# A Riccati-Lyapunov Approach to Nonfeedback Capacity of MIMO Gaussian
Channels Driven by Stable and Unstable Noise

Charalambos D. Charalambous and Stelios Louka  University of Cyprus
chadcha<EMAIL_ADDRESS>
###### Abstract

In this paper it is shown that the nonfeedback capacity of multiple-input
multiple-output (MIMO) additive Gaussian noise (AGN) channels, when the noise
is nonstationary and unstable, is characterized by an asymptotic optimization
problem that involves, a generalized matrix algebraic Riccati equation (ARE)
of filtering theory, and a matrix Lyapunov equation of stability theory of
Gaussian systems. Furthermore, conditions are identified such that, the
characterization of nonfeedback capacity corresponds to the uniform asymptotic
per unit time limit, over all initial distributions, of the characterization
of a finite block or transmission without feedback information (FTwFI)
capacity, which involves, two generalized matrix difference Riccati equations
(DREs) and a matrix difference Lyapunov equation.

##  I Introduction, Problem, and Main Results

Shannon’s information theoretic definition of nonfeedback capacity of additive
Gaussian noise (AGN) channels with memory, is a fundamental mathematical tool
for analysis and synthesis of reliable data transmission over noisy
communication channels. The characterization of nonfeedback capacity in
frequency-domain, for stationary or asymptotically stationary channels, i.e.,
when the channel noise inputs and outputs are asymptotically stationary, gave
rise to the so-called water-filling solution, which is documented in [1, 2, 3,
4] and in several research papers, such as Tsybakov [5]. The analysis of
channel capacity for asymptotically equivalent matrices for MIMO Gaussian
channels, is found in [6, 7] and more recently in [8]. For Gaussian channels
with intersymbol interference in [9].

Bounds on nonfeedback capacity for single-input single-output (SISO) AGN
channel with stable noise with memory, are derived in [10, 11, 12], as well as
comparisons to feedback capacity. Recently, sequential time-domain
characterizations of nonfeedback capacity for SISO AGN channels with unstable,
finite-memory autoregressive noise are presented in [13]. Sequential
characterization of nonfeedback capacity for SISO AGN channels with general
unstable noise with memory, are obtained in [14], and lower bounds in [15,
Corollary II.1]. Additional lower bounds are discussed in [16], [17], which
are equivalent to the Cover and Pombra [18] nonfeeback capacity formula.
Equivalent characterizations of Cover and Pombra $n-$finite transmission, or
block length for MIMO AGN channels with memory are presented in [19].

The purpose of this paper is to derive new results on nonfeedback capacity for
multiple-input multiple-output (MIMO) AGN channels, driven by unstable,
nonstationary and nonergodic noise in time-domain, described by

$\displaystyle Y_{t}=H_{t}X_{t}+V_{t},\ \ t=1,\ldots,n,\ \ \frac{1}{n}{\bf
E}\Big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\Big{\\}}\leq\kappa$
(I.1)

where $\kappa\in[0,\infty)$,
$X_{t}:\Omega\rightarrow{\mathbb{X}}\stackrel{{\scriptstyle\triangle}}{{=}}{\mathbb{R}}^{n_{x}}$,
$Y_{t}:\Omega\rightarrow{\mathbb{Y}}\stackrel{{\scriptstyle\triangle}}{{=}}{\mathbb{R}}^{n_{y}}$,
and
$V_{t}:\Omega\rightarrow{\mathbb{V}}\stackrel{{\scriptstyle\triangle}}{{=}}{\mathbb{R}}^{n_{y}}$,
are the channel input, channel output and noise random variables (RVs),
respectively, $(n_{x},n_{y})$ are finite positive integers,
$H_{t}\in{\mathbb{R}}^{n_{y}\times n_{x}}$ is nonrandom and the distribution
of the sequence $V^{n}=\\{V_{1},\ldots,V_{n}\\}$, i.e., ${\bf
P}_{V^{n}}\stackrel{{\scriptstyle\triangle}}{{=}}{\mathbb{P}}\\{V_{1}\leq
v_{1},\ldots,V_{n}\leq v_{n}\\}$, is jointly Gaussian.
The main fundamental difference from previous characterizations of nonfeedback
capacity found in the literature, i.e., [1, 2, 3, 4, 5], is that the
consideration of nonergodic, time-varying and unstable noise, gives higher
achievable rates compared to stable noise (see [14, 16, 17, 13]).

