# Macroscopic Wave Propagation for 2D Lattice with Random Masses

Joshua A. McGinnis

###### Abstract

We consider a simple 2D harmonic lattice with random, independent and
identically distributed masses. Using the methods of stochastic
homogenization, we show that solutions with long wave initial data converge in
an appropriate sense to solutions of an effective wave equation. The
convergence is strong and almost sure. In addition, the role of the lattice’s
dimension in the rate of convergence is discussed. The technique combines
energy estimates with powerful classical results about sub-Gaussian random
variables.

##  1 Introduction

We prove an almost sure convergence result for solutions of the following two-
dimensional spatially discrete, linear harmonic lattice with random masses in
the long wave limit i.e. as the wavelength becomes longer:

$m({\bf{j}})\ddot{u}({\bf{j}},t)=\Delta u({\bf{j}},t)$ (1.1)

where ${\bf{j}}=(j_{1},j_{2})$. The discrete Laplacian is defined as

$\Delta
u({\bf{j}},t)\coloneqq-4u({\bf{j}},t)+\sum_{i=1}^{2}u({\bf{j}}+{\bf{e}}_{i},t)+u({\bf{j}}-{\bf{e}}_{i},t),$
(1.2)

where ${\bf{e}}_{1}=(1,0)$ and ${\bf{e}}_{2}=(0,1).$

Here, $(j_{1},j_{2})\in{\bf{Z}}^{2},t\in{\bf{R}},$ and $\Delta$ is the usual
discrete Laplacian on the integer lattice ${\bf{Z}}^{2}$. The $m(j_{1},j_{2})$
are independent and identically distributed random variables (i.i.d.)
contained almost surely in some interval $[a,b]\in{\bf{R}}^{+}$. There is a
long history of studying such lattices, especially when the masses or
constants of elasticity vary periodically [1], since they can provide
simplified models for investigating physical phenomena arising in more complex
systems. We are interested in understanding to what degree approximations to
the wave equation are still valid, since random coefficients could in
principle be used to model impurities. The system is well understood when the
masses are either constant or periodic with respect to $(j_{1},j_{2})$ [12],
but for a 2D lattice most of what is known for the random problem is for weak
randomness [10] or numerical [11]. Much has already been said about the
continuous versions of (1.1). For an extensive resource, see [8]. Note that
rates of convergence are rarely addressed. Furthermore almost sure convergence
results in the discrete setting require different techniques. In the
continuous setting convergence can be achieved more directly through the law
of large numbers. In our setting, one must define what one means mean by
convergence, and this is why we prove convergence through “coarse-graining”,
which is used in [12] to prove convergence in the periodic problem. To achieve
a rigorous rate of convergence, one must have bounds on the stochastic error
terms. In [14], the law of the iterated logarithm was used. Here we use the
theory of sub-Gaussian random variables.

For initial conditions whose wavelength and amplitude is $O(\epsilon^{-1})$
with $\epsilon$ a small positive number, we prove that the $\ell^{2}$ norm of
the differences between true solutions and appropriately scaled solutions to
the wave equation is almost surely $O(\epsilon^{-1-\sigma})$, where $\sigma$
is any small positive number. While such an absolute error diverges as
$\epsilon\to 0^{+},$ it happens that this is enough to establish an almost
sure convergence of the macroscopic dynamics within the coarse-graining
setting.

The article [14] studies a similar problem on a 1D lattice. There the
constants of elasticity also vary randomly and the system is studied in the
relative coordinates. The so called multiscale method of homogeniziation, a
by-now classical tool with a long history in PDE for deriving effective
equations, see [2], is employed. Our results should be contrasted with those
in 1D. In [14], it is shown that that the coarse-graining converges at a rate
of $\sqrt{\epsilon\log\log(\epsilon^{-1})}$, and this is thought to be sharp
based on numerical evidence. Below, we show that in 2D, the rate of
convergence is no slower than $\epsilon^{1-\sigma}$ when the masses are i.i.d.
However the rate of convergence is different for layered masses i.e. where
masses are only random with respect to $j_{1}$ and constant with respect to
$j_{2}$. We explore this in Section 7.2 and in the numerical experiments in
Section 9. In such a case we show that the convergence rate is comparable to
the rate in the 1D lattice. We do not believe our estimates are sharp like
those in the 1D setting because the analysis requires the use of a cut-off
function, which demands slightly greater regularity of the initial conditions.
Still, numerical evidence suggests they are probably close to being sharp and
likely only off a logarithmic term.

Although large parts of the formal derivation are the same in 2D as they are
in 1D, difficulty arises in the fact that, as far as we can tell, in 2D no
known explicit formulation of a solution for (2.19) exists. It can however be
solved on a finite domain. Therefore a cut-off function involving $\epsilon$
is introduced and although it eventually disappears from the final Theorems
7.1 and 8.1, several new arguments need to be made, most important of which
are almost sure estimates for solutions of (3.5) involving tails of sequences
of sub-Gaussian random variables. In Section 2, we derive the effective
equations. The core probability theory is in Section 3. A common energy
estimate, akin to those found in [3], [9], and [16] is given in Section 4.
These help bound the error in terms of the “residuals” defined in (2.1).
Elementary theory of the energy of the effective wave, such as that in [7], is
given in Section 5. This theory is used to derive estimates of various norms
needed in the following Section 6. Finally, Section 8 contains the main
convergence result. The appendix contains technical proofs of several
statements that are believable enough to be skipped in a first read through.

##  2 Deriving the Effective Equations

###  2.1 Expansions

Define the “residual” to be

$\operatorname{Res}\widetilde{u}({\bf{j}},t)=m({\bf{j}})\ddot{\widetilde{u}}-\Delta\widetilde{u}({\bf{j}},t).$
(2.1)

We hope to find an approximate solution to (1.1) whose residual is as small as
possible. We look for approximate solutions of the form

$\widetilde{u}({\bf{j}},t)=\epsilon^{-1}U_{0}({\bf{j}},\epsilon{\bf{j}},\epsilon
t)+\epsilon U_{2}({\bf{j}},\epsilon{\bf{j}},\epsilon t).$ (2.2)

The amplitude of the ansatz is $O(\epsilon^{-1})$ so that the relative
displacement of the masses is $O(1)$. The relative displacement of the masses
is given by

$r_{i}({\bf{j}},t)\coloneqq u({\bf{j}}+{\bf{e}}_{i},t)-u({\bf{j}},t).$ (2.3)

These are the same initial conditions used in [12]. Here

$U_{i}:{\bf{Z}}^{2}\times{\bf{R}}^{2}\times{\bf{R}}\to{\bf{R}},$ (2.4)

and we keep track of its arguments by letting ${\bf{X}}=\epsilon{\bf{j}}$ and
$\tau=\epsilon t,$ in which case ${\bf{X}}=(X_{1},X_{2})\in{\bf{R}}^{2}.$ In
order to plug (2.2) into (1.1), we need to understand how $\Delta$ acts on
functions of this form. Note that

$\Delta
U({\bf{j}},{\bf{X}})=\sum_{i=1}^{2}U({\bf{j}}+{\bf{e}}_{i},\epsilon{\bf{j}}+\epsilon{\bf{e}}_{i})+U({\bf{j}}-{\bf{e}}_{i},\epsilon{\bf{j}}-\epsilon{\bf{e}}_{i})-2U({\bf{j}},\epsilon{\bf{j}}).$
(2.5)

The addends,

$U({\bf{j}}+{\bf{e}}_{i},\epsilon{\bf{j}}+\epsilon{\bf{e}}_{i})+U({\bf{j}}-{\bf{e}}_{i},\epsilon{\bf{j}}-\epsilon{\bf{e}}_{i})$
(2.6)

can be Taylor expanded in powers of $\epsilon$. Using this expansion in (2.5),
we have

$\Delta U={\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U+\epsilon{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{1}U+\epsilon^{2}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{2}U,$
(2.7)

where

${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U({\bf{j}},{\bf{X}})\coloneqq-2U({\bf{j}},{\bf{X}})+\sum_{i=1}^{2}U({\bf{j}}+{\bf{e}}_{i},{\bf{X}})+U({\bf{j}}-{\bf{e}}_{i},{\bf{X}}),$
(2.8) ${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{1}U({\bf{j}},{\bf{X}})\coloneqq\sum_{i=1}^{2}\partial_{X_{i}}U({\bf{j}}+{\bf{e}}_{i},{\bf{X}})-\partial_{X_{i}}U({\bf{j}}-{\bf{e}}_{i},{\bf{X}}),$
(2.9)

and

${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{2}U({\bf{j}},{\bf{X}})\coloneqq\frac{1}{2}\sum_{i=1}^{2}\partial_{X_{i}X_{i}}U({\bf{j}}+{\bf{e}}_{i},{\bf{X}})+\partial_{X_{i}X_{i}}U({\bf{j}}-{\bf{e}}_{i},{\bf{X}}).$
(2.10)

Everything remaining in the expansion is $O(\epsilon^{3})$ so we neglect it
for now. Now we calculate $\operatorname{Res}{\widetilde{u}}$ using (2.7).
Again grouping terms by powers of $\epsilon$ yields

$\displaystyle\operatorname{Res}\widetilde{u}({\bf{j}},{\bf{X}},\tau)=$
$\displaystyle-\epsilon^{-1}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U_{0}({\bf{j}},{\bf{X}},\tau)$
(2.11) $\displaystyle-{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{1}U_{0}({\bf{j}}-{\bf{e}}_{i},{\bf{X}},\tau)$
$\displaystyle+\epsilon
m({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{j}},{\bf{X}},\tau)-\epsilon\sum_{i=0}^{1}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{2i}U_{2-2i}({\bf{j}},{\bf{X}},\tau)$
$\displaystyle+O(\epsilon^{2}).$

Since we want $\operatorname{Res}\widetilde{u}$ to be small, we solve for
$U_{0}$ and $U_{2}$ so that formally the residual is $O(\epsilon^{2}).$
Assuming we seek bounded solutions, the $O(\epsilon^{-1})$ term implies

${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U_{0}({\bf{j}},{\bf{X}},\tau)=0.$
(2.12)

We can meet (2.12) by putting

$U_{0}({\bf{j}},{\bf{X}},\tau)=\bar{U}_{0}({\bf{X}},\tau)$ (2.13)

i.e. $U_{0}$ does not depend on the microscopic component. From (2.13), it is
seen that

${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{1}U_{0}\equiv
0$ (2.14)

and that

${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{2}U_{0}=\Delta_{\bf{X}}U_{0}\coloneqq\sum_{i=1}^{2}\partial_{X_{i}X_{i}}U_{0}.$
(2.15)

With $U_{0}$ constant in ${\bf{j}}$, the residual in (2.11) simplifies to

$\displaystyle\operatorname{Res}\widetilde{u}({\bf{j}},{\bf{X}},\tau)=$
$\displaystyle\epsilon
m({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\epsilon\Delta_{{\bf{X}}}U_{0}({\bf{X}},\tau)-\epsilon{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{2}U_{0}({\bf{j}},{\bf{X}},\tau)+O(\epsilon^{2}).$
(2.16)

To make the $O(\epsilon)$ terms vanish, we need

$m({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\Delta_{\bf{X}}U_{0}({\bf{X}},\tau)={\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U_{2}({\bf{j}},{\bf{X}},\tau).$
(2.17)

Let $z({\bf{j}})\coloneqq m({\bf{j}})-\bar{m}$, and
$\bar{m}\coloneqq{\mathbb{E}}[m({\bf{j}})]$ where ${\mathbb{E}}$ is the
expected value with respect to the probability measure of the i.i.d. lattice
of masses. One way to solve (2.17) is to write it as

$\bar{m}\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\Delta_{\bf{X}}U_{0}({\bf{X}},\tau)={\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U_{2}({\bf{j}},{\bf{X}},\tau)-z({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau),$
(2.18)

and then pick $U_{0}$ and $U_{2}$ to force the left hand side and right hand
side to vanish independently. From the left hand side, we find $U_{0}$ solves
an effective wave equation. In this case, the wave speed $\bar{m}$ is guessed
to be the correct one. For a more probabilistic derivation, see Subsection 2.2
below.

Solving the right hand side by picking $U_{2}$ would be easy if only we had
$\chi:{\bf{Z}}^{2}\to{\bf{R}}$ s.t.

$\Delta\chi({\bf{j}})=z({\bf{j}}).$ (2.19)

With such a solution we have

$U_{2}({\bf{j}},{\bf{X}},\tau)=\chi({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{j}},\tau).$
(2.20)

From this, the approximate solution would be

$\widetilde{u}({\bf{j}},{\bf{X}},\tau)=\epsilon^{-1}U_{0}({\bf{j}},\tau)+\epsilon\chi({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau).$
(2.21)

In order to control the magnitude of the residual, and to prove that
$\epsilon^{-1}U_{0}$ is the dominant term in the approximation, we need to
have asymptotic bounds on $\chi$. However, we do not even know how to solve
for such a $\chi$. To circumnavigate this issue, we resort to solving
$\eqref{Chi Easy}$ where the support of $z$ is on a finite domain dependent
upon $\epsilon$. We call this restricted solution $\chi_{r}$. Justifying that

$\widetilde{u}({\bf{j}},{\bf{X}},\tau)=\epsilon^{-1}U_{0}({\bf{j}},\tau)+\epsilon\chi_{r}({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau),$
(2.22)

where $U_{0}$ solves the wave equation, is a valid approximation by finding
almost sure asymptotic bounds on $\chi_{r}$ in $\epsilon$ is the main novelty
of this paper.

###  2.2 Averaging

Another way to get at (2.17) is by averaging. We make another ad hoc
assumption, this time on $U_{2}$. We require that

$\lim_{|{\bf{j}}|_{\infty}\to\infty}\dfrac{U_{2}({\bf{j}},{\bf{X}},\tau)}{\lvert{\bf{j}}\rvert_{\infty}}=0$
(2.23)

where

$\lvert{\bf{j}}\rvert_{\infty}\coloneqq\max\\{|j_{1}|,|j_{2}|\ |\
{\bf{j}}=(j_{1},j_{2})\\}.$ (2.24)

We have introduced the $\infty$ norm here for technical reasons. Henceforth
${\bf{0}}$ is the vector $(0,0).$ We define disks

$D({\bf{j}}_{0},r)\coloneqq\\{{\bf{j}}\in{\bf{Z}}^{2}\ |\
|{\bf{j}}-{\bf{j}}_{0}|_{\infty}\leq r\\}.$ (2.25)

The boundary of these disks are given by

$\delta D({\bf{j}}_{0},r)=\\{{\bf{j}}\ |\
|{\bf{j}}-{\bf{j}}_{0}|_{\infty}=r\\}.$ (2.26)

For any set $U\subset{\bf{Z}}^{2}$, $\lvert U\rvert$ means the number of
elements in the set. With the assumption (2.23), one may argue that for
$U_{2}$ satisfying (2.17) to exist for all ${\bf{X}}$ and $\tau$, it must be
that its spatial average is $0$ i.e.

$\lim_{r\to\infty}\frac{1}{|D({\bf{0}},r)|}\sum_{{\bf{j}}\in
D({\bf{0}},r)}m({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\Delta_{{\bf{X}}}U_{0}({\bf{X}},\tau)=0,$
(2.27)

but by the law of large numbers

$\displaystyle\lim_{r\to\infty}\frac{1}{|D({\bf{0}},r)|}\sum_{{\bf{j}}\in
D({\bf{0}},r)}m({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\Delta_{{\bf{X}}}U_{0}({\bf{X}},\tau)$
(2.28) $\displaystyle=$
$\displaystyle\bar{m}\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\Delta_{\bf{X}}U_{0}({\bf{X}},\tau).$

Combining (2.27) with (2.28) yields the same effective wave equation as seen
in the left hand side of 2.18,

$\bar{m}\partial_{\tau\tau}U_{0}({\bf{X}},\tau)-\Delta_{\bf{X}}U_{0}({\bf{X}},\tau)=0.$
(2.29)

We now subtract (2.29) from (2.17) yielding

$z({\bf{j}})\partial_{\tau\tau}U_{0}({\bf{X}},\tau)={\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\displaystyle\Delta$}}}}{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=8.33336pt}$}}}}\cr\hbox{$\textstyle\Delta$}}}}{{\ooalign{\hbox{\raise
6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=5.83334pt}$}}}}\cr\hbox{$\scriptstyle\Delta$}}}}{{\ooalign{\hbox{\raise
5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=4.16667pt}$}}}}\cr\hbox{$\scriptscriptstyle\Delta$}}}}}_{0}U_{2}({\bf{j}},{\bf{X}},\tau)$
(2.30)

which is the right hand side of (2.18).

