
the water column, but are limited by their sophisticated nature. Expert
engineers are typically needed to deploy these costly systems, which may not
have access to satellite positioning. Due to the lack of accessible, highly
maneuverable, and cost-effective methods for sampling from the water column
across a body of water, scientists often fall back to laborious manual data
acquisition campaigns [9].

##  II Background

Unmanned surface vehicles have found increasing prominence in near-shore
surface observation operations [10, 11, 12]. These inexpensive and intuitive
systems fill various niches such as persistent monitoring [13], collecting in
situ collections of both physical samples [14, 15, 16] and observational
samples [17, 18].

Sensors deployed from ships have a long history of extending observational
capabilities from manned research vessels by deploying tethered sensors [19].
ASVs have also been utilized to manage tethered underwater vehicles [20],
effectively serving as a local localization and communication station for
advanced vehicles.

We are interested in developing a budget-friendly resource for scientists to
autonomously capture data from below the surface. This makes the ASV a natural
platform from which to transport and deliver our sensing platform. Others have
employed a similar approach. In [21], the authors develop an advanced probe
which is capable of controlling its position within the water column. The
design in [22] presents a winch and probe system for collecting physical water
samples along the water column at up to $50~{}m$ depth on the [10] platform.

Our system, inspired by both [22] and [21], is a simpler and modern
alternative which utilizes commercially available components to deliver a
probe to controllable near-surface depths. We envision that this flexible and
approachable mechanical design will increase accessibility for near-shore
science missions.

Figure 3: Displacement map from Finite Element Analysis, showing highest
concentrations at the centre of each plate

(a) Controller diagram

(b) Unity simulation

(c) Configured ASV

Figure 4: Fig. 4(a) details the control system which governs the depth control
for our winch. A fully-controllable version of our system is available in our
simulator, shown in Fig. 4(b). Finally, 4(c) shows our fully-configured system
deployed.

##  III System Design

This autonomous probing system was designed with the goal of demonstrating an
approachable configuration for collecting in situ samples at depth from
surface vehicles. Although we present field results on a small catamaran-style
ASV, our modular design is compatible with many platforms.

We approached the problem of designing the probe deployment system with the
following objectives:

  * •

Size: small and lightweight to enable transport by $2-3$ persons and
transportation by commercial airline.

  * •

Compatibility: the probe is interchangeable.

  * •

Safety: the winch controller should prioritize safety of the entire system and
fail in a non-catastrophic way.

  * •

Highly Configurable: the frame and mounting system should be adaptable as new
equipment is added or old equipment is removed. This controller can be easily
integrated with any ROS-based system.

  * •

Reproducible and Open The system is developed from commercially available
parts and the winch control software is available open-source.

The system is sized to be installed on a small (human-portable) boat. The
design was built to be highly maneuverable and re-configurable so that it can
be transported both by air and in personal vehicles. The main components,
consisting of a frame, winch, and probe are detailed in Fig. 2.

###  III-A Frame

The mounting frame is constructed from the highly-configurable and anti-
corrosive $80-20$ T-slot aluminum profiles. In our experiments, the $80-20$
frame is mounted on two $0.0642$ $m^{3}$ Polyethylene pontoons. In this
configuration, the flotation is limited to $106.7~{}kg$ in the fresh water
(assumed density of 997 $kg/m^{3}$) with a safety factor of $1.2$ based on the
buoyancy force as described by Eq. 1.

$F_{b}=\rho_{water}\times g\times V_{pvc}=1256.8N$ (1)

$W_{overall}=\dfrac{F_{b}}{S.F}\div g=106.76kg$ (2)

There are several key advantages in the design of the mounting frame. The
horizontal T-slot profiles attach to the vertical T-slots, meaning the number
and height of frame layers is flexible. This design choice was imperative, as
we find it is often necessary to adjust the ASV’s configuration as new
instruments are included in the system. With the $80/20$ design, we can easily
add additional $80/20$-compatible T-slots, fasteners, or T-studs when new
configurations are needed. The frame attaches to two aluminum rods which
assemble into the $1.5~{}in$ native holes in our pontoons (which are actually
commercially-available kayak stability floats). To secure the rods to the
floats, we reconfigure $1.5~{}in$ T-shaped female PVC into _endcaps_ by
drilling $8mm$ bolts into the T intersection. To attach the endcaps we press-
fit $8mm$ nuts into the inside of the aluminum rods. This ergonomic endcap
makes the boat easy to assemble, sturdy, and enables fast replacement of a
pontoon in the water if necessary.