Operational Nonfeedback Code. The code for the MIMO AGN channel (I.1), is
denoted by $\\{(n,{\cal M}^{(n)},\epsilon_{n},\kappa):n=0,1,\dots\\}$ and
consists of (a) a set of uniformly distributed messages
$M:\Omega\rightarrow{\cal
M}^{(n)}\stackrel{{\scriptstyle\triangle}}{{=}}\\{1,\ldots,M^{(n)}\\}$, known
to the encoder and decoder, (b) a set of encoder strategies mapping messages
$M=m$ and past channel inputs into current inputs, defined by111The
superscript on expectation operator ${\bf`E}^{g}$ indicates that the
corresponding distribution ${\bf P}={\bf P}^{g}$ depends on the encoding
strategy $g$.

$\displaystyle{\cal E}_{n}(\kappa)\triangleq\Big{\\{}g_{i}:{\cal
M}^{(n)}\times{\mathbb{X}}^{i-1}\rightarrow{\mathbb{X}}_{i},\>x_{1}=g_{1}(m),\;x_{2}=g_{2}(m,x_{1}),$
$\displaystyle\ldots,x_{n}=g_{n}(m,x^{n-1})\;\Big{|}\;\frac{1}{n}{\bf
E}^{g}\Big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\Big{\\}}\leq\kappa\Big{\\}}$
(I.2)

where $g_{i}(\cdot)$ are measurable maps and (c) a decoder
$d_{n}(\cdot):{\mathbb{Y}}^{n}\rightarrow{\cal M}^{(n)}$, with average
probability of decoding error

$\displaystyle{\bf P}_{error}^{(n)}\triangleq$
$\displaystyle\frac{1}{M^{(n)}}\sum_{m\in{\cal M}^{(n)}}{\bf
P}^{g}\Big{\\{}d_{n}(Y^{n})\neq m\big{|}M=m\Big{\\}}\leq\epsilon_{n}.$ (I.3)

Note that ${\bf P}_{error}^{(n)}$ depends on the distribution of $V_{1}$,
i.e., ${\bf P}_{V_{1}}$. The messages $M:\Omega\rightarrow{\cal M}^{(n)}$ are
assumed independent of the noise sequences $V^{n}$, that is ${\bf
P}_{V^{n}|M}={\bf P}_{V^{n}}$. The code rate is defined by
$r_{n}\triangleq\frac{1}{n}\log M^{(n)}$. A rate $R$ is called an achievable
rate, if there exists an encoder and decoder sequence satisfying
$\lim_{n\longrightarrow\infty}{\epsilon}_{n}=0$ and
$\liminf_{n\longrightarrow\infty}\frac{1}{n}\log{M}^{(n)}\geq R$. The
operational definition of the nonfeedback capacity is
$C^{OP}(\kappa)\triangleq\sup\\{R\big{|}R\>\>\mbox{is
achievable}\\}\;\forall\;{\bf P}_{V_{1}}$, i.e., it does not depend on ${\bf
P}_{V_{1}}$.

###  I-A Main Results of the Paper

Throughout this paper, we consider the noise of Definition I.1.