##  3 Probabilistic Estimates

The intuition for the next step is that we need only solve (2.19) on a domain
which is accessible to the wave i.e. since we only care about $|t|\leq
T\epsilon^{-1}$, the approximation works without having a solution to (2.19)
outside a radius proportional to $T\epsilon{-1}$. Given enough regularity of
the initial conditions, only a negligible amount of the wave will have
traveled outside this radius. Therefore we can use a spatial cutoff and well
known results of sub-Gaussian random variables to obtain bounds on $\chi$. The
form of the estimates in this section are motivated by the error estimates in
the following sections, specifically Section 6.

###  3.1 Poisson Problem with Random Source

Let $I:{\bf{Z}}^{2}\to\\{0,1\\}$ s.t.

$I({\bf{j}})=\begin{cases}1&{\bf{j}}=0\\\ 0&{\bf{j}}\neq 0\end{cases}$ (3.1)

and $\varphi:{\bf{Z}}^{2}\to{\bf{R}}$ s.t.

$\Delta\varphi({\bf{j}})=I({\bf{j}}).$ (3.2)

Then $\varphi$ is the fundamental solution and it is known that
$\varphi({\bf{0}})=0$, and for ${\bf{j}}\neq 0$

$\varphi({\bf{j}})=\frac{1}{2\pi}\log|{\bf{j}}|+C_{0}+O(|{\bf{j}}|^{-2}),\
|{\bf{j}}|\to\infty$ (3.3)

where $|{\bf{j}}|\coloneqq\sqrt{j_{1}^{2}+j_{2}^{2}}.$ The proof of the
existence and uniqueness of $\varphi$ can be found in [5]. In (3.3), $\log$
may be replaced with $\log^{+}$ 111$\log^{+}(x)=\max\\{0,\log(x)\\}$ and
$\log^{+}(0)=0$., so all logs in the sequel should be thought of as
$\log^{+}$, even if we neglect the plus sign.

Define

$\chi_{r}({\bf{j}})\coloneqq\sum_{{\bf{k}}\in
D({\bf{0}},r)}\varphi({\bf{j}}-{\bf{k}})z({\bf{k}})$ (3.4)

where the $z({\bf{k}})$ and $D({\bf{0}},r)$ are defined in Subsection 2.1. The
following is true

$\Delta\chi_{r}({\bf{j}})=\begin{cases}z({\bf{j}})&{\bf{j}}\in
D({\bf{0}},r)\\\ 0&{\bf{j}}\in{\bf{Z}}^{2}/D({\bf{0}},r)\end{cases}.$ (3.5)

For any generic linear operator $\mathcal{L}$ acting on functions of
${\bf{Z}}^{2}$, we have

$\mathcal{L}(\chi_{r})({\bf{j}})=\sum_{{\bf{k}}\in
D({\bf{0}},r)}\mathcal{L}(\varphi)({\bf{j}}-{\bf{k}})z({\bf{k}}).$ (3.6)

###  3.2 Estimates on the Solution

Recall that $z({\bf{j}})\in[a-\bar{m},b-\bar{m}]$ almost surely and are mean
$0$. Therefore Hoeffding’s Lemma [13] tells us that for all $s\in{\bf{R}}$

${\mathbb{E}}\left[\exp({s{\bf{z}}({\bf{j}})})\right]\leq\exp\left({\frac{s^{2}(a-b)^{2}}{8}}\right).$
(3.7)

Moreover

${\mathbb{E}}\left[\exp\left({s\mathcal{L}(\varphi)({\bf{j}}-{\bf{k}}){\bf{z}}({\bf{j}})}\right)\right]\leq\exp\left({\frac{s^{2}(\mathcal{L}(\varphi)({\bf{j}}-{\bf{k}}))^{2}(a-b)^{2}}{8}}\right).$
(3.8)

By the independence of $z({\bf{j}})$

${\mathbb{E}}\left[\exp\left(s\mathcal{L}(\chi_{r})({\bf{j}})\right)\right]\leq\exp\left(\frac{s^{2}\left\|\mathcal{L}\varphi\right\|_{D({\bf{j}},r)}^{2}(a-b)^{2}}{8}\right)$
(3.9)

where

$\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}\coloneqq\sum_{{\bf{k}}\in
D({\bf{j}},r)}(\mathcal{L}(\varphi)({\bf{k}}))^{2}=\sum_{{\bf{k}}\in
D(0,r)}(\mathcal{L}(\varphi)({\bf{j}}-{\bf{k}}))^{2}.$ (3.10)

From (3.9), it follows that the $\chi_{r}({\bf{j}})$ are sub-Gaussian, with
proxy variance given by
$\frac{(a-b)^{2}}{4}\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}.$
Letting $P$ be the probability measure of the random masses, a common estimate
on sub-Gaussian random variables, also found in [13], says

$P(|\mathcal{L}(\chi_{r})({\bf{j}})|\geq t)\leq
2\exp\left(\frac{-2t^{2}}{(a-b)^{2}\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}}\right).$
(3.11)

To make use of (3.11), we order the $\chi_{r}({\bf{j}})$ into a sequence.
First, restrict $r$ to take on only values ${\bf{N}}^{+}/\\{1\\}$. We map $r$
and ${\bf{j}}$ to a set of indices using a bijection
$I:{\bf{Z}}^{2}\times{\bf{N}}^{+}/\\{1\\}\to{\bf{N}}^{+}$ which satisfies the
inequality

$I({\bf{j}},r)\leq C\max\\{r^{3},|{\bf{j}}|^{3}\\}.$ (3.12)

Such a bijection is possible by Lemma 11.1. We let

$\mathcal{L}(\chi)(n)\coloneqq\mathcal{L}(\chi_{r})({\bf{j}})\ \text{and}\
\left\|\mathcal{L}\varphi\right\|^{2}_{D(n)}\coloneqq\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}$
(3.13)

when $I({\bf{j}},r)=n$. One sees we still have (3.11) as

$P(|\mathcal{L}(\chi)(n)|\geq t)\leq
2\exp\left(\frac{-2t^{2}}{\left\|\mathcal{L}\varphi\right\|^{2}_{D(n)}(a-b)^{2}}\right).$
(3.14)

Using the Borel-Cantelli Lemma, [6], we have that for all but finitely many
$n$

$|\mathcal{L}(\chi)(n)|\leq\sqrt{(a-b)^{2}\left\|\mathcal{L}\varphi\right\|^{2}_{D(n)}\log(n)}$
(3.15)

almost surely, but (3.15) and (3.12) imply that there exists $C_{\omega}$
almost surely s.t.

$|\mathcal{L}(\chi_{r})({\bf{j}})|\leq
C_{\omega}\sqrt{\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}\left(\log(|{\bf{j}}|)+\log(r)\right)}.$
(3.16)

For $\varphi$ given by (3.3) we find that for some constant $C$

$\left\|\varphi\right\|^{2}_{D({\bf{j}},r)}\leq Cr^{2}\log(|{\bf{j}}|+r)^{2}.$
(3.17)

A quartet of $\mathcal{L}$ operators we need to consider is

$S_{i}^{\pm}(\xi)({\bf{j}})\coloneqq\xi({\bf{j}}\pm{\bf{e}}_{i}).$ (3.18)

Notice that for $S_{i}^{\pm}$, there exists a $C$ s.t.

$\left\|S^{\pm}_{i}\varphi\right\|^{2}_{D({\bf{j}},r)}\leq
Cr^{2}\log(|{\bf{j}}|+r)^{2}.$ (3.19)

Another pair of $\mathcal{L}$ operators we need to consider is $\delta_{i}$
which we define as

$\delta_{i}(\varphi)({\bf{j}})\coloneqq\varphi({\bf{j}}+{\bf{e}}_{i})-\varphi({\bf{j}}-{\bf{e}}_{i}).$
(3.20)

From Lemma 11.3, there exists a $C$ s.t.

$|\delta_{i}(\varphi)({\bf{j}})|\leq\frac{C}{|{\bf{j}}|+1}.$ (3.21)

From Lemma 11.4 there exists another constant $C$ s.t.

$\left\|\delta_{i}\varphi\right\|^{2}_{D({\bf{j}},r)}\leq
C\log(|{\bf{j}}|+r).$ (3.22)

From (3.16), (3.17), and (3.22) we get the following theorem.

######  Theorem 3.1.

Let $\\{z({\bf{j}})\\}_{{\bf{j}}\in{\bf{Z}}^{2}}$ be independent, mean zero
random variables with $z({\bf{j}})\in[a-\bar{m},b-\bar{m}]\subset R$ a.s. Let
$r\geq 2$ be any real number, and $\chi_{r}({\bf{j}})$ be as defined in (3.4)
with $\varphi$ defined by (3.2). Then there almost surely exists a constant
$C_{\omega}$ s.t.

$|\chi_{r}({\bf{j}})|\leq
C_{\omega}r\left(\log(|{\bf{j}}|)^{\frac{3}{2}}+\log(|r|)^{\frac{3}{2}}\right),$
(3.23) $|S_{i}^{\pm}(\chi_{r})({\bf{j}})|\leq
C_{\omega}r\left(\log(|{\bf{j}}|)^{\frac{3}{2}}+\log(|r|)^{\frac{3}{2}}\right)$
(3.24)

and

$|\delta_{i}(\chi_{r})({\bf{j}})|\leq
C_{\omega}\left(\log(|{\bf{j}}|)+\log(|r|)\right).$ (3.25)

######  Remark 1.

Note that the definition in (3.4) implies $\chi_{r}=\chi_{\lfloor r\rfloor}$,
and thus the theorem holds for all real $r\geq 2$, even though $r$ was
originally an integer in the bijection in (3.12).

######  Remark 2.

The $\omega$ indicates that the constant depends on the realization of the
masses.

###### Proof.

Insert (3.19) and (3.22) into (3.16). After sweeping any constants into
$C_{\omega}$, one has

$|\chi_{r}({\bf{j}})|\leq
C_{\omega}r\log(|{\bf{j}}|+r)\sqrt{\left(\log(|{\bf{j}}|)+\log(r)\right)}$
(3.26)

and

$|\delta_{i}(\chi_{r})({\bf{j}})|\leq
C_{\omega}\sqrt{\log(|{\bf{j}}|+r)\left(\log(|{\bf{j}}|)+\log(r)\right)}.$
(3.27)

An elementary argument in Lemma 11.2 yields

$|\chi_{r}({\bf{j}})|\leq
C_{\omega}r\left(\log(|{\bf{j}}|)+\log(r)\right)^{\frac{3}{2}}$ (3.28)

and

$|\delta_{i}(\chi_{r})({\bf{j}})|\leq
C_{\omega}\left(\log(|{\bf{j}}|)+\log(r)\right).$ (3.29)

A typical argument using convexity of $(\cdot)^{\frac{3}{2}}$ and sweeping
constants into $C_{\omega}$ now gives the result. ∎

##  4 Lattice Energy Argument

###  4.1 Error Estimate

We want the approximate solution to be good for $t$ s.t.
$|t|\leq\frac{T}{\epsilon}$. Recall that the wave travels at an effective
speed given by

$c\coloneqq(\bar{m})^{-\frac{1}{2}}$ (4.1)

according to (2.29). Therefore, let

$R_{\epsilon}\coloneqq\frac{cT+\epsilon^{-\sigma}}{\epsilon}+1$ (4.2)

where $\sigma$ is any small positive real number. Recall that $\tau=\epsilon
t$ and ${\bf{X}}=\epsilon{\bf{j}}$. Then the approximate solution has the form
of (2.22) and is

$\widetilde{u}({\bf{j}},t)=\epsilon^{-1}U({\bf{X}},\tau)+\epsilon\chi_{R_{\epsilon}}({\bf{j}})\partial_{\tau\tau}U({\bf{X}},\tau),$
(4.3)

where $U$ satisfies (2.29) and $\chi_{R_{\epsilon}}$ is defined by (3.4) so
that $\chi_{R_{\epsilon}}$ solves (3.5). Define the “error”

$\xi({\bf{j}},t)\coloneqq\widetilde{u}({\bf{j}},t)-u({\bf{j}},t).$ (4.4)

Using (1.1) we find

$m({\bf{j}})\ddot{\xi}({\bf{j}},t)=\Delta\xi({\bf{j}},t)+\operatorname{Res}{\widetilde{u}}({\bf{j}},t)$
(4.5)

where $\operatorname{Res}{\widetilde{u}}$ is given by $\eqref{Res}.$ Before
going further it is useful to define some other operators similar to
$\delta_{i}$ given by (3.20). We also have the two partial difference
operators

$\delta_{i}^{\pm}(\xi)({\bf{j}})\coloneqq\pm(\xi({\bf{j}}\pm{\bf{e}}_{i})-\xi({\bf{j}})).$
(4.6)

The energy of the error is given by

$H(t)\coloneqq\frac{1}{2}\sum_{{\bf{j}}\in{\bf{Z}}^{2}}m({\bf{j}})\dot{\xi}({\bf{j}},t)^{2}+(\delta^{+}_{1}\xi({\bf{j}},t))^{2}+(\delta^{+}_{2}\xi({\bf{j}},t))^{2}.$
(4.7)

Differentiating gives

$\dot{H}(t)=\sum_{{\bf{j}}\in{\bf{Z}}^{2}}\left(m({\bf{j}})\dot{\xi}({\bf{j}},t)\ddot{\xi}({\bf{j}},t)+\sum_{i=1}^{2}\delta^{+}_{i}\xi({\bf{j}},t)\delta^{+}_{i}\dot{\xi}({\bf{j}},t)\right).$
(4.8)

We insert (4.5)

$\dot{H}(t)=\sum_{{\bf{j}}\in{\bf{Z}}^{2}}\left(\dot{\xi}({\bf{j}},t)(\Delta\xi({\bf{j}},t)+\operatorname{Res}{\widetilde{u}}({\bf{j}},t))+\sum_{i=1}^{2}\delta^{+}_{i}\xi({\bf{j}},t)\delta^{+}_{i}\dot{\xi}({\bf{j}},t)\right).$
(4.9)

Via summation by parts

$\sum_{{\bf{j}}\in{\bf{Z}}^{2}}\left(\dot{\xi}({\bf{j}},t)\Delta\xi({\bf{j}},t)+\sum_{i=1}^{2}\delta^{+}_{i}\xi({\bf{j}},t)\delta^{+}_{i}\dot{\xi}({\bf{j}},t)\right)=0.$
(4.10)

Hence (4.9) with Cauchy-Schwarz becomes

$\dot{H}(t)=\sum_{{\bf{j}}\in{\bf{Z}}^{2}}\dot{\xi}({\bf{j}},t)\operatorname{Res}{\widetilde{u}}({\bf{j}},t)\leq\left\|\dot{\xi}(\cdot,t)\right\|_{\ell^{2}}\left\|\operatorname{Res}{\widetilde{u}}(\cdot,t)\right\|_{\ell^{2}}$
(4.11)

The $\ell^{2}$ norm,
$\left\|\dot{\xi},\delta^{+}_{1}\xi,\delta^{+}_{2}\xi\right\|_{\ell^{2}}^{2},$
and the energy, $H$ are equivalent, and so there exists a constant $C$
depending on $a$ and $b$ from (1.1) s.t. (4.11) becomes