Finite Element Analysis (FEA) was performed in Siemens’ NX software to ensure
the overall structure can withstand the applied loads. Our winch located at
the back upper layer is modeled as a box with the dimension of
$~{}40.6~{}\times~{}30.5~{}\times~{}20.3~{}cm$ and weights $5.9~{}kg$
according to the user manual. Two golf-cart batteries that weigh $50~{}kg$ in
total were placed on the bottom plate. The two $1.2192~{}m$ long aluminum
supporting rods are assumed rigid. The force exerted on the winch by the
tether and the probe and gravitational force acting on all parts were taken
into account in this analysis.

The simulated stresses on the frame, according to our described load, reaches
a maximum stress well below the ultimate strength of the material ($120~{}MPa$
for $6051$ aluminum); this indicates high reliability ($8.16$ safety factor
calculated via Eq. 3) and the option to add additional payload. We found that
the maximum stress in the system of $14.7~{}MPa$ is located where the aluminum
rods mate with the pontoons. The mount experiences the largest displacement at
the bottom plate, where the maximum displacement is $0.25~{}mm$. See Fig. 3
for more details.

$S.F=\frac{Yield\;Strength}{Maximum\;Stress}=\frac{120}{14.7}=8.16$ (3)

###  III-B Probe and Winch

The central feature of our design is the sensing probe and the winch which
controls the position of the probe in the $z$ axis. The probe facilitates
water sample collection and provides real-time observational feedback for
users, high-level controllers on the ASV (which may be important for adaptive
sampling missions), and the winch controller. The system is designed so that
the probe is interchangeable though we do require that the selected probe be
capable of communicating its observed depth through our tether. To be
compatible with the winch, the probe assembly must be negatively buoyant but
weigh less than $11~{}kg$ in air.

In our experiments, a YSI EXO1 sonde [23] (see Fig. 2(d)) is utilized as our
probe. The sonde is connected to one end of BlueRobotic’s Fathom tether [24]
by a 6-pin waterproof male-female connector. The tether provides power and a
communication interface for the sonde. To reduce external drag forces on the
cable, we install a cable support grip made of stainless steel onto end mated
to the probe. The tether length is limited to $10m$ by the diameter of the
winch spool. Future designs should consider employing either a larger spool or
a thinner cable to facilitate the collection of deeper samples.

Figure 5: This figure depicts a histogram of the observed science values when
normalized and compared to depth. All of the parameters we measured show
reliable change over depth, emphasizing the importance of measuring throughout
the water column.

The probe can be configured via the winch into two modes: _underway mode_ and
_deployed mode_. The probe is retrieved so that it sits just below the water
in a vertical orientation in the _underway_ configuration. In the _deployed_
configuration, the ASV is commanded to remain stationary to reduce strain on
the cable as the the winch controls the vertical position of the probe.

We utilise the low-cost and lightweight, _Trac G3 Fisherman $25$_ Electric
Anchor Winch (see Fig. 2(b)) for probe deployment and retrieval. The winch
ships with a nylon rope which we replace with $10~{}m$ BlueRobotics’ Fathom
tether [24] to serve as both a communication channel and a physical tether.
The cable is neutrally buoyant, has $158kg$ breaking strength, and is embedded
with water-blocking fibers to seal any leaks. The tether carries four
unshielded twisted pairs (UTP) of $26~{}AWG$ wire, which makes it compatible
with several sensors. A slip ring (see Fig. 2(c)) is used to transmit
electrical signals from a rotating cable to the on-board computer.

Fig. 2(b) shows the configuration of the winch from our _winchcam_ , a ROS-
enabled camera which captures a live view of the probe deployment system at
$30~{}fps$.

| Maximum
---
Payload (kg)
| Payout
---
Speed (m/min)
| Retrieval
---
Speed (m/min)
| Operating
---
Voltage (V)
11.340 | 21.336 | 19.812 | 12

TABLE I: APS Specifications

The APS system assumes the vehicle stops and maintains its station to deploy
and recover the probe. The controller considers the probe’s state which is
found from fusing the GPS estimate from onboard the ASV and a depth-pressure
from the the probe itself. Our closed-loop controller (see Fig. 4(a)) enables
precise depth control of the sensor by deploying or retrieving the cable
connecting the sensor to the ASV. Control signals are sent from the ASV’s
onboard computer to the an Arduino are sent through a two-channel relay to
control actuation. Our system is capable of $19.812~{}m/min$ payout.

###  III-C Electronics, Control, and ASV Integration

The electronic components of the APS enable seamless communication between the
probe, winch, and the central computing platform. In our experiments, we
powered both the winch and the sonde on independent, external batteries,
negating the need to integrate with shipboard power. The winch is controlled
using a relay module with two mechanical relays that allow ON-OFF-ON style
control in either direction. This binary system allows power to flow from
external batteries when in the _on_ position. The relays are controlled by an
Arduino-compatible microcontroller (MCU), which is exposed on the ASV’s
internal ROS network over a serial connection.