######  Definition I.1.

A time-varying partially observable state space (PO-SS) realization of the
Gaussian noise $V^{n}$, is defined by

$\displaystyle S_{t+1}=A_{t}S_{t}+B_{t}W_{t},\hskip 7.22743ptt=1,\ldots,n-1$
(I.4) $\displaystyle V_{t}=C_{t}S_{t}+N_{t}W_{t},\hskip
7.22743ptt=1,\ldots,n,$ (I.5) $\displaystyle S_{1}\in
G(\mu_{S_{1}},K_{S_{1}}),\hskip 7.22743ptK_{S_{1}}\succeq 0,$ (I.6)
$\displaystyle W_{t}\in G(0,K_{W_{t}}),\hskip 7.22743ptK_{W_{t}}\succ 0,\hskip
7.22743ptt=1\ldots,n,$ (I.7) $\displaystyle
S_{t}:\Omega\rightarrow{\mathbb{R}}^{n_{s}},\
W_{t}:\Omega\rightarrow{\mathbb{R}}^{n_{w}},\
V_{t}:\Omega\rightarrow{\mathbb{R}}^{n_{y}},$ (I.8) $\displaystyle
R_{t}\stackrel{{\scriptstyle\triangle}}{{=}}N_{t}K_{W_{t}}N_{t}^{T}\succ
0,\hskip 7.22743ptt=1,\ldots,n$ (I.9)

where $W_{t},t=1\ldots,n$ is an independent Gaussian process, independent of
$S_{1}$, $n_{y},n_{s},n_{w}$ are arbitrary positive integers,
$(A_{t},B_{t},C_{t},N_{t},K_{S_{1}},K_{W_{t}})$ are nonrandom, $X\in
G(\mu_{X},K_{X})$ means $X$ is Gaussian distributed with mean $\mu_{X}$ and
covariance $K_{X}$ and for any matrix $Q\in{\mathbb{R}}^{n\times n}$. The
notation $Q\succeq 0$ (resp. $Q\succ 0$) means $Q$ is symmetric positive
semidefinite (resp. definite).

Converse Coding Theorem. Suppose there exists a sequence of achievable
nonfeedback codes with error probability ${\bf P}_{error}^{(n)}\rightarrow 0$,
as $n\rightarrow\infty$. Then
$R\leq\lim_{n\longrightarrow\infty}\frac{1}{n}C_{n}(\kappa,{\bf P}_{Y_{1}})$,
where $C_{n}(\kappa,{\bf P}_{Y_{1}})$ is the sequential characterization of
the $n-$finite block length, or transmission without feedback information
($n$-FTwFI) capacity formula [19, Section I, III], given as follows.

$\displaystyle C_{n}(\kappa,{\bf P}_{Y_{1}}$
$\displaystyle)=\sup_{\frac{1}{n}{\bf
E}\big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\big{\\}}\leq\kappa}\sum_{t=1}^{n}I(X_{t},V^{t-1};Y_{t}|Y^{t-1})$
(I.10) $\displaystyle=$ $\displaystyle\sup_{\frac{1}{n}{\bf
E}\big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\big{\\}}\leq\kappa}H(Y^{n})-H(V^{n})\in[0,\infty]$
(I.11)

where (I.11) follows from the channel definition (I.1) (provided the
probability density functions exist) and the supremum is over ${\bf
P}_{X_{t}|X^{t-1}},t=1,\ldots,n$ induced by,

$\displaystyle X_{t}=\Lambda_{t}{\bf X}^{t-1}+Z_{t},\hskip
7.22743ptX_{1}=Z_{1},$ (I.12) $\displaystyle{\bf
X}^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\left[\begin{array}[]{cccc}X_{1}^{T}&X_{2}^{T}&\ldots&X_{n}^{T}\end{array}\right]^{T},$
(I.14) $\displaystyle Z_{t}\in G(0,K_{Z_{t}}),\;K_{Z_{t}}\succeq
0,\;t=1,\ldots,n,\mbox{ indep. Gaussian},$ (I.15) $\displaystyle
Z_{t}\;\mbox{independent of}\;(V^{t-1},X^{t-1},Y^{t-1},Z^{t-1}),t=1,\ldots,n,$
(I.16)
$\displaystyle{\Lambda}_{t}\in{\mathbb{R}}^{n_{x}\times(t-1)n_{x}},\;\mbox{$\forall
t$ is nonrandom}.$ (I.17)