$\dot{H}(t)(H(t))^{-\frac{1}{2}}\leq
C\left\|\operatorname{Res}{\widetilde{u}}(\cdot,t)\right\|_{\ell^{2}}.$ (4.12)

Integrating from $0$ to $t$ for any $|t|\leq\frac{T}{\epsilon}$ gives

$\left(H(t)\right)^{-\frac{1}{2}}\leq\left(H(0)\right)^{\frac{1}{2}}+\epsilon^{-1}CT\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}(\cdot,t)\right\|_{\ell^{2}}.$
(4.13)

By the equivalence of norms and (4.13)

$\left\|\dot{\xi}(\cdot,t),\delta_{1}^{+}\xi(\cdot,t),\delta_{2}^{+}\xi(\cdot,t)\right\|_{\ell^{2}}\leq
C\left\|\dot{\xi}(\cdot,0),\delta_{1}^{+}\xi(\cdot,0),\delta_{2}^{+}\xi(\cdot,0)\right\|_{\ell^{2}}+\epsilon^{-1}CT\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}(\cdot,t)\right\|_{\ell^{2}}$
(4.14)

The bound (4.14) is valid for all $t$ s.t. $|t|\leq\frac{T}{\epsilon}$. Let us
suppose that

$\left\|\dot{\xi}(\cdot,0),\delta_{1}^{+}\xi(\cdot,0),\delta_{2}^{+}\xi(\cdot,0)\right\|_{\ell^{2}}\leq\epsilon^{-1}CT\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}(\cdot,t)\right\|_{\ell^{2}}.$
(4.15)

Furthermore let

$p({\bf{j}},t)\coloneqq\dot{u}({\bf{j}},t)\ \text{and}\
r_{i}({\bf{j}},t)\coloneqq\delta_{i}^{+}u({\bf{j}},t)$ (4.16)

with the analogous definitions of $\widetilde{p}$ and $\widetilde{r}_{i}$.
These are the velocities and relative displacements of the masses. Then, for
some constant $C$ dependent on $T$, (4.14) becomes

$\sup_{|t|\leq\epsilon^{-1}T}\left\|p-\widetilde{p},r_{1}-\widetilde{r}_{1},r_{2}-\widetilde{r}_{2}\right\|_{\ell^{2}}\leq\epsilon^{-1}C\left(\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}\right\|_{\ell^{2}}\right).$
(4.17)

From (4.17) and (4.16), we also have by integrating that

$\sup_{|t|\leq\epsilon^{-1}T}\left\|u-\widetilde{u}\right\|_{\ell_{2}}\leq\epsilon^{-2}C\left(\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}\right\|_{\ell^{2}}\right).$
(4.18)

Controlling the residual is essential to proving our main theorem. Thus we are
concerned with proving the following proposition throughout the next couple of
sections.

######  Proposition 4.1.

Suppose the $m(j)$ are i.i.d. random variables contained in some interval
$[a,b]\subset{\bf{R}}^{+}$ almost surely. Then almost surely there exists a
constant $C_{\omega}(T,a,b,\bar{m})$ s.t. (4.19)

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}\widetilde{u}\right\|\leq\epsilon^{1-\sigma}C_{\omega}\log(\epsilon^{-1-\sigma})^{\frac{3}{2}}\left\|\Delta\phi,\psi\right\|_{H^{5}_{\sigma}}.$
(4.19)

The norm $\left\|\Delta\phi,\psi\right\|_{H^{5}_{\sigma}}$ is a norm on the
initial conditions of $U({\bf{X}},\tau)$ given in Subsection 5.1. The norm is
defined in Subsection 5.4. Crucially, it does not depend on $\epsilon$.

###  4.2 Calculating the Residual

Now we calculate (2.1) for our approximate solution given by (4.3). We do not
write the arguments of functions when we do not need, but recall that
${\bf{X}}=\epsilon j$ and $\tau=\epsilon t.$ Recall the operators defined in
(3.18), (3.20), and (4.6). We also have

$\Delta_{i}u\coloneqq S_{i}^{+}u-2u+S_{i}^{-}u$ (4.20)

and note that

$\Delta u=\Delta_{1}u+\Delta_{2}u.$ (4.21)

Next calculate using Lemma 11.5 that

$\Delta\widetilde{u}=\epsilon^{-1}\Delta
U+\epsilon\Delta(\chi_{R_{\epsilon}})U_{\tau\tau}+\epsilon\sum_{i=1}^{2}\delta_{i}(\chi_{R_{\epsilon}})\delta_{i}^{-}(U_{\tau\tau})+S_{i}^{+}(\chi_{R_{\epsilon}})\Delta_{i}(U_{\tau\tau}).$
(4.22)

For $D({\bf{0}},R_{\epsilon})\subset{\bf{Z}}^{2}$, we denote the indicator of
$D({\bf{0}},R_{\epsilon})$ as $\mathbb{I}_{D({\bf{0}},R_{\epsilon})}$. The
complement of $D({\bf{0}},R_{\epsilon})$ is $D({\bf{0}},R_{\epsilon})^{c}.$
Using (3.5) in (4.22)

$\Delta\widetilde{u}=\epsilon^{-1}\Delta U+\epsilon
z\mathbb{I}_{D({\bf{0}},R_{\epsilon})}U_{\tau\tau}+\epsilon\sum_{i=1}^{2}\delta_{i}(\chi_{R_{\epsilon}})\delta_{i}^{-}(U_{\tau\tau})+S_{i}^{+}(\chi_{R_{\epsilon}})\Delta_{i}(U_{\tau\tau}).$
(4.23)

Recall that ${\bf{z}}({\bf{j}})=m({\bf{j}})-\bar{m}$ and that $U$ solves
(2.29). Then (4.23) becomes

$\Delta\widetilde{u}=\epsilon^{-1}\Delta
U-\epsilon\Delta_{\bf{X}}U+\epsilon\left(mU_{\tau\tau}-z\mathbb{I}_{D({\bf{0}},R_{\epsilon})^{c}}U_{\tau\tau}+\sum_{i=1}^{2}\delta_{i}(\chi_{R_{\epsilon}})\delta_{i}^{-}(U_{\tau\tau})+S_{i}^{+}(\chi_{R_{\epsilon}})\Delta_{i}(U_{\tau\tau})\right)$
(4.24)

where $D({\bf{0}},R_{\epsilon})^{c}$ denotes the complement of the original
set. Plugging into (2.1) we have

$\displaystyle\operatorname{Res}{\widetilde{u}}=$
$\displaystyle\epsilon^{-1}(\epsilon^{2}\Delta_{\bf{x}}U-\Delta
U)+\epsilon\left(z\mathbb{I}_{D({\bf{0}},R_{\epsilon})^{c}}U_{\tau\tau}-\sum_{i=1}^{2}\delta_{i}(\chi_{R_{\epsilon}})\delta_{i}^{-}(U_{\tau\tau})+S_{i}^{+}(\chi_{R_{\epsilon}})\Delta_{i}(U_{\tau\tau})\right)$
(4.25) $\displaystyle+\epsilon^{3}m\chi_{R_{\epsilon}}U_{\tau\tau\tau\tau}.$

##  5 The Effective Wave

###  5.1 Initial Conditions

We specify the initial conditions for (1.1). For smooth enough functions
$\phi,\psi:{\bf{R}}^{2}\to{\bf{R}}$, we let

$u({\bf{j}},0)=\epsilon^{-1}\phi(\epsilon{\bf{j}})\ \text{and}\
\dot{u}({\bf{j}},0)=\psi(\epsilon{\bf{j}}).$ (5.1)

With these initial conditions, the initial relative displacements and velocity
as defined in (4.16) are roughly $\epsilon^{-1}$ in $\ell^{2}$. If the initial
conditions for $\widetilde{u}$ are defined analogously to (5.1) with
$\widetilde{\phi}$ and $\widetilde{\psi},$ then in order to satisfy (4.15), we
need informally that

$\left\|\phi-\widetilde{\phi},\psi-\widetilde{\psi}\right\|_{\ell^{2}}\leq\epsilon
C.$ (5.2)

Let us suppose that (4.15) is satisfied so that we do not need to worry about
the disparity of the initial conditions. Therefore, to save on writing, $\phi$
and $\psi$ can simply refer to the initial conditions of either the true or
approximate solution for now.

Suppose we have the following

$U({\bf{X}},0)=\phi({\bf{X}})\ \text{and}\
\partial_{\tau}U({\bf{X}},0)=\psi({\bf{X}})$ (5.3)

Then $\epsilon^{-1}U({\bf{X}},\tau)$ satisfies (2.29) and so according to
(4.3) excluding the higher order term, which we show later is small enough to
ignore, we have

$\widetilde{u}({\bf{j}},t)=\epsilon^{-1}U({\bf{X}},\tau).$ (5.4)

This is the correct approximate solution in that it satisfies (2.29) and
(5.1).

###  5.2 The Energy

Recall that $c$ is the effective wave speed defined in (4.1). In order to
bound the terms in (4.25), we need to control $L^{2}$ norms of derivatives of
$U$ in terms of the initial conditions. This can be done using arguments
involving the energy of $U$. For
$V({\bf{X}},\tau):{\bf{R}}^{2}\times{\bf{R}}\to{\bf{R}}^{2^{k}}$, let $DV$ be
the total derivative of $V$ with respect to its $2$ spatial components. In
particular $D$ is the gradient, $\nabla$, of scalar valued functions. For each
$j\in\\{0,1\dots,2^{k}\\},$ define

$E(V_{j})(\tau)\coloneqq\frac{1}{2}\int_{{\bf{R}}^{2}}\left(\partial_{\tau}V_{j}\right)^{2}+c^{2}\left|\nabla\left(V_{j}\right)\right|^{2}d{\bf{X}}.$
(5.5)

Then $E(V)(\tau)$ represents a $2^{k}$ dimensional vector of energies. For $U$
satisfying (2.29) with initial conditions (5.3), it is well known, see for
instance [4] or [7], that for all $\tau$ that

$\left|E\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq
C\left(\left\|\nabla\phi\right\|_{H^{i+k}}^{2}+\left\|\psi\right\|_{H^{i+k}}^{2}\right).$
(5.6)

The proof is also provided in Lemma 11.6 in the Appendix. The energy of
derivatives of $U$ (5.5) is equivalent to $L^{2}$ norms of derivatives of $U$.

$\sum_{j=1}^{k}\left\|D^{j}\frac{\partial^{i}U}{\partial\tau^{i}}\right\|^{2}_{L^{2}}\leq
C\sum_{j=0}^{k-1}\left|E\left(D^{j}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq
C\left(\left\|\nabla\phi\right\|^{2}_{H^{i+k-1}}+\left\|\psi\right\|^{2}_{H^{i+k-1}}\right).$
(5.7)

The constant depends on $\bar{m}$ and we see that $\nabla\phi\in H^{i+k-1}$
and $\psi\in H^{i+k-1}.$

###  5.3 Weighted Energy

In (4.25), there are terms where $U$ is “weighted” by some version of $\chi$.
By 3.24, the spatially varying aspect of these can be bounded by functions
involving logarithms. We need a way to deal with these weights in a continuous
setting. We can obviously find a smooth function
$w({\bf{X}}):{\bf{R}}^{d}\to{\bf{R}}$ s.t.

$w({\bf{X}})=\log(|{\bf{X}}|+1)^{\frac{3}{2}}+1\ \forall\ |{\bf{X}}|\geq 1\
\text{and}\ w({\bf{X}})\geq 1,$ (5.8)

and for all $n\in{\bf{N}}^{+}$ there exists a constant $B$ s.t.

$\left|D^{n}w\right|\leq B\quad\text{and}\quad\left|D(w^{2})\right|\leq B.$
(5.9)

In the same context as (5.2), the possibly vector valued “weighted energy” is
given by

$E_{w}(V_{j})(\tau)\coloneqq\frac{1}{2}\int_{{\bf{R}}^{2}}w^{2}\left(\partial_{\tau}V_{j}\right)^{2}+c^{2}w^{2}\left|\nabla\left(V_{j}\right)\right|^{2}d{\bf{X}}.$
(5.10)

According to Lemma 11.7, for all $\tau$

$\left|E_{w}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq|\tau|C\left(\left\|\nabla\phi\right\|^{2}_{H_{w}^{i+k}}+\left\|\psi\right\|^{2}_{H_{w}^{i+k}}\right),$
(5.11)

where from (5.8)

$\left\|\psi\right\|_{H^{k}}\leq\left\|\psi\right\|_{H^{k}_{w}}\coloneqq\sum_{j=0}^{k}\left\|wD^{j}\psi\right\|_{L^{2}}.$
(5.12)

The constant depends on $\bar{m}$ as well as $B$. Assumptions (5.8) and (5.9)
on $w$ also give us

$\left\|D^{k}\left(w\frac{\partial^{i}U}{\partial\tau^{i}}\right)\right\|_{L^{2}}\leq
C\sum_{j=0}^{k}\left\|wD^{j}\frac{\partial^{i}U}{\partial\tau^{i}}\right\|_{L^{2}}\leq
C\sum_{j=0}^{k}\sqrt{\left|E_{w}\left(D^{j}\frac{\partial^{i-1}U}{\partial\tau^{i-1}}\right)\right|}.$
(5.13)

One sees from (5.11) and (5.13) that for $i\geq 1$

$\left\|w\frac{\partial^{i}U}{\partial\tau^{i}}\right\|_{H^{k}}\leq
C\sqrt{|\tau|}\left(\left\|\nabla\phi\right\|_{H_{w}^{i+k-1}}+\left\|\psi\right\|_{H_{w}^{i+k-1}}\right)$
(5.14)

which holds for all $\tau.$

###  5.4 Tail Energy

We have already defined disks, so to be very clear, we let

$B(r)=\\{{\bf{X}}\in{\bf{R}}^{2}\ |\ |{\bf{X}}|<r\\}.$ (5.15)

Recall that $U$ satisfies (2.29) and travels with speed $c$. In the same
context as (5.2), the energy at the tails of the function is given by

$\widetilde{E}(V)(\tau)\coloneqq\int_{B(c|\tau|+\epsilon^{-\sigma})^{c}}\left(\partial_{\tau}V_{j}\right)^{2}+c^{2}\left|\nabla\left(V_{j}\right)\right|^{2}d{\bf{X}}.$
(5.16)

Lemma 11.8 shows that for $i\geq 1$

$\left|\widetilde{E}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq
C\left(\left\|\nabla\phi\right\|^{2}_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}+\left\|\psi\right\|^{2}_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}\right)$
(5.17)

where $C$ depends only on $\bar{m}.$ We have

$\left\|\frac{\partial^{i}}{\partial\tau^{i}}U\right\|_{H^{k}(B(c\tau+\epsilon^{-\sigma})^{c})}\leq
C\left(\left\|\nabla\phi\right\|_{H^{i+k-1}(B(\epsilon^{-\sigma})^{c})}+\left\|\psi\right\|_{H^{i+k-1}(B(\epsilon^{-\sigma})^{c})}\right).$
(5.18)

It follows from Lemma 11.9, that for any $\sigma>0$

$\left\|\psi\right\|_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}\leq\epsilon\left\|\psi\right\|_{H^{i+k}_{\sigma}}\coloneqq\epsilon\sum_{j=0}^{i+k}\left\|\left(1+\left|\
\cdot\ \right|\right)^{\sigma^{-1}}\left|D^{j}\psi\right|\right\|_{L^{2}}.$
(5.19)

Therefore as long as $i\geq 1$, (5.18) becomes

$\left\|\frac{\partial^{i}}{\partial\tau^{i}}U\right\|_{H^{k}(B(c|\tau|+\epsilon^{-\sigma})^{c})}\leq\epsilon
C\left(\left\|\nabla\phi\right\|_{H^{i+k-1}_{\sigma}}+\left\|\psi\right\|_{H^{i+k-1}_{\sigma}}\right),$
(5.20)

which holds for all $\tau$.

##  6 Residual Estimates

We bound the terms from left to right in (4.25) by the initial conditions
given by (5.1).