The MCU, running its own distributed ROS node, provides functionality for both
autonomous and manual control of the winch. Manual control is accomplished
through open-loop service calls to raise or lower the winch in finite steps.
In autonomous control mode, the system integrates real-time depth positions as
reported by the probe to determine winch direction using closed-loop control.

All interfaces utilize standard connectors. The tether to the probe provides a
powered connection to the boat and supports several interfaces such as RS-232
and some Ethernet standards. The serial interface to the MCU is managed over
universal serial bus (USB). These standardized connections mean that the
entire system is ”plug-and-play” and ROS compatible with any ASV that
satisfies the payload requirements.

##  IV Control Simulator

In order to develop and test our controller without deploying the boat, we
built a ROS-compatible simulation environment in Unity (see Fig:4(b)). We
release this system as an open-source codebase on  github. Our simulated
environment consists of a fully controllable ASV and winch equipped with a
probe capable of reporting its depth. Although useful for controller
evaluation, there are several limitations.

Our simulated winch assumes a constant speed of payout/retraction. We attempt
to make the probe’s simulation realistic by considering gravity, buoyancy, and
a drag force. Accordingly, the probe accelerates during payout until the
external forces are balanced. When the winch recovers the probe, it raises to
the water surface with a constant retrieval speed of 19.812$m/min$.

##  V Experiments

We validate our system in Lake Hertel $(45.54437,-73.15212)$ at the McGill,
Gault Nature Reserve in Mont-Saint-Hilaire, Quebec, Canada. For the
experiments in this paper, we demonstrate the APS with a YSI EXO
multiparameter water quality sonde, equipped sensors for turbidity,
temperature, salinity, PH, ORP, dissolved oxygen, conductivity, and depth.
When the ASV is operating in _underway mode_ , our mounting system allows
continuous data collection by positioning the sensor probe just below the
waterline. We deployed the ASV to desirable locations and deployed the probe
throughout the water column. See Fig. 6(a) for an example profile. Data from
underway experiments are presented in Fig. 6(b).

Our design was motivated by the scientific imperative to collect samples from
throughout the water column from a mobile, low-cost surface vehicle. Data
collected from the experiment site (see Fig. 5) demonstrates that our system
is capable of collecting samples of phenomena which varies both spatially
across the surface of the water and by depth. The samples are collected
efficiently and without requiring personnel to be deployed on a boat, or even
to the site given internet networking is available.

(a) Dissolved Oxygen captured from our APS in four consecutive probes.

(b) Dissolved Oxygen captured in the _underway_ mode.

Figure 6: Data resulting from field experiments at Lake Hertel. These plots
depict one of the water quality parameters measured by our sonde probe. The
data clearly varies over the surface of the lake as well as throughout the
water column.

##  VI Limitations

Although the APS allows us to reach locations that are unattainable from a
traditional ASV, there are some limitations. As there is no active depth
control on the probe itself, we can only indirectly control the probe’s
position in the $z$ plane by releasing or retracting the tether. Several times
during our field deployments, the probe settled on top of vegetation rather
than sinking to the bottom. Without an active probe, there is little our
system can do to force the probe past the vegetation and down to its target
depth.

We are also limited in our localization of the probe, since we only have its
depth and an estimate of the payout of the tether. In the current system, we
simple assign the ASV’s latitude and longitude positions to sonde readings
along with the measured depth of the probe. Future work could try to estimate
the full 6DOF pose of the probe based on images or inertial measurement
estimates.

On the controls side, the current system has no behavior for realizing when
the probe has become entangled. Instead, it employs a timeout function,
whereby if the sensor’s depth doesn’t change after a configurable number of
seconds of the winch operating in a direction, the controller assumes that the
command must be aborted. Future versions of our system should consider a more
complete version of the state and attempt to reason through possible causes of
the sensor’s stall (such as becoming entangled in vegetation).

##  VII Conclusion

This paper presents a robotic system capable of collecting in situ samples
from throughout the water column from an autonomous surface vehicle. Our
solution consists of open-source mechanical, electrical, and software design
to solve the problem of collecting samples at-depth. We demonstrate the
efficacy of the system in lake trials, capturing data that further motivates
our system. Finally, we discuss limitations and suggest future improvements.

We believe this systems fills an important niche, enabling science-critical
access to the data below the surface without requiring expensive installations
or sophisticated equipment.

##  VIII Acknowledgements

This work was generously supported by the NSERC Canadian Robotics Network
(NCRN). We would also like to thank Chris Jing and Khaled Al Masaid for their
assistance in the early development of the mechanical design.

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