By recursive substitution of ${\bf X}^{t-1}$ into the right hand side of
(I.12), we obtain $X_{t}=\overline{Z}_{t}$, where $\overline{Z}_{t}\in
G(0,K_{\overline{Z}_{t}}),t=1,\ldots,n$, is a correlated Gaussian noise, as
given in [18]. However, for the purpose of our asymptotic analysis, we prefer
(I.12)-(I.17), because it is much easier to analyse.

We emphasize that the consideration of unstable noise $V^{n}$ implies $Y^{n}$
is also unstable and therefore for our asymptotic analysis, we need to use the
two innovations processes of $V^{n}$ and $Y^{n}$, as in [14, 16, 17, 13],
giving rise to the following characterization of ${C}_{n}(\kappa,{\bf
P}_{Y_{1}})\in[0,\infty]$ given by,

$\displaystyle{C}_{n}(\kappa,{\bf
P}_{Y_{1}})=\sup_{(\Lambda_{t},K_{\overline{Z}_{t}}),t=1,\ldots,n,\frac{1}{n}{\bf
E}\big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\big{\\}}\leq\kappa}\sum_{t=1}^{n}\Big{\\{}H(I_{t})-H(\hat{I}_{t})\Big{\\}}$
(I.18)

where $I_{t},\hat{I}_{t}$ are the innovations processes of $Y^{n},V^{n}$,

$\displaystyle I_{t}\stackrel{{\scriptstyle\triangle}}{{=}}Y_{t}-{\bf
E}\big{\\{}Y_{t}\Big{|}Y^{t-1}\big{\\}},\hskip
14.45377pt\hat{I}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}V_{t}-{\bf
E}\big{\\{}V_{t}\Big{|}V^{t-1}\big{\\}}.$ (I.19)

Clearly, the convergence properties of
$\lim_{n\longrightarrow\infty}\frac{1}{n}C_{n}(\kappa,{\bf P}_{Y_{1}})$ and
its independent of all ${\bf P}_{Y_{1}}$, are directly related to the
convergence properties of $(I_{t},\hat{I}_{t},X_{n}),t=1,2,\ldots,n$, as
$n\rightarrow\infty$.

State Space Realization of Channel Input. For the analysis of the limit
$\lim_{n\longrightarrow\infty}\frac{1}{n}C_{n}(\kappa,{\bf P}_{Y_{1}})$, we
consider the alternative, state space realization of the Gaussian input
$X^{n}$.

$\displaystyle\Xi_{t+1}=F_{t}\Xi_{t}+G_{t}Z_{t},\hskip
7.22743ptt=1,\ldots,n-1,$ (I.20) $\displaystyle
X_{t}=\Gamma_{t}\Xi_{t}+D_{t}Z_{t},\hskip 7.22743ptt=1,\ldots,n,$ (I.21)
$\displaystyle\Xi_{1}\in G(\mu_{\Xi_{1}},K_{\Xi_{1}}),\;K_{\Xi_{1}}\succeq 0,$
(I.22) $\displaystyle Z_{t}\in G(0,K_{Z_{t}}),\hskip 7.22743ptK_{Z_{t}}\succeq
0,\hskip 7.22743ptt=1\ldots,n,$ (I.23) $\displaystyle\mbox{$Z^{n}$ indep.
seq.},\hskip 7.22743pt\mbox{$(\Xi_{1},Z^{n},W^{n})$ mutually indep.}$ (I.24)
$\displaystyle\Xi_{t}:\Omega\rightarrow{\mathbb{R}}^{n_{\xi}},\
Z_{t}:\Omega\rightarrow{\mathbb{R}}^{n_{z}},\
X_{t}:\Omega\rightarrow{\mathbb{R}}^{n_{x}}$ (I.25)