######  Lemma 6.1.

For $U$ satisfying (2.29) with sufficiently smooth initial conditions given by
(5.1), there exists a constant $C$ depending only on $\bar{m}$ s.t.

$\sup_{|t|\leq\epsilon^{-1}T}\epsilon^{-1}\left\|\Delta
U(\epsilon\cdot,\epsilon
t)-\epsilon^{2}\Delta_{\bf{X}}U(\epsilon\cdot,\epsilon
t)\right\|_{\ell^{2}}\leq\epsilon^{2}4C\left(\left\|\nabla\phi\right\|_{H^{5}}+\left\|\psi\right\|_{H^{5}}\right)$
(6.1)

###### Proof.

Recall the definition of $\Delta_{\bf{X}}U$ in (2.15). For each $i=1,2$, by
Taylor’s Theorem with remainder, there exists a $\widehat{{\bf{j}}}$ with
$\widehat{j}_{i}\in[j_{i}-1,j_{i}+1]$ s.t

$\Delta_{i}U(\epsilon{\bf{j}})-\epsilon^{2}\partial_{X_{i}X_{i}}(U)(\epsilon{\bf{j}})=\epsilon^{4}\partial_{X_{i}X_{i}X_{i}X_{i}}(U)(\epsilon\widehat{{\bf{j}}}).$
(6.2)

By convexity of $(\cdot)^{2}$,

$\left(\Delta
U(\epsilon{\bf{j}})-\epsilon^{2}\Delta_{\bf{x}}(U)(\epsilon{\bf{j}})\right)^{2}\leq
2\sum_{i=1}^{2}\left(\Delta_{i}U(\epsilon{\bf{j}})-\epsilon^{2}\partial_{X_{i}X_{i}}(U)(\epsilon{\bf{j}})\right)^{2}.$
(6.3)

Combining (6.2) and (6.3) yields

$\left(\Delta
U(\epsilon{\bf{j}})-\epsilon^{2}\Delta_{\bf{x}}(U)(\epsilon{\bf{j}})\right)^{2}\leq\epsilon^{8}2\sum_{i=1,2}\left(\partial_{X_{i}X_{i}X_{i}X_{i}}(U)(\epsilon\widehat{{\bf{j}}})\right)^{2}.$
(6.4)

Summing over ${\bf{j}}$ and using Corollary 11.11 we find

$\left\|\Delta U(\epsilon\cdot,\epsilon
t)-\epsilon^{2}\Delta_{\bf{X}}U(\epsilon\cdot,\epsilon
t)\right\|_{\ell^{2}}\leq\epsilon^{3}4\left\|D^{4}U\right\|_{H^{2}}.$ (6.5)

Now we apply (5.7) to get a constant $C$ which depends only on $\bar{m}$ s.t.

$\left\|\Delta U(\epsilon\cdot,\epsilon
t)-\epsilon^{2}\Delta_{\bf{X}}U(\epsilon\cdot,\epsilon
t)\right\|_{\ell^{2}}\leq\epsilon^{3}4C\left(\left\|\nabla\phi\right\|_{H^{5}}+\left\|\psi\right\|_{H^{5}}\right).$
(6.6)

∎

######  Lemma 6.2.

For $U$ satisfying (2.29) with sufficiently smooth initial conditions given by
(5.1), there exists a constant $C$ depending on $a,b,$ and $\bar{m}$ s.t.

$\sup_{|t|\leq\epsilon^{-1}T}\epsilon\left\|z\mathbb{I}_{D({\bf{0}},R_{\epsilon})^{c}}U_{\tau\tau}\right\|_{\ell^{2}}\leq\epsilon
C\left(\left\|\nabla\phi\right\|_{H^{3}_{\sigma}}+\left\|\psi\right\|_{H^{3}_{\sigma}}\right)$
(6.7)

###### Proof.

First recall that $z({\bf{j}})\in[a-\bar{m},b-\bar{m}]$. There exists a $C$
that depends on $a,b$ or $\bar{m}$ s.t.

$\left\|z\mathbb{I}_{D({\bf{0}},R_{\epsilon})^{c}}U_{\tau\tau}\right\|_{\ell^{2}}\leq
C\left\|\mathbb{I}_{D({\bf{0}},R_{\epsilon})^{c}}U_{\tau\tau}\right\|_{\ell^{2}}.$
(6.8)

Recall the definition of $R_{\epsilon}$ in (4.2). Note that $R_{\epsilon}$ has
the form $\epsilon^{-1}\widetilde{R}(T)+1$ where $\widetilde{R}(\tau)\coloneqq
c\tau+\epsilon^{-\sigma}$. According to Lemma 11.12

$\left\|\mathbb{I}_{D({\bf{0}},R_{\epsilon})^{c}}U_{\tau\tau}\right\|_{\ell^{2}}\leq
2\epsilon^{-1}\left\|U_{\tau\tau}\right\|_{H^{2}(B(\widetilde{R}(T))^{c})}.$
(6.9)

Since $|\tau|\leq T,$

$\left\|U_{\tau\tau}\right\|_{H^{2}(B(\widetilde{R}(T)^{c})}\leq\left\|U_{\tau\tau}\right\|_{H^{2}(B(\widetilde{R}(\tau)^{c}))}.$
(6.10)

It follows from (5.20) that

$\left\|U_{\tau\tau}\right\|_{H^{2}(B(\widetilde{R}(\tau))^{c})}\leq\epsilon
C\left(\left\|\nabla\phi\right\|_{H^{3}_{\sigma}}+\left\|\psi\right\|_{H^{3}_{\sigma}}\right).$
(6.11)

Stringing these inequalities together and taking $\sup$ yields the result. ∎

######  Lemma 6.3.

Let $U$ satisfy (2.29) with sufficiently smooth initial conditions given by
(5.1). For each $\omega$ in a set of probability one, there exists a constant
$C_{\omega}$ s.t.

$\sup_{|t|\leq\epsilon^{-1}T_{0}}\epsilon\left\|\delta_{i}(\chi)\delta_{i}^{-}(U_{\tau\tau})\right\|_{\ell^{2}}\leq\epsilon
C_{\omega}\left(\left(\sqrt{T}+\log(R_{\epsilon})\right)\left(\left\|\nabla\phi\right\|_{H_{w}^{4}}+\left\|\psi\right\|_{H_{w}^{4}}\right)\right)$
(6.12)

###### Proof.

Recall the bound given by (3.25). We therefore have that

$\left\|\delta_{i}(\chi_{R_{\epsilon}})\delta_{i}^{-}(U_{\tau\tau})\right\|_{\ell^{2}}\leq
C_{\omega}\left(\left\|\log^{+}(\cdot)\delta_{i}^{-}\left(U_{\tau\tau}\right)\right\|_{\ell^{2}}+\log(R_{\epsilon})\left\|\delta_{i}^{-}(U_{\tau\tau})\right\|_{\ell^{2}}\right).$
(6.13)

Recalling the definition of $w$ in (5.8)

$\log^{+}(\cdot)\leq\log(\epsilon^{-1})\log(\epsilon\cdot+1)\leq\log(R_{\epsilon})w_{\epsilon}(\cdot).$
(6.14)

where

$w_{\epsilon}(\cdot)\coloneqq w(\epsilon\cdot).$ (6.15)

Then

$\left\|\delta_{i}^{-}(U_{\tau\tau})\right\|_{\ell^{2}}\leq\left\|w_{\epsilon}\delta_{i}^{-}(U_{\tau\tau})\right\|_{\ell^{2}}\
\text{and}\
\left\|\log^{+}(\cdot)\delta_{i}^{-}\left(U_{\tau\tau}\right)\right\|_{\ell^{2}}\leq\log(R_{\epsilon})\left\|w_{\epsilon}\delta_{i}^{-}(U_{\tau\tau})\right\|_{\ell^{2}}.$
(6.16)

Applying Corollary 11.10 we have

$\left\|w_{\epsilon}\delta_{i}^{-}\left(U_{\tau\tau}\right)\right\|_{\ell^{2}}\leq\epsilon^{-1}\left\|w(\cdot)\left(U_{\tau\tau}(\cdot)-U_{\tau\tau}(\cdot-\epsilon{\bf{e}}_{i})\right)\right\|_{H^{2}}.$
(6.17)

Note that $\delta_{\epsilon i}^{-}(U)(\cdot,\tau)\coloneqq
U(\cdot,\tau)-U(\cdot-\epsilon{\bf{e}}_{i},\tau)$ solves the same wave
equation $U$ does with initial conditions

$\delta_{\epsilon i}^{-}U({\bf{X}},0)=\delta_{\epsilon i}^{-}(\phi)({\bf{X}})\
\text{and}\ \delta_{\epsilon i}^{-}U_{\tau}({\bf{X}},0)=\delta_{\epsilon
i}^{-}(\phi)({\bf{X}}).$ (6.18)

Hence we can apply (5.14)

$\left\|w(\cdot)\left(U_{\tau\tau}(\cdot)-U_{\tau\tau}(\cdot-\epsilon{\bf{e}}_{i})\right)\right\|_{H^{2}}\leq
C\sqrt{|\tau|}\left(\left\|\delta^{-}_{\epsilon
i}(\nabla\phi)\right\|_{H^{3}_{w}}+\left\|\delta_{\epsilon
i}^{-}(\psi)\right\|_{H^{3}_{w}}\right).$ (6.19)

By Lemma 11.13

$\left\|w(\cdot)\left(U_{\tau\tau}(\cdot)-U_{\tau\tau}(\cdot-\epsilon{\bf{e}}_{i})\right)\right\|_{H^{2}}\leq\epsilon
C\sqrt{|\tau|}\left(\left\|\nabla\phi\right\|_{H^{4}_{w}}+\left\|\psi\right\|_{H^{4}_{w}}\right).$
(6.20)

Stringing the inequalities together and taking $\sup$ gives us the result. ∎

######  Lemma 6.4.

Let $U$ satisfy (2.29) with sufficiently smooth initial conditions given by
(5.1). For each $\omega$ in a set of probability one, there exists a constant
$C_{\omega}$ s.t.

$\sup_{|t|\leq\epsilon^{-1}T}\epsilon\left\|S_{i}^{+}(\chi_{R_{\epsilon}})\Delta_{i}(U_{\tau\tau})\right\|_{\ell^{2}}\leq\epsilon^{2}R_{\epsilon}C_{\omega}\left(\left(\sqrt{T}+\log(R_{\epsilon})^{\frac{3}{2}}\right)\left(\left\|\nabla\phi\right\|_{H_{w}^{5}}+\left\|\psi\right\|_{H_{w}^{5}}\right)\right)$
(6.21)

###### Proof.

The proof is essentially the same as the proof of the previous theorem but now
we use (3.24) instead of (3.24) and Corollary 11.14 instead of Lemma 11.13. ∎

######  Lemma 6.5.

Let $U$ satisfy (2.29) with sufficiently smooth initial conditions given by
(5.1). For each $\omega$ in a set of probability one, there exists a constant
$C_{\omega}$ s.t.

$\sup_{|t|\leq\epsilon^{-1}T}\epsilon^{3}\left\|m\chi_{R_{\epsilon}}U_{\tau\tau\tau\tau}\right\|_{\ell^{2}}\leq\epsilon^{2}R_{\epsilon}C_{\omega}\left(\left(\sqrt{T}+\log(R_{\epsilon})^{\frac{3}{2}}\right)\left(\left\|\nabla\phi\right\|_{H_{w}^{5}}+\left\|\psi\right\|_{H_{w}^{5}}\right)\right).$
(6.22)

###### Proof.

Since $m({\bf{j}})\in[a,b]$, there exists a constant either $a$ or $b$ s.t.

$\left\|m\chi_{R_{\epsilon}}U_{\tau\tau\tau\tau}\right\|_{\ell^{2}}\leq
C\left\|\chi_{R_{\epsilon}}U_{\tau\tau\tau\tau}\right\|_{\ell^{2}}.$ (6.23)

The steps are now very similar to those found in Lemma 6.3 but with (3.23)
instead of (3.24). We then use Lemma 11.10 and (5.14). ∎

##  7 Residual Bound and Discussion

###  7.1 Proof of Proposition 4.1

###### Proof.

Recall the calculation for $\operatorname{Res}\widetilde{u}$ in (4.25). We
have bounded each of the terms, in order, using Lemmas 6.1, 6.2, 6.3, 6.4 and
6.5. We obtain Proposition (4.1) by using the largest parts of each of the
bounds. ∎

Lemma 6.1 bounds a completely deterministic term that would appear no matter
how the masses are chosen. Lemma 6.2 bounds a term that arises due to our use
of a cut-off function. Recall we need to use a cut-off in order to work around
solving (2.19). In the case where this can be solved, say for example where
the masses vary periodically, then this term would not appear. The norm,
$H^{3}_{\sigma}$, in this bound is the largest norm. As $\sigma$ becomes
smaller, this norm becomes larger. The term in Lemma 6.3 is the first term
that must be bounded using probabilistic arguments. The constant $C_{\omega}$
exists almost surely but depends on the actual realization of masses. It
therefore could be arbitrarily large, since there is always a small
probability that (3.15) holds only for $n$ extremely large. Thus $C_{\omega}$
may be worthy of statistical quantification in a follow up work. Recalling the
definition of $R_{\epsilon}$ in (4.2), the bound for the term in Lemma (6.4)
is the dominant one in $\epsilon$. It is
$O(\epsilon^{1-\sigma}\log(\epsilon^{-1-\sigma})^{3/2})$. This term also
requires the most smoothness of the initial condition. The final term in Lemma
(6.5) is also $O(\epsilon^{1-\sigma}\log(\epsilon^{-1-\sigma})^{3/2})$ for
similar reasons. It may be conjectured that $\sigma$ here is artificial as it
arises from our inability to solve (2.19). We could analyse more; for example,
we could find the dependence of $T,\bar{m},a$ or $b$, but what we are most
interested in is tracking $\epsilon$.

###  7.2 Other Masses

The method we have employed is robust enough to consider other ways of
realizing the masses. For example, we may consider periodic masses by which we
mean there exists a positive integer $k$ s.t.
$m(j_{1},j_{2})=m(j_{1}+k,j_{2}+k)$. We also may consider masses which are all
identical. In such a case, the only term which appears in the residual is the
one bounded in Lemma 6.1. This improves the accuracy of approximate solution
substantially as the residual would be $O(\epsilon^{2})$. Another
generalization we can make is that masses need not be identically distributed,
so long as they all fall into some interval $[a,b]\in{\bf{R}}^{+}$ and have
the same expected value. Since our methods did not use any other features of
the masses being identically distributed, e.g. equal variances or
probabilities, our result extends to this case without modification.

Another important example is layered media. Suppose that

$\\{m(j_{1},j_{2})\\}_{j_{1}=-\infty}^{\infty}.$ (7.1)

is random i.i.d. sequence of masses and that for all $j_{1}$

$\cdots=m(j_{1},-1)=m(j_{1},0)=m(j_{1},1)=\cdots$ (7.2)

In this case, it is actually possible to solve (2.19) and thereby not use a
cutoff, but we proceed using the same tools we have developed, since such
tools can be utilized in higher dimensions. In this case, (3.9) does not
immediately hold since we do not have complete independence. Reconsider (3.6)

$\displaystyle\mathcal{L}(\chi_{R_{\epsilon}})(j_{1},j_{2})$
$\displaystyle=\sum_{|k_{1}|\leq r}\sum_{|k_{2}|\leq
r}\mathcal{L}(\varphi)(j_{1}-k_{1},j_{2}-k_{2})z(k_{1},k_{2})$ (7.3)
$\displaystyle=\sum_{k_{1}=j_{1}-r}^{j_{1}+r}\sum_{k_{2}=j_{2}-r}^{j_{2}+r}\mathcal{L}(\varphi)(k_{1},k_{2})z(j_{1}-k_{1},j_{2}-k_{2}).$

Let $z(k_{1})\coloneqq z(k_{1},\cdot).$ This is well defined because of (7.2).
Thus

$\mathcal{L}(\chi_{r})(j_{1},j_{2})=\sum_{k_{1}=j_{1}-r}^{j_{1}+r}z(j_{1}-k_{1})\sum_{k_{2}=j_{2}-r}^{j_{2}+r}\mathcal{L}(\varphi)(k_{1},k_{2}).$
(7.4)

Let

$\mathcal{L}(\varPhi_{r})(j_{2},k_{1})\coloneqq\sum_{k_{2}=j_{2}-r}^{j_{2}+r}\mathcal{L}(\varphi)(k_{1},k_{2})$
(7.5)

so

$\mathcal{L}(\chi_{R_{\epsilon}})(j_{1},j_{2})=\sum_{k_{1}=j_{1}-r}^{j_{1}+r}z(j_{1}-k_{1})\mathcal{L}(\varPhi_{r})(j_{2},k_{1}).$
(7.6)

This is a sum involving independent random variables so we get a version of
(3.9). Let

$\left\|\mathcal{L}(\varPhi_{r})(j_{1},j_{2})\right\|^{2}\coloneqq\sum_{k_{1}=j_{1}-r}^{j_{1}+r}\mathcal{L}(\varPhi_{r})(j_{2},k_{1})^{2}$
(7.7)

Then we have

$\exp(s\chi_{r}(j_{1},j_{2}))\leq\exp\left(\frac{s^{2}\left\|\mathcal{L}(\varPhi_{r})(j_{1},j_{2})\right\|^{2}(a-b)^{2}}{8}\right).$
(7.8)

Now the same argument works as is made in Section 3.2 but the relevant
quantity to calculate is the square root of (7.7). Take $\mathcal{L}$ to be
any of the operators in Section 3.2. Then

$\left\|\mathcal{L}(\varPhi_{r})(j_{1},j_{2})\right\|^{2}=\sum_{k_{1}=j_{1}-r}^{j_{1}+r}\left(\sum_{k_{2}=j_{2}-r}^{j_{2}+r}\mathcal{L}(\varphi)(k_{1},k_{2})\right)^{2}.$
(7.9)

It is possible to use Jensen’s inequality to obtain

$\left\|\mathcal{L}(\varPhi_{r})(j_{1},j_{2})\right\|^{2}\leq(2r+1)\sum_{k_{1}=j_{1}-r}^{j_{1}+r}\sum_{k_{2}=j_{2}-r}^{j_{2}+r}\mathcal{L}(\varphi)(k_{1},k_{2})^{2}=(2r+1)\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}.$
(7.10)

One notices that $\left\|\mathcal{L}\varphi\right\|^{2}_{D({\bf{j}},r)}$ are
norms we have already computed in Section 3.2, see (3.17) for example. The $r$
out front provides an extra half power of $\epsilon^{-1-\sigma}$ when one
considers (4.2) and after taking square roots as in (3.16). Therefore, in
contrast with (4.19), we have

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}\widetilde{u}\right\|_{\ell^{2}}\leq\epsilon^{\frac{1}{2}-\frac{3}{2}\sigma}C_{\omega}\log(\epsilon^{-1-\sigma})^{\frac{3}{2}}\left\|\phi,\psi\right\|_{H^{5}_{\sigma}}.$
(7.11)

This example sheds some light on the complexity introduced when considering
random masses. In contrast, for periodic masses, even if (7.2) holds, then
(2.19) is solvable and the solution is periodic so most importantly bounded.
Therefore the residual is always $O(\epsilon).$

###  7.3 Main Estimate Result

######  Theorem 7.1.

Suppose $\psi,\phi\in H_{\sigma}^{5}$. Let $u$ satisfy (4.10) with

$u({\bf{j}},0)=\epsilon^{-1}\phi(\epsilon{\bf{j}})\ \text{and}\
\dot{u}({\bf{j}},0)=\psi(\epsilon{\bf{j}}).$ (7.12)

and let $U$ satisfy (2.29) with

$U({\bf{X}},0)=\phi({\bf{X}})\ \text{and}\
\partial_{\tau}U({\bf{X}},0)=\psi({\bf{X}}).$ (7.13)

With

$\widehat{u}({\bf{j}},t)=\epsilon^{-1}U(\epsilon{\bf{j}},\epsilon t)$ (7.14)

where $\chi_{R_{\epsilon}}$ is given by (3.4) and $R_{\epsilon}$ by (4.2).
Then there exists a $C_{\omega}(a,b,\bar{m},T)$ a.s. s.t.

$\sup_{|t|\leq\epsilon^{-1}T}\left\|u-\widehat{u}\right\|_{\ell^{2}}\leq\epsilon^{-1-\sigma}C^{*}_{\omega}\log(\epsilon^{-1-\sigma})^{\frac{3}{2}}\left\|\phi,\psi\right\|_{H^{5}_{\sigma}}.$
(7.15)

and

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\dot{u}-\dot{\widehat{u}}\right\|_{\ell^{2}}\leq\epsilon^{-\sigma}C^{*}_{\omega}\log(\epsilon^{-1-\sigma})^{\frac{3}{2}}\left\|\phi,\psi\right\|_{H^{5}_{\sigma}}.$
(7.16)

######  Remark 3.

Although the right hand side of (7.15) and (7.16) appear to grow, the size of
$u$ and $\dot{u}$ is roughly $\epsilon^{-2}$ and $\epsilon^{-1}$, so the error
is relatively small.

###### Proof.

Let

$\widetilde{u}({\bf{j}},t)=\epsilon^{-1}U(\epsilon{\bf{j}},\epsilon
t)+\epsilon\chi_{R_{\epsilon}}({\bf{j}})U_{\tau\tau}(\epsilon{\bf{j}},\epsilon
t).$ (7.17)

This is what has typically been our $\widetilde{u}$ as defined in (4.3). Since
we have taken the initial conditions for $u$ and $\widehat{u}$ to be
identical, (4.15) holds as long as we have

$\epsilon\left\|\epsilon\chi_{R_{\epsilon}}(\cdot)U_{\tau\tau\tau}(\cdot,0),\delta_{1}^{+}\left(\chi_{R_{\epsilon}}U_{\tau\tau}\right)(\cdot,0),\delta_{2}^{+}\left(\chi_{R_{\epsilon}}U_{\tau\tau}\right)(\cdot,0)\right\|\leq\epsilon^{-1}C\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}\right\|_{\ell^{2}}$
(7.18)

which can be proven using the same kind of arguments given in Section 6.

Then we have by (4.17) and (4.18) that for a constant depending on depending
on $a,b$ and $T$ s.t.

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\dot{u}-\dot{\widetilde{u}}\right\|_{\ell^{2}}\leq\epsilon^{-1}C\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}\right\|_{\ell^{2}}$
(7.19)

and

$\sup_{|t|\leq\epsilon^{-1}T}\left\|u-\widetilde{u}\right\|_{\ell^{2}}\leq\epsilon^{-2}C\sup_{|t|\leq\epsilon^{-1}T}\left\|\operatorname{Res}{\widetilde{u}}\right\|_{\ell^{2}}.$
(7.20)

Now we use (4.19) to obtain the correct right hand side in (7.15) and (7.16)
but for $\widetilde{u}$ instead of $\widehat{u}.$ Thus it remains to analyse
the difference between the two which is

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\dot{\widehat{u}}-\dot{\widetilde{u}}\right\|_{\ell^{2}}\leq\sup_{|t|\leq\epsilon^{-1}T}\left\|\epsilon^{2}\chi_{R_{\epsilon}}(\cdot)U_{\tau\tau\tau}(\epsilon\cdot)\right\|_{\ell^{2}}$
(7.21)

and

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\widehat{u}-\widetilde{u}\right\|_{\ell^{2}}\leq\sup_{|t|\leq\epsilon^{-1}T}\left\|\epsilon\chi_{R_{\epsilon}}(\cdot)U_{\tau\tau}(\epsilon\cdot)\right\|_{\ell^{2}}.$
(7.22)

Both of these can be bounded using steps similar to those in Lemmas 6.3 and
6.4. One finds

$\sup_{|t|\leq\epsilon^{-1}T}\left\|\dot{\widehat{u}}-\dot{\widetilde{u}}\right\|_{\ell^{2}}\leq\epsilon
C_{\omega}R_{\epsilon}\left((\sqrt{T}+\log(R_{\epsilon})^{\frac{3}{2}})(\left\|\phi\right\|_{H_{w}^{3}}+\epsilon\left\|\psi\right\|_{H_{w}^{2}})\right)$
(7.23)
$\sup_{|t|\leq\epsilon^{-1}T}\left\|\widehat{u}-\widetilde{u}\right\|_{\ell^{2}}\leq
C_{\omega}R_{\epsilon}\left((\sqrt{T}+\log(R_{\epsilon})^{\frac{3}{2}})(\left\|\phi\right\|_{H_{w}^{2}}+\epsilon\left\|\psi\right\|_{H_{w}^{1}})\right).$
(7.24)

Recalling the definition of $R_{\epsilon}$ in (4.2), these bounds have the
correct power of $\epsilon$. Thus with the appropriate constant
$C^{\star}_{\omega}$ everything can be dominated by the right hand side in
(7.15) and (7.16). ∎

##  8 Coarse Graining

The Theorem 7.1 says that the macroscopic behavior of the system is wavelike
i.e. it evolves according to an effective wave equation. We formalize this
notion by proving a convergence result in the macroscopic setting. We have a
number of operators to introduce. The discrete Fourier Transform is given by

$F[f]({\bf{y}})=\frac{1}{(2\pi)^{2}}\sum_{{\bf{j}}\in{\bf{Z}}^{2}}\exp\left(-i{\bf{j}}\cdot{\bf{y}}\right)f({\bf{j}})$
(8.1)

Here $y\in{\bf{R}}^{2}$. Its inverse is

$F^{-1}[g]({\bf{j}})=\int_{[-\pi,\pi]^{2}}\exp\left(i{\bf{y}}\cdot{\bf{j}}\right)g({\bf{y}})d{\bf{y}}.$
(8.2)

Let $\mathcal{F}$ and $\mathcal{F}^{-1}$ be the Fourier Transform and its
inverse, defined analogously to the discrete versions (i.e. replacing the sum
with an integral). The sampling operator is

$\mathcal{S}(u)({\bf{j}})=u({\bf{j}}).$ (8.3)

A cutoff operator is

$\theta_{\phi}({\bf{y}})=\begin{cases}1&{\bf{y}}\in[-\phi,\phi]^{2}\\\
0&\text{else}.\end{cases}$ (8.4)

Finally, we define a low pass interpolator. For a continuous variable
${\bf{x}}\in{\bf{R}}^{2}$

$\mathcal{L}[f]({\bf{x}})=\mathcal{F}^{-1}[\theta_{\pi}(\cdot)F[f](\cdot)]({\bf{x}}).$
(8.5)

The following theorem says that the same abstract diagram found in [12] holds
in this setting almost surely.

###  8.1 Main Convergence Result

######  Theorem 8.1.

Let $U$ solve (2.29) with initial conditions given by (5.3). Let $u$ solve
(1.1) with initial conditions given by (5.1). Put

$U_{\epsilon}({\bf{X}},\tau)=\epsilon\mathcal{L}(u(\cdot,\tau/\epsilon))({\bf{X}}/\epsilon)$
(8.6)

so that

$\partial_{\tau}U_{\epsilon}({\bf{X}},\tau)=\mathcal{L}(\dot{u}(\cdot,\tau/\epsilon))({\bf{X}}/\epsilon).$
(8.7)

Then

$\lim_{\epsilon\to 0}\sup_{|\tau|\leq
T}\left\|U_{\epsilon}-U,\partial_{\tau}U_{\epsilon}-\partial_{\tau}U\right\|_{L^{2}}=0$
(8.8)

almost surely.

.

######  Remark 4.

The rate of convergence is
$\epsilon^{1-\sigma}\log(\epsilon^{-1-\sigma})^{\frac{3}{2}}$ as seen in the
proof.

###### Proof.

$\left\|U_{\epsilon}-U\right\|_{L^{2}}\leq\left\|\epsilon\mathcal{L}(u(\cdot,\tau/\epsilon))(\cdot/\epsilon)-\mathcal{LS}(U(\epsilon\cdot,\tau))(\cdot/\epsilon)\right\|_{L^{2}}+\left\|\mathcal{LS}(U(\epsilon\cdot,\tau))(\cdot/\epsilon)-U(\cdot,\tau)\right\|_{L^{2}}.$
(8.9)

According to Lemma 11.15, the second term goes to $0$ uniformly in $\tau$ at a
rate of $\epsilon$. From the first term, we have

$\left\|\epsilon\mathcal{L}(u(\cdot,\tau/\epsilon))(\cdot/\epsilon)-\mathcal{LS}U(\epsilon\cdot,\tau)(\cdot/\epsilon)\right\|_{L^{2}}=\epsilon\left\|\epsilon\mathcal{L}(u(\cdot,\tau/\epsilon))(\cdot)-\mathcal{LS}U(\epsilon\cdot,\tau)(\cdot)\right\|_{L^{2}}.$
(8.10)

By Plancherel

$\epsilon\left\|\epsilon\mathcal{L}(u(\cdot,\tau/\epsilon))({\bf{X}})-\mathcal{LS}U(\epsilon\cdot,\tau)(\cdot)\right\|_{L^{2}}\leq\epsilon
C\left\|\epsilon
u(\cdot,\tau/\epsilon)-\mathcal{S}U(\epsilon\cdot,\tau)\right\|_{L^{2}}.$
(8.11)

Now we use that $U(\epsilon\cdot,\tau)=\epsilon\widehat{u}(\cdot,\epsilon t)$
as in Theorem 7.1 and the result follows from that same theorem. Recall
$\tau=\epsilon t$. Similarly

$\epsilon\left\|\mathcal{L}(\dot{u}(\cdot,\tau/\epsilon))({\bf{X}})-\mathcal{LS}U_{\tau}(\epsilon\cdot,\tau)(\cdot)\right\|_{L^{2}}\leq\epsilon\left\|\dot{u}(\cdot,\tau/\epsilon)-\mathcal{S}U_{\tau}(\epsilon\cdot,\tau)\right\|_{\ell^{2}}.$
(8.12)

Again we use Theorem 7.1 where
$U_{\tau}(\epsilon\cdot,\tau)=\dot{\widehat{u}}(\cdot,t).$ ∎

##  9 Numerical Results

Our numerical results focus on confirming the upper bounds found in Theorem
7.1.We refer to the left hand side of (7.15) as the absolute error of the
displacement (a.e.d.) and the left hand side of (7.16) as the absolute error
of the velocity (a.e.v). For the next two experiments, we have chosen

$\phi(X_{1},X_{2})=\operatorname{sech}\left(\frac{1}{2}(X_{1}-1)^{2}+(X_{2}-1)^{2}\right),\
\psi(X_{1},X_{2})=\operatorname{sech}\left((X_{1}+1)^{2}+\frac{1}{2}(X_{2}+1)^{2}\right)$
(9.1)

and looked at $\epsilon$ over $\\{1/2,1/4,1/8,1/16,1/32\\}.$ Every $m(j)$ is
sampled from $\\{1/2,3/2\\}$ and for the first experiment the masses are
chosen to be i.i.d. As $\epsilon$ varies, this grid of randomly chosen masses
remains fixed.

Figure 1: The left (right) panel shows a $\log$-$\log$ graph of a.e.d. (a.e.v)
versus $\epsilon$. Since the masses are chosen randomly, ten realization of
masses were tested. The distribution of the error is plotted with a box and
whisker plot at each value of $\epsilon$. The slope of the line of best fit
numerically approximates the power of $\epsilon$ in (7.15) ((7.16)).

The upper bound for the a.e.v. obtained in Theorem 7.1 is
$\epsilon^{-\sigma}\log(\epsilon^{-1-\sigma})^{\frac{3}{2}}$ for the a.e.v.
(Recall that $\sigma$, is any arbitrarily small positive number.) The slopes
seen in Figure 1 are in agreement with the bounds obtained in the theorem,
i.e. the slopes reflect to what power of $\epsilon$ the error depends. In fact
the estimate is close to sharp. We can thus think of such bounds as giving an
analytic prediction on the size of the error in many cases.

For the second experiment, we use the setup for the masses given by (7.1) and
(7.2) i.e. the masses are layered. According to (7.11), we expect the slopes
to be no more than a half power less than those seen in Figure 1. This is
indeed what we see in Figure 2. Again, the numerical error is close to the
predicted error.

Figure 2: These panels reflect the same measurements as those in Figure 1, but
now the masses have been chosen according to (7.1) and (7.2)

Finally we compare these results to what happens in three deterministic cases.
In one case we choose the masses to be constant in which case we would expect
the slope for the a.e.v and the a.e.d. to be no worse than $1$ and 0
respectively. In a second, the masses are periodically layered i.e. they only
vary periodically with a period of 2 along one of the coordinate axis and
along the other, (7.2) holds. For the third case, masses are chosen to vary
periodically along both coordinate axes with period of 2. In both cases we
expect the slope of the a.e.v. and a.e.d. to be no worse than $0$ and $1$. In
both cases the upper bound holds; however, unlike in all the previous
experiments, numerically computed a.e.d. seems to be substantially better than
the analytic upper bound, since the slope for both periodic cases is closer to
$0$ than to $1$.

Figure 3: Masses have been chosen deterministically in three different ways.

One important observation is that when the randomly chosen masses are layered,
the approximation to the wave equation does not converge as quickly as it does
when they are not layered. The main physical explanation we propose for the
difference in the observed slope between Figures 1 and 2 is that in the second
experiment, reflections caused by the random masses manifest as long ripples
transverse to the direction in which the masses are randomly changing. This is
in contrast to the first experiment where reflections manifest as localized
disturbances. Figure (4) gives some empirical evidence for this phenomenon.

One possible heuristic explanation for why we don’t get improvement for
periodic masses is that there is always a direction in ${\bf{R}}^{2}$ along
which the averages of masses in lines transverse to that direction are varying
(unless the masses are all constant).This produces a kind of layering that
cannot occur if all the masses are chosen to be i.i.d. since masses along any
line average to the same value. Hence, in this sense the homogenization occurs
faster, albeit the amplitudes of localized fluctuations are larger in the
i.i.d. case.

Figure 4: The left panel shows a wave traveling through masses which are
i.i.d. and the right shows a wave traveling through layered random masses. One
can spot the long transverse ripples in the right panel, whereas in the left
panel, deviations appear more localized.

Finally, we have performed a number of similar experiments in 1D, the results
of which are summarized in 1. The predicted values can all be proven using
essentially the same method as what has been demonstrated or discussed in
Section 7.2 and throughout the rest of the paper. Even though our predicted
values are upper bounds, we see that in most cases our estimates are close to
sharp. The exception is when the masses are chosen periodically, where the
predicted a.e.d. often overshoots by close to a full power of $\epsilon$. This
indicates that integrating $p$, and then applying a triangle inequality to
obtain an upper bound on $u$ is not efficient in the periodic case.

For both the constant coefficient and periodic cases, a half power of
$\epsilon$ is lost with each increase in dimension. This is due to the length
scaling and is negated when one considers the coarse-graining limit. We see no
reason this trend should not continue into higher dimensions. On the other
hand, there is not a decrease in the power of $\epsilon$ in the i.i.d. case.
Heuristically, one can see this by comparing the variance of $\chi$ in 1D and
in 2D. The fundamental solution $\varphi$ of $\Delta$ plays an important role
in the growth rate. In 1D, this fundamental solution is given by
$\varphi(\cdot)=\frac{1}{2}|\cdot|.$ Taking the expectation of $\chi_{r}^{2}$
where $\chi_{r}$ is defined by (3.4) yields that $\chi_{r}^{2}$ is
approximately the size of $r^{3}$ in $\ell^{\infty}$. The same procedure in 2D
yields $r^{2}\log(r)^{2}$. This accounts for an additional half power of
$\epsilon$ after taking square roots. The argument using the ideas of sub-
Gaussian random variables is one way to formalize this intuition and obtain an
almost sure estimate. Ultimately, this half power is negated by the increase
in dimension, and since the random term in the residual is still the dominant
one, we find that the size of the absolute error (as dependent upon
$\epsilon$) roughly does not change from 1D to 2D. Therefore the coarse-
graining limit converges faster in 2D.

Rates for: | a.e.v. | a.e.d.
---|---|---
Mass/Dim. | Predicted | Simulated | Predicted | Simulated
1D Const. Coeff. | $1.5$ | $1.5$ | $0.5$ | $0.5$
1D Period. Coeff. | $0.5$ | $0.7$ | $-0.5$ | $0.5$
1D Rando. Coeff. | $-\sigma$ | $0.2$ | $-1-\sigma$ | $-0.8$
2D Const. Coeff | $1$ | $1.1$ | $0$ | $0.0$
2D Period. Coeff | $0$ | $0.1$ | $-1$ | $0.0$
2D Rando. Coeff | $-\sigma$ | $0.1$ | $-1-\sigma$ | $-1.0$
2D Layered Coeff | $-0.5-\sigma$ | $-0.4$ | $-1.5-\sigma$ | $-1.5$
Table 1: This table summarizes the epsilon dependence of the absolute errors.
The numbers indicate the power of $\epsilon$. $\sigma$ is a small and positive
constant.

##  10 Conclusion

We have rigorously justified the wave equation as descriptor for the
macroscopic dynamics of a 2D square lattice composed of i.i.d. masses. We have
given analytic as well as numerical evidence that this description is more
accurate in 2D than it is in 1D. We conjecture that in 3D, the a.e.d. and
a.e.v. is roughly the same as it is in 2D for the i.i.d. case. This means it
would be roughly only a half power of epsilon larger than in the constant
case. We think such a result is modest evidence that waves propagate more
easily in a disordered lattice in higher dimensions. An important exception
that was seen to this occurs in the case of layered random masses, where the
error became larger from 2D to 1D in the same way it did for periodic or
constant masses. Although there are results in the continuous setting that are
similar to ours [8], as far as we can tell, this is the first result that
provides a rigorous almost sure bounds on the rate of convergence for
lattices, and we think that these rates of convergence provide insight into
the effects of dimensionality of the disorder on wave propagation. Finally,
the techniques introduced, especially the use of sub-Gaussian random variables
can probably be used to access similar results for various other discrete
systems with different kinds of disorder.

##  11 Appendix

###  11.1 Probabilistic Estimates A

######  Lemma 11.1.

There exists a bijection

$I:{\bf{Z}}^{2}\times{\bf{N}}^{+}/\\{1\\}\to{\bf{N}}^{+}$ (11.1)

such that

$I({\bf{j}},r)\leq C\max\\{{\bf{j}}^{3},r^{3}\\}.$ (11.2)

###### Proof.

Notice the set

$B(n)\coloneqq\\{({\bf{j}},r)\ |\ |{\bf{j}}|\leq n\ ,\ r\leq n\\}$ (11.3)

is a subset of the top half of a ball in ${\bf{Z}}^{3}.$ If we require

$I({\bf{j}}_{1},r_{1})<I({\bf{j}}_{2},r_{2})$ (11.4)

whenever there exists an integer $n$ s.t.

$({\bf{j}}_{1},r_{1})\in B(n)\ \ \text{but}\ \ ({\bf{j}}_{2},r_{2})\notin
B(n)$ (11.5)

we have at worst that

$I({\bf{j}},r)\leq|B(\max\\{{\bf{j}},r\\})|\leq
C\max\\{{\bf{j}}^{3},r^{3}\\}.$ (11.6)

∎

######  Lemma 11.2.

For $r\geq 2$, and ${\bf{j}}\in{\bf{Z}}^{2},$ there exists a constant $C$ s.t.

$\log^{+}(|{\bf{j}}|+r)\leq C(\log^{+}(|{\bf{j}}|)+\log^{+}(r))$ (11.7)

###### Proof.

The inequality is trivial for $|{\bf{j}}|=0$. For $|{\bf{j}}|\neq 0$, we can
prove the inequality for $\log$. By the concavity of $\log$, $\log(\cdot+1)$
is sub-additive.

$\log(|{\bf{j}}|+r-1+1)\leq\log(|{\bf{j}}|+1)+\log(r)\leq
C(\log(|{\bf{j}}|)+\log(r)).$ (11.8)

∎

######  Lemma 11.3.

For $\varphi$ given by (3.3) and $\delta_{i}$ defined in (3.20) we have for
some $C$

$|\delta_{i}(\varphi)({\bf{j}})|\leq\frac{C}{|{\bf{j}}|+1}$ (11.9)

###### Proof.

By (3.3), we have

$\delta_{i}(\varphi)({\bf{j}})=\frac{1}{2\pi}\log^{+}|{\bf{j}}+{\bf{e}}_{i}|-\frac{1}{2\pi}\log^{+}|{\bf{j}}-{\bf{e}}_{i}|+O(|{\bf{j}}|^{-2})$
(11.10)

Note that when $|{\bf{j}}|=0$, we are left with only small $O(|j|^{-2})$
terms. For $|{\bf{j}}|=1$ we have

$\left|\log^{+}|{\bf{j}}+{\bf{e}}_{i}|-\log^{+}|{\bf{j}}-{\bf{e}}_{i}|\right|\leq\log(2).$
(11.11)

For $|{\bf{j}}|>1$, we have

$\displaystyle\left|\log|{\bf{j}}+{\bf{e}}_{i}|-\log|{\bf{j}}-{\bf{e}}_{i}|\right|$
$\displaystyle\leq\left|\log\left(\frac{|{\bf{j}}|+|{\bf{e}}_{i}|}{|{\bf{j}}|-|{\bf{e}}_{i}|}\right)\right|$
(11.12)
$\displaystyle=\left|\log\left(\frac{|{\bf{j}}|-1+2}{|{\bf{j}}|-1}\right)\right|$
$\displaystyle\leq\left|\log\left(1+\frac{2}{|{\bf{j}}|-1}\right)\right|$
$\displaystyle\leq\left|\log\left(1+\frac{2}{|{\bf{j}}|+1}\frac{|{\bf{j}}|+1}{|{\bf{j}}|-1}\right)\right|$
$\displaystyle\leq\left|\log\left(1+\frac{4}{|{\bf{j}}|+1}\right)\right|$
$\displaystyle\leq\frac{4}{|{\bf{j}}|+1}.$

Thus we obtain the result. ∎

######  Lemma 11.4.

For $\varphi$ given by (3.3) and $\delta_{i}$ defined in (3.20), we have for
some $C$

$\left\|\delta_{i}\phi\right\|^{2}_{D({\bf{j}},r)}\leq C\log(|{\bf{j}}|+r)$
(11.13)

###### Proof.

By Lemma 11.3

$\left\|\delta_{i}\varphi\right\|^{2}_{D({\bf{j}},r)}\leq C\sum_{{\bf{k}}\in
D({\bf{j}},r)}\left(\frac{1}{|{\bf{k}}|+1}\right)^{2}.$ (11.14)

The largest magnitude in $D({\bf{j}},r)$ is $|{\bf{j}}|+r$ is. Also note that
there is some constant $C$ s.t. the number of elements of some magnitude
$r^{\prime}$ is less than $C(r^{\prime}+1)$. Thus there exists a constant $C$
s.t.

$\displaystyle\sum_{{\bf{k}}\in
D({\bf{j}},r)}\left(\frac{1}{|{\bf{k}}|+1}\right)^{2}$ $\displaystyle\leq
C\sum_{r^{\prime}=0}^{|{\bf{j}}|+r}(r^{\prime}+1)\left(\frac{1}{r^{\prime}+1}\right)^{2}=C\sum_{r^{\prime}=0}^{|{\bf{j}}|+r}\left(\frac{1}{r^{\prime}+1}\right)$
(11.15)

A common bound on the harmonic series yields

$\sum_{{\bf{k}}\in D({\bf{j}},r)}\left(\frac{1}{|{\bf{k}}|+1}\right)^{2}\leq
C\log(|{\bf{j}}|+r+2),$ (11.16)

which yields the result. ∎

###  11.2 Lattice Energy Argument A

######  Lemma 11.5.

Let $f,g:{\bf{Z}}^{2}\to{\bf{R}}$. Recall the definition of the operators in
(3.20), (4.6), (3.18), and (4.20). Then

$\Delta_{i}(fg)=\Delta_{i}(f)g+S_{i}^{+}(f)\Delta_{i}(g)+\delta_{i}(f)\delta_{i}^{-}(g).$
(11.17)

###### Proof.

Starting from the definition in (4.20) we have

$\displaystyle\Delta_{i}(fg)=$ $\displaystyle S_{i}^{+}(fg)-2fg+S_{i}^{-}(fg)$
(11.18) $\displaystyle=$ $\displaystyle
S_{i}^{+}(f)g-2fg+S_{i}^{-}(f)g+S_{i}^{+}(fg)+S_{i}^{-}(fg)-S_{i}^{+}(f)g-S_{i}^{-}(f)g$
$\displaystyle=$
$\displaystyle\Delta_{i}(f)g+S_{i}^{+}(fg)-S_{i}^{+}(f)g+S_{i}^{-}(fg)-S_{i}^{-}(f)g$
$\displaystyle=$
$\displaystyle\Delta_{i}(f)g+S_{i}^{+}(fg)-2S_{i}^{+}(f)g+S_{i}^{+}(f)S_{i}^{-}(g)$
$\displaystyle+S_{i}^{+}(f)g-S_{i}^{+}(f)S_{i}^{-}(g)+S_{i}^{-}(fg)-S_{i}^{-}(f)g$
$\displaystyle=$
$\displaystyle\Delta_{i}(f)g+S_{i}^{+}(f)\Delta_{i}(g)+S_{i}^{+}(f)(g-S_{i}^{-}(g))-S_{i}^{-}(f)(g-S_{i}^{-}(g))$
$\displaystyle=$
$\displaystyle\Delta_{i}(f)g+S_{i}^{+}(f)\Delta_{i}(g)+\delta_{i}(f)\delta^{-}_{i}(g).$

∎

###  11.3 The Effective Wave A

######  Lemma 11.6.

Let $U$ satisfy (2.29) with sufficiently smooth initial conditions given by
(5.3). Then

$\left|E\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)\right|\leq
C\left(\left\|\nabla\phi\right\|_{H^{i+k}}^{2}+\left\|\psi\right\|_{H^{i+k}}^{2}\right)$
(11.19)

where $E\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)$ is defined
by (5.5). Here the constant $C$ depends on the wave speed $c$ which in turn is
entirely dependent on $\bar{m}.$

###### Proof.

Note that
$D^{k}U({\bf{X}},\tau):{\bf{R}}^{2}\times{\bf{R}}\to{\bf{R}}^{2^{k}}.$
Consider the $j^{\text{th}}$ component of the $2^{k}$ dimensional vector of
$D^{k}U$ denoted by $D^{k}U_{j}$. Consider a ball of radius $r$ in
${\bf{R}}^{d}$ about the origin denoted $B(r)$.

$\dot{E}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=\lim_{r\to\infty}\int_{B(r)}\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)\left(\frac{d^{i+2}D^{k}U_{j}}{d\tau^{i+2}}\right)+c^{2}\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\cdot\nabla\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)d{\bf{X}}$
(11.20)

Using integration by parts we find

$\displaystyle\dot{E}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)=$
$\displaystyle\lim_{r\to\infty}\int_{B(r)}\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)\left(\frac{d^{i+2}D^{k}U_{j}}{d\tau^{i+2}}\right)-c^{2}\Delta\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)d{\bf{X}}$
(11.21) $\displaystyle+\lim_{r\to\infty}\int_{\partial
B(r)}\left(\frac{\partial\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}}{\partial\nu}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)dS.$

Since $U$ solves (2.29), swapping the order of partials yields,

$\left(\frac{d^{i+2}D^{k}U_{j}}{d\tau^{i+2}}\right)-c^{2}\Delta\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)=0.$
(11.22)

We are left with

$\dot{E}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=\lim_{r\to\infty}\int_{\partial
B(r)}\left(\frac{\partial\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}}{\partial\nu}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)dS,$
(11.23)

which is $0$ for all $\tau$ due to the finite propagation speed of the wave.
Note that at time zero, when $i$ is even,

$\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}=c^{2}D^{k}\Delta_{\bf{X}}^{\frac{i}{2}}U_{j}=c^{2}D^{k}\Delta_{\bf{X}}^{\frac{i}{2}}\phi_{j}$
(11.24)

while, when $i$ is odd,

$\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}=c^{2}D^{k}\Delta_{\bf{X}}^{\frac{i-1}{2}}\frac{d_{j}U}{d\tau}=c^{2}D^{k}\Delta_{\bf{X}}^{\frac{i-1}{2}}\psi_{j}.$
(11.25)

Since (11.21) is $0$ and holds for all $j\in\\{1,2,\cdots,2^{k}\\}$, we have
using either (11.24) or (11.25) in (5.5) that

$\left|E\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\right|=\left|E\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(0)\right|\leq
C\left(\left\|\nabla\phi\right\|_{H^{i+k}}^{2}+\left\|\psi\right\|_{H^{i+k}}^{2}\right).$
(11.26)

∎

######  Lemma 11.7.

Let $U$ satisfy (2.29) with initial conditions given by (5.3). Suppose that
$w$ satisfies (5.8) and (5.9). Then

$\left|E_{w}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq|\tau|C\left(\left\|\phi\right\|^{2}_{H_{w}^{i+k+1}}+\left\|\psi\right\|^{2}_{H_{w}^{i+k}}\right).$
(11.27)

The constant $C$ depends upon $\bar{m}$ as well as $B$ found in (5.9).

###### Proof.

Note that
$D^{k}U({\bf{X}},\tau):{\bf{R}}^{2}\times{\bf{R}}\to{\bf{R}}^{2^{k}}.$
Consider the $j^{\text{th}}$ component of the $2^{k}$ dimensional vector of
$D^{k}U$ denoted by $D^{k}U_{j}$. Consider a ball of radius $r$ in
${\bf{R}}^{d}$ about the origin denoted $B(r)$.

$\dot{E}_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=\lim_{r\to\infty}\int_{B(r)}w^{2}\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)\left(\frac{d^{i+2}D^{k}U_{j}}{d\tau^{i+2}}\right)+c^{2}w^{2}\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\cdot\nabla\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)d{\bf{X}}$
(11.28)

Using integration by parts we find

$\displaystyle\dot{E}_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=$
$\displaystyle\lim_{r\to\infty}\int_{B(r)}w^{2}\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)\left(\frac{d^{i+2}D^{k}U_{j}}{d\tau^{i+2}}\right)-c^{2}\nabla\cdot\left(w^{2}\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)d{\bf{X}}$
(11.29) $\displaystyle+\lim_{r\to\infty}\int_{\partial
B(r)}w^{2}c^{2}\left(\frac{\partial\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}}{\partial\nu}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)dS.$

The boundary term vanishes in the limit due to finite propagation speed of the
wave. We need to calculate

$\nabla\cdot\left(w^{2}\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right)=\left(\nabla
w^{2}\right)\cdot\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)+w^{2}\Delta\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right).$
(11.30)

Since $D^{k}U_{j}$ satisfies (2.29), we are left with

$\dot{E}_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=\lim_{r\to\infty}\int_{B(r)}\nabla
w^{2}\cdot\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)d{\bf{X}}.$
(11.31)

Using the assumption on $w$ in (5.9) and Cauchy-Schwarz, we have

$\left|\nabla
w^{2}\cdot\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right|\leq
C\left|\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right|.$ (11.32)

Therefore

$\dot{E}_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq\lim_{r\to\infty}C\int_{B(r)}\left|\nabla\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right|\left|\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right|.d{\bf{X}}$
(11.33)

Using Cauchy-Schwarz once and swapping derivatives

$\dot{E}_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq
C\left\|D^{k+1}\left(\frac{d^{i}U_{j}}{d\tau^{i}}\right)\right\|_{L^{2}}\left\|\frac{D^{k}d^{i+1}U_{j}}{d\tau^{i+1}}\right\|_{L^{2}}.$
(11.34)

From (5.7), we know how to bound such beasts.

$\dot{E}_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq
C\left(\left\|\nabla\phi\right\|^{2}_{H^{i+k}}+\left\|\psi\right\|^{2}_{H^{i+k}}\right).$
(11.35)

Integrating yields

$E_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq|\tau|C\left(\left\|\phi\right\|^{2}_{H^{i+k+1}}+\left\|\psi\right\|^{2}_{H^{i+k}}\right)+E_{w}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(0).$
(11.36)

Using (11.24) or (11.25) and the definition (5.12)

$E_{w}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(0)\leq
C\left(\left\|\nabla\phi\right\|_{H^{i+k}_{w}}^{2}+\left\|\psi\right\|_{H^{i+k}_{w}}^{2}\right).$
(11.37)

This holds for all $j\in\\{1,2,\dots,2^{k}\\}.$ Thus

$\left|E_{w}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq|\tau|C\left(\left\|\phi\right\|^{2}_{H_{w}^{i+k}}+\left\|\psi\right\|^{2}_{H_{w}^{i+k}}\right).$
(11.38)

∎

######  Lemma 11.8.

Let $U$ satisfy (2.29) with initial conditions given by (5.3)

$\left|\widetilde{E}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(\tau)\right|\leq
C\left(\left\|\nabla\phi\right\|^{2}_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}+\left\|\psi\right\|^{2}_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}\right).$
(11.39)

The constant $C$ depends only on $\bar{m}.$

###### Proof.

Note that
$D^{k}U({\bf{X}},\tau):{\bf{R}}^{2}\times{\bf{R}}\to{\bf{R}}^{2^{k}}.$
Consider the $j^{\text{th}}$ component of the $2^{k}$ dimensional vector of
$D^{k}U$ denoted by $D^{k}U_{j}$. We apply Leibniz’s rule

$\displaystyle\dot{\widetilde{E}}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=$
$\displaystyle\int_{B(c|\tau|+\epsilon^{-\sigma})^{c}}\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)\left(\frac{d^{i+2}D^{k}U_{j}}{d\tau^{i+2}}\right)+c^{2}D\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\cdot\left(D\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)d{\bf{X}}$
(11.40) $\displaystyle-\operatorname{sgn}(\tau)\frac{1}{2}\int_{\partial
B(c\tau+\epsilon^{-\sigma})}c\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)^{2}+c^{3}\left|D\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right|^{2}dS.$

As we have seen two times already, since $D^{k}U_{j}$ satisfies (2.29),
integration by parts yields

$\displaystyle\dot{\widetilde{E}}^{(i)}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)=$
$\displaystyle-\int_{\partial
B(c\tau+\epsilon^{-\sigma})}c^{2}\left(\frac{\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}}{\partial\nu}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)$
(11.41)
$\displaystyle+\operatorname{sgn}(\tau)\left(\frac{1}{2}c\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)^{2}+\frac{1}{2}c^{3}\left|D\left(\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right)\right|^{2}\right)dS.$

The first term in the integrand is bounded

$c^{2}\left|\left(\frac{\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}}{\partial\nu}\right)\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)\right|\leq
c^{2}\left|D\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right|\left|\frac{d^{i+1}V}{d\tau^{i+1}}\right|\leq\frac{1}{2}c\left(\frac{d^{i+1}D^{k}U_{j}}{d\tau^{i+1}}\right)^{2}+\frac{1}{2}c^{3}\left|D\frac{d^{i}D^{k}U_{j}}{d\tau^{i}}\right|^{2}$
(11.42)

Therefore

$\dot{\widetilde{E}}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq
0$ (11.43)

and thus

$\widetilde{E}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq\widetilde{E}\left(D^{k}\frac{\partial^{i}U}{\partial\tau^{i}}\right)(0).$
(11.44)

Using (11.24) or (11.25)

$\widetilde{E}\left(\frac{\partial^{i}D^{k}U_{j}}{\partial\tau^{i}}\right)(\tau)\leq
C\left(\left\|\nabla\phi\right\|^{2}_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}+\left\|\psi\right\|^{2}_{H^{i+k}(B(\epsilon^{-\sigma})^{c})}\right).$
(11.45)

This holds for all $j\in\\{1,\dots,2^{k}\\}$ therefore yielding the result. ∎

######  Lemma 11.9.

For $\sigma>0$

$\left\|\phi\right\|_{H^{k}(B(\epsilon^{-\sigma})^{c})}\leq\epsilon\left\|\phi\right\|_{H^{k}_{\sigma}}.$
(11.46)

###### Proof.

Consider

$\left\|D^{j}\phi\right\|^{2}_{L^{2}(B(\epsilon^{-\sigma})^{c})}=\int_{B(\epsilon^{-\sigma})^{c}}\left|D^{j}\phi({\bf{X}})\right|^{2}d{\bf{X}}=\int_{B(1)^{c}}\left|D^{j}\phi(\epsilon^{-\sigma}{\bf{X}})\right|^{2}\epsilon^{-2\sigma}d{\bf{X}}.$
(11.47)

Therefore

$\left\|D^{j}\phi\right\|^{2}_{L^{2}(B(\epsilon^{-\sigma})^{c})}=\int_{B(1)^{c}}\frac{\left|\epsilon^{\sigma}{\bf{X}}\right|^{2\sigma^{-1}}}{\left|\epsilon^{\sigma}{\bf{X}}\right|^{2\sigma^{-1}}}\left|D^{k}\phi(\epsilon^{-\sigma}{\bf{X}})\right|^{2}\epsilon^{-2\sigma}d{\bf{X}}\leq\epsilon^{2}\int_{B(\epsilon^{-\sigma})^{c}}\left|{\bf{X}}\right|^{2\sigma^{-1}}\left|D^{k}\phi({\bf{X}})\right|^{2}d{\bf{X}}.$
(11.48)

Thus

$\left\|D^{j}\phi\right\|^{2}_{L^{2}(B(\epsilon^{-\sigma})^{c})}\leq\epsilon^{2}\left\|\left(1+\left|\
\cdot\ \right|\right)^{\sigma^{-1}}D^{j}\phi(\cdot)\right\|^{2}_{L^{2}}.$
(11.49)

Taking square roots, summing over $j$ and comparing with the definition of
$H^{k}_{w}$ in (5.19) yields the result. ∎

###  11.4 Residual Estimates A

######  Lemma 11.10.

Let $f({\bf{x}}):{\bf{R}}^{2}\to{\bf{R}}$ be in $H^{2}$ and let
${\bf{x}}^{({\bf{j}})}\in{\bf{R}}^{2}\cap\prod_{i=1}^{2}[j_{1},j_{1}+1]$ where
${\bf{j}}\in\mathcal{Q}\subset{\bf{Z}}^{2}$. Then

$\sum_{{\bf{j}}\in\mathcal{Q}}f({\bf{x}}^{({\bf{j}})})^{2}\leq\sum_{{\bf{j}}\in\mathcal{Q}}4\left\|f\right\|^{2}_{H^{2}(\prod_{i=1}^{2}[{\bf{j}},{\bf{j}}+{\bf{e}}_{i}])}.$
(11.50)

###### Proof.

Note that $f$ is continues by Sobolev embedding. Let
$\underline{{\bf{x}}}^{({\bf{j}})}$ denote the minimizer of $f$ over
$[j_{1},j_{1}+1]\times\\{x^{({\bf{j}})}_{2}\\}$. We can write

$f({\bf{x}}^{({\bf{j}})})^{2}=f(\underline{{\bf{x}}}^{({\bf{j}})})^{2}+\int_{\underline{x}^{({\bf{j}})}_{1}}^{x^{({\bf{j}})}_{1}}\frac{d}{dx_{1}}f(x_{1},x^{({\bf{j}})}_{2})^{2}dx_{1}.$
(11.51)

We can apply Cauchy-Schwarz to get

$f({\bf{x}}^{({\bf{j}})})^{2}\leq
f(\underline{{\bf{x}}}^{({\bf{j}})})^{2}+2\left|\int_{\underline{x}^{({\bf{j}})}_{1}}^{x^{({\bf{j}})}_{1}}f(x_{1},x^{({\bf{j}})}_{2})^{2}dx_{1}\right|^{\frac{1}{2}}\left|\int_{\underline{x}^{({\bf{j}})}_{1}}^{x^{({\bf{j}})}_{1}}\left(\frac{d}{dx_{1}}f(x_{1},x^{({\bf{j}})}_{2})\right)^{2}dx_{1}\right|^{\frac{1}{2}},$
(11.52)

which becomes

$f({\bf{x}}^{({\bf{j}})})^{2}\leq
f(\underline{{\bf{x}}}^{({\bf{j}})})^{2}+\left|\int_{\underline{x}^{({\bf{j}})}_{1}}^{x^{({\bf{j}})}_{1}}f(x_{1},x^{({\bf{j}})}_{2})^{2}dx_{1}\right|+\left|\int_{\underline{x}^{({\bf{j}})}_{1}}^{x^{({\bf{j}})}_{1}}\left(\frac{d}{dx_{1}}f(x_{1},x^{({\bf{j}})}_{2})\right)^{2}dx_{1}\right|.$
(11.53)

Since the integrands are non-negative

$f({\bf{x}}^{({\bf{j}})})^{2}\leq
f(\underline{{\bf{x}}}^{({\bf{j}})})^{2}+\int_{j_{1}}^{j_{1}+1}f(x_{1},x^{({\bf{j}})}_{2})^{2}dx_{1}+\int_{j_{1}}^{j_{1}+1}\left(\frac{d}{dx_{1}}f(x_{1},x^{({\bf{j}})}_{2})\right)^{2}dx_{1}.$
(11.54)

The first term on the right hand side is now less than the first integral.

$f({\bf{x}}^{({\bf{j}})})^{2}\leq
2\int_{j_{1}}^{j_{1}+1}f(x_{1},x^{({\bf{j}})}_{2})^{2}dx_{1}+\int_{j_{1}}^{j_{1}+1}\left(\frac{d}{dx_{1}}f(x_{1},x^{({\bf{j}})}_{2})\right)^{2}dx_{1}.$
(11.55)

Let

$F(x_{2})^{2}\coloneqq
2\int_{j_{1}}^{j_{1}+1}f(x_{1},x_{2})^{2}dx_{1}+\int_{j_{1}}^{j_{1}+1}\left(\frac{d}{dx_{1}}f(x_{1},x_{2})\right)^{2}dx_{1}.$
(11.56)

Since we are done with the old $\underline{x}^{({\bf{j}})}_{2}$, we use it to
denote the minimizer of $F$ over $[j_{2},j_{2}+1].$ Now similar to before we
get that

$F(x_{2}^{({\bf{j}})})^{2}=F(\underline{x}^{({\bf{j}})}_{2})^{2}+\int_{\underline{x}^{({\bf{j}})}_{2}}^{x_{2}^{({\bf{j}})}}\frac{d}{dx_{2}}F(x_{2})^{2}dx_{2}.$
(11.57)

Now

$\frac{d}{dx_{2}}F(x_{2})^{2}=\int_{j_{1}}^{j_{1}+1}4f(x_{1},x_{2})\frac{\partial}{\partial
x_{2}}f(x_{1},x_{2})+2\frac{\partial}{\partial
x_{1}}f(x_{1},x_{2})\frac{\partial^{2}}{\partial
x_{2}x_{1}}f(x_{1},x_{2})dx_{1}.$ (11.58)

One application of Cauchy-Schwarz to the integrand gives

$\left|\frac{d}{dx_{2}}F(x_{2})^{2}\right|\leq
4\int_{j_{1}}^{j_{1}+1}\left(f^{2}+\left(\frac{\partial}{\partial
x_{1}}f\right)^{2}\right)^{\frac{1}{2}}\left(\left(\frac{\partial}{\partial
x_{2}}f\right)^{2}+\left(\frac{\partial^{2}}{\partial
x_{2}x_{1}}f\right)^{2}\right)^{\frac{1}{2}}dx_{1},$ (11.59)

which becomes

$\left|\frac{d}{dx_{2}}F(x_{2})^{2}\right|\leq
2\int_{j_{1}}^{j_{1}+1}f^{2}+\left(\frac{\partial}{\partial
x_{1}}f\right)^{2}+\left(\frac{\partial}{\partial
x_{2}}f\right)^{2}+\left(\frac{\partial^{2}}{\partial
x_{2}x_{1}}f\right)^{2}dx_{1}.$ (11.60)

Therefore

$\int_{\underline{x}^{({\bf{j}})}_{2}}^{x_{2}^{({\bf{j}})}}\left|\frac{d}{dx_{2}}F^{2}\right|dx_{2}\leq
2\int_{j_{2}}^{j_{2}+1}\int_{j_{1}}^{j_{1}+1}f^{2}+\left(\frac{\partial}{\partial
x_{1}}f\right)^{2}+\left(\frac{\partial}{\partial
x_{2}}f\right)^{2}+\left(\frac{\partial^{2}}{\partial
x_{2}x_{1}}f\right)^{2}dx_{1}dx_{2}.$ (11.61)

Also

$F(\underline{x}_{2}^{({\bf{j}})})^{2}\leq\int_{j_{2}}^{j_{2}+1}F(x_{2})^{2}dx_{2}.$
(11.62)

By the (11.56)

$F(\underline{x}_{2}^{({\bf{j}})})^{2}\leq
2\int_{j_{2}}^{j_{2}+1}\int_{j_{1}}^{j_{1}+1}f(x_{1},x_{2})^{2}+\left(\frac{d}{dx_{1}}f(x_{1},x_{2})\right)^{2}dx_{1}dx_{2}.$
(11.63)

Using (11.61) and (11.63) in (11.57), we have

$F(x_{2}^{({\bf{j}})})^{2}\leq\int_{j_{2}}^{j_{2}+1}\int_{j_{1}}^{j_{1}+1}4f^{2}+4\left(\frac{\partial}{\partial
x_{1}}f\right)^{2}+2\left(\frac{\partial}{\partial
x_{2}}f\right)^{2}+2\left(\frac{\partial^{2}}{\partial
x_{2}x_{1}}f\right)^{2}dx_{1}dx_{2}.$ (11.64)

From (11.55), we have

$f({\bf{x}}^{({\bf{j}})})^{2}\leq\int_{j_{2}}^{j_{2}+1}\int_{j_{1}}^{j_{1}+1}4f^{2}+4\left(\frac{\partial}{\partial
x_{1}}f\right)^{2}+2\left(\frac{\partial}{\partial
x_{2}}f\right)^{2}+2\left(\frac{\partial^{2}}{\partial
x_{2}x_{1}}f\right)^{2}dx_{1}dx_{2}.$ (11.65)

Summing over ${\bf{j}}\in\mathcal{Q}$ we get

$\sum_{{\bf{j}}\in\mathcal{Q}}f({\bf{x}}^{({\bf{j}})})^{2}\leq
4\sum_{{\bf{j}}\in\mathcal{Q}}\left\|f\right\|^{2}_{H^{2}(\prod_{i=1}^{2}[{\bf{j}},{\bf{j}}+{\bf{e}}_{i}])}.$
(11.66)

∎

######  Corollary 11.11.

In the same context as Lemma 11.10 with $\mathcal{Q}={\bf{Z}}^{2},$ we have

$\sum_{{\bf{j}}\in{\bf{Z}}^{2}}f(\epsilon{\bf{x}}^{({\bf{j}})})^{2}\leq\epsilon^{-2}4\left\|f\right\|^{2}_{H^{2}}.$
(11.67)

######  Lemma 11.12.

In the same context as Lemma 11.10 with
$\mathcal{Q}=D({\bf{0}},\epsilon^{-1}R+1)^{c},$ we have

$\sum_{{\bf{j}}\in\mathcal{Q}}f(\epsilon{\bf{x}}^{({\bf{j}})})^{2}\leq\epsilon^{-2}4\left\|f\right\|^{2}_{H^{2}(B(R)^{c})}.$
(11.68)

###### Proof.

From Lemma 11.10

$\sum_{{\bf{j}}\in
D({\bf{0}},\epsilon^{-1}R+1)^{c}}f(\epsilon{\bf{x}}^{({\bf{j}})})^{2}\leq
4\sum_{{\bf{j}}\in
D({\bf{0}},\epsilon^{-1}R+1)^{c}}\left\|f(\epsilon\cdot)\right\|^{2}_{H^{2}(\prod_{i=1}^{2}[{\bf{j}},{\bf{j}}+{\bf{e}}_{i}])}.$
(11.69)

Note that

$\bigcup_{{\bf{j}}\in
D({\bf{0}},\epsilon^{-1}R+1)^{c}}\prod_{i=1}^{2}[{\bf{j}},{\bf{j}}+{\bf{e}}_{i}]\subset
B(\epsilon^{-1}R)^{c}.$ (11.70)

Hence

$\sum_{{\bf{j}}\in
D({\bf{0}},\epsilon^{-1}R+1)^{c}}\left\|f(\epsilon\cdot)\right\|^{2}_{H^{2}(\prod_{i=1}^{2}[{\bf{j}},{\bf{j}}+{\bf{e}}_{i}])}\leq\left\|f(\epsilon\cdot)\right\|^{2}_{H^{2}(B(\epsilon^{-1}R)^{c})}.$
(11.71)

Written in polar coordinates

$\displaystyle\left\|f(\epsilon\cdot)\right\|^{2}_{H^{2}(B(\epsilon^{-1}R)^{c})}=$
$\displaystyle\int_{0}^{2\pi}\int_{\epsilon^{-1}R}^{\infty}f(\theta,\epsilon
r)^{2}rdrd\theta$ (11.72) $\displaystyle=$
$\displaystyle\epsilon^{-2}\int_{0}^{2\pi}\int_{R}^{\infty}f(\theta,s)^{2}sdsd\theta$
$\displaystyle=$
$\displaystyle\epsilon^{-2}\left\|f\right\|^{2}_{H^{2}(B(R)^{c})}.$

∎

######  Lemma 11.13.

Let $\delta^{\pm}_{\epsilon
i}\phi({\bf{X}})\coloneqq\pm(\phi({\bf{X}})-\phi({\bf{X}}-\epsilon{\bf{e}}_{i})).$
Let $w$ be as in assumptions $\eqref{Assumption 1 on Weight}$ and (5.9). Then

$\left\|\delta^{\pm}_{\epsilon i}(\phi)\right\|_{H^{k}_{w}}\leq\epsilon
C\left\|\phi\right\|_{H^{k+1}_{w}}.$ (11.73)

###### Proof.

We do the proof for $i=1$. Let $\partial$ be some generic partial derivative
of order $k$. By the Fundamental Theorem of Calculus

$\partial\delta^{+}_{\epsilon i}\phi({\bf{X}})=\delta^{+}_{\epsilon
i}\partial\phi({\bf{X}})=\int_{X_{1}}^{X_{1}+\epsilon}\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})dX_{1}^{\prime}.$
(11.74)

Now we calculate

$\displaystyle\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}w(X_{1},X_{2})^{2}\left(\partial\delta_{\epsilon
i}\phi(X_{1},X_{2})^{2}\right)^{2}dX_{1}dX_{2}$ (11.75)
$\displaystyle=\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}w(X_{1},X_{2})^{2}\left(\int_{X_{1}}^{X_{1}+\epsilon}\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})dX_{1}^{\prime}\right)^{2}dX_{1}dX_{2}$

An application of Jensen’s Inequality yields

$\displaystyle\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}\left(\int_{X_{1}}^{X_{1}+\epsilon}\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})dX_{1}^{\prime}\right)^{2}dX_{1}dX_{2}$
(11.76)
$\displaystyle=\epsilon\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}w(X_{1},X_{2})^{2}\int_{X_{1}}^{X_{1}+\epsilon}\left(\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})\right)^{2}dX_{1}^{\prime}dX_{1}dX_{2}.$

Now we flip the $X_{1}$ and $X_{1}^{\prime}$

$\displaystyle\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}w(X_{1},X_{2})^{2}\int_{X_{1}}^{X_{1}+\epsilon}\left(\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})\right)^{2}dX_{1}^{\prime}dX_{1}dX_{2}$
(11.77)
$\displaystyle=\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}\left(\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})\right)^{2}\int_{X_{1}^{\prime}-\epsilon}^{X_{1}^{\prime}}w(X_{1},X_{2})^{2}dX_{1}dX_{1}^{\prime}dX_{2}$

From our definition of $w$ in (5.8)

$\displaystyle\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}\left(\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})\right)^{2}\int_{X_{1}^{\prime}-\epsilon}^{X_{1}^{\prime}}w(X_{1},X_{2})^{2}dX_{1}dX_{1}^{\prime}dX_{2}$
(11.78)
$\displaystyle\leq\epsilon\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}\left(\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})\right)^{2}w(X_{1}^{\prime},X_{2})^{2}dX_{1}^{\prime}dX_{2}.$

∎

Stringing these inequalities together, we find

$\displaystyle\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}w(X_{1},X_{2})^{2}\left(\partial\delta_{\epsilon
i}\phi(X_{1},X_{2})^{2}\right)^{2}dX_{1}dX_{2}$ (11.79)
$\displaystyle\leq\epsilon^{2}\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}\left(\partial_{X^{\prime}_{1}}\partial\phi(X_{1}^{\prime},X_{2})\right)^{2}w(X_{1}^{\prime},X_{2})^{2}dX_{1}^{\prime}dX_{2},$

which implies

$\left\|\delta^{+}_{\epsilon
i}\phi\right\|_{H^{k}_{w}}\leq\epsilon\left\|\phi\right\|_{H^{k+1}_{w}}.$
(11.80)

Now let $S^{-}_{\epsilon
i}\phi({\bf{X}})\coloneqq\phi({\bf{X}}-\epsilon{\bf{e}}_{i})$. Thus
$\delta_{\epsilon i}^{-}\phi=S_{\epsilon i}^{-}\delta^{+}_{\epsilon i}\phi$.
We now show that $S_{\epsilon i}^{-}$ is a bounded operator. Consider any
$f\in L^{2}$, then

$\int_{{\bf{R}}^{2}}w({\bf{X}})^{2}f({\bf{X}}-\epsilon{\bf{e}}_{i})^{2}d{\bf{X}}=\int_{{\bf{R}}^{2}}w({\bf{X}}+\epsilon{\bf{e}}_{i})^{2}f({\bf{X}})^{2}d{\bf{X}}.$
(11.81)

From the definition of the weight in (5.8), there exists a constant $C$ s.t.

$w({\bf{X}}+\epsilon{\bf{e}}_{i})^{2}\leq C^{2}w({\bf{X}})^{2}.$ (11.82)

Therefore the operator $S_{\epsilon i}^{-}$ is bounded by this constant $C.$
Thus

$\left\|\delta^{-}_{\epsilon i}\phi\right\|_{H^{k}_{w}}\leq\epsilon
C\left\|\phi\right\|_{H^{k+1}_{w}}.$ (11.83)

######  Corollary 11.14.

Let
$\Delta_{\epsilon}\phi({\bf{X}})\coloneqq\phi({\bf{X}}+\epsilon{\bf{e}}_{i})-2\phi({\bf{X}})+\phi({\bf{X}}-\epsilon{\bf{e}}_{i})$
and $w$ as in the previous lemma. Then

$\left\|\Delta_{\epsilon}\phi\right\|_{H_{w}^{k}}\leq\epsilon^{2}C\left\|\phi\right\|_{H^{k+2}_{w}}.$
(11.84)

###  11.5 Coarse Graining A

######  Lemma 11.15.

Let $U:{\bf{R}}^{2}\to{\bf{R}}^{2}$ be in $H^{s}$ with $s>1$. Put
$U_{\epsilon}({\bf{x}})\coloneqq U(\epsilon{\bf{x}}).$ Then

$\lim_{\epsilon\to
0^{+}}\left\|\mathcal{LS}[U_{\epsilon}](\cdot/\epsilon)-U\right\|_{L^{2}}=0$
(11.85)

where $\mathcal{L}$ and $\mathcal{S}$ are defined by (8.5) and (8.3).

###### Proof.

From their definitions we have

$\mathcal{LS}[U_{\epsilon}]({\bf{x}})=\frac{1}{4\pi}\int_{[-\pi,\pi]^{2}}\left(\sum_{{\bf{j}}\in{\bf{Z}}}e^{-i{\bf{y}}\cdot{\bf{j}}}U(\epsilon{\bf{j}})\right)e^{i{\bf{y}}\cdot{\bf{x}}}d{\bf{y}}.$
(11.86)

Changing variables from ${\bf{y}}\to\epsilon{\bf{y}}$ gives

$\mathcal{LS}[U_{\epsilon}]({\bf{x}})=\frac{\epsilon^{2}}{4\pi}\int_{[-\pi/\epsilon,\pi/\epsilon]^{2}}\left(\sum_{{\bf{j}}\in{\bf{Z}}}e^{-i\epsilon{\bf{y}}\cdot{\bf{j}}}U(\epsilon{\bf{j}})\right)e^{i\epsilon{\bf{y}}\cdot{\bf{x}}}d{\bf{y}}.$
(11.87)

Now swap the integral and the sum and integrate to get

$\mathcal{LS}[U_{\epsilon}]({\bf{x}})=\sum_{{\bf{j}}\in{\bf{Z}}^{2}}U(\epsilon{\bf{j}})\operatorname{sinc}(x_{1}-j_{1})\operatorname{sinc}(x_{2}-j_{2}),$
(11.88)

where $\operatorname{sinc}$ is the normalized $\operatorname{sinc}$ function,
$\frac{\sin(\pi x)}{\pi x}.$ Subbing ${\bf{x}}\to{\bf{X}}/\epsilon$ yields

$\mathcal{LS}[U_{\epsilon}]({\bf{X}}/\epsilon)=\sum_{{\bf{j}}\in{\bf{Z}}^{2}}U(\epsilon{\bf{j}})\operatorname{sinc}(X_{1}/\epsilon-
j_{1})\operatorname{sinc}(X_{2}/\epsilon-j_{2}).$ (11.89)

This series is exactly equal to the band-limited approximation of
$U_{\epsilon}$ given by

$\mathcal{LS}[U_{\epsilon}]({\bf{X}}/\epsilon)=\widetilde{U}_{\epsilon}({\bf{X}})\coloneqq\mathcal{F}^{-1}[\theta_{\pi/\epsilon}\mathcal{F}[U]]({\bf{X}}).$
(11.90)

See [15] for details regarding band-limited functions. Now we only need to
compare $U$ and $\widetilde{U}_{\epsilon}$, but by use of Plancherel’s
Theorem, this is equivalent to looking at

$\displaystyle\left\|\theta_{\pi/\epsilon}\mathcal{F}[U]-\mathcal{F}[U]\right\|^{2}_{L^{2}}$
$\displaystyle=\int_{|{\bf{y}}|_{\infty}>\pi/\epsilon}\widehat{U}({\bf{y}})^{2}d{\bf{y}}$
(11.91)
$\displaystyle\leq\sup_{|{\bf{y}}|_{\infty}\geq\pi/\epsilon}\left(\frac{1}{|{\bf{y}}|^{2s}}\right)\int_{|{\bf{y}}|_{\infty}>\pi/\epsilon}|{\bf{y}}|^{2s}\widehat{U}({\bf{y}})^{2}d{\bf{y}}$
$\displaystyle\leq C\epsilon^{2s}\left\|U\right\|^{2}_{H^{s}}.$

Since we have taken $s>1$, the result is proved and the rate of convergence is
greater than $\epsilon.$

∎

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