where $n_{\xi},n_{z}$ are arbitrary positive integers and
$(F_{t},G_{t},\Gamma_{t},D_{t},K_{\Xi_{1}},K_{Z_{t}})$ are nonrandom matrices
$\forall t$.
Note that any finite-memory $AR$ input
$X_{t}=\sum_{j=1}^{M}\Lambda_{t,j}X_{t-j}+Z_{t}$, with arbitrary large $M$,
approximates (I.12), and this is a special case of (I.20)-(I.25). The
performance of such inputs with $M=1$ discussed in [13, Section IV] and [16,
Theorem III.3] for IID input. Asymptotic Characterization of Capacity. The
purpose of this paper is to analyze the two problems listed below.
Problem #1. Identify conditions, such that asymptotic limit exists and does
not depend on ${\bf P}_{Y_{1}}$,

$\displaystyle C^{o}(\kappa,{\bf
P}_{Y_{1}})\stackrel{{\scriptstyle\triangle}}{{=}}\lim_{n\longrightarrow\infty}\frac{1}{n}C_{n}(\kappa,{\bf
P}_{Y_{1}})=C(\kappa)\in[0,\infty),\hskip 7.22743pt\forall{\bf P}_{Y_{1}}$
(I.26)

and characterize $C(\kappa)$, which is independent of ${\bf P}_{Y_{1}}$.

######  Assumptions I.1.

Considered for the asymptotic analysis are the two cases of noise and channel
input realizations.
Case 1: Time-invariant,

$\displaystyle(A_{n},B_{n},C_{n},N_{n},K_{W_{n}})=(A,B,C,N,K_{W}),\;K_{W}\succ
0,\hskip 7.22743pt\forall n$ (I.27)
$\displaystyle(F_{n},G_{n},\Gamma_{n},D_{n},K_{Z_{n}})=(F,G,\Gamma,D,K_{Z}),K_{Z}\succeq
0,\hskip 7.22743pt\forall n.$ (I.28)

Case 2: Asymptotically time-invariant,

$\displaystyle\lim_{n\longrightarrow\infty}(A_{n},B_{n},C_{n},N_{n},K_{W_{n}})=(A,B,C,N,K_{W}),\>K_{W}\succ
0,$ (I.29)
$\displaystyle\lim_{n\longrightarrow\infty}(F_{n},G_{n},\Gamma_{n},D_{n},K_{Z_{n}})=(F,G,\Gamma,D,K_{Z}),\>K_{Z}\succeq
0$ (I.30)

where the limits are element wise. For both Cases 1 and 2, the time-invariant
realizations are assumed of minimal dimensions.

For Cases 1 and 2, we identify conditions on 1) channel model matrices
$(H,A,B,C,N,K_{W})$ and 2) channel input matrices
$(F,G,K_{Z},\Gamma,D),K_{Z}\succeq 0$, such that the limit in (I.26) exists,
is independent of ${\bf P}_{Y_{1}}$ and is characterized by

$\displaystyle C^{o}(\kappa,{\bf
P}_{Y_{1}})=C(\kappa)\stackrel{{\scriptstyle\triangle}}{{=}}\sup\frac{1}{2}\ln\big{\\{}\frac{\det\big{(}{\bf
C}\Pi{\bf C}^{T}+{\bf D}K_{\overline{W}}{\bf D}^{T}\big{)}}{\det\big{(}C\Sigma
C^{T}+NK_{W}N^{T}\big{)}}\big{\\}}^{+}$ (I.31) $\displaystyle\mbox{the
supremum is over}\;(F,G,\Gamma,D,K_{Z})\;\;\mbox{and}$ $\displaystyle
K_{Z}\succeq 0,\;tr(\Gamma P\Gamma^{T}+DK_{Z}D^{T})\leq\kappa,$ (I.32)
$\displaystyle\Pi\succeq 0,\;\Sigma\succeq 0\;\mbox{satisfy matrix Algebraic
Riccati Eqns},$ (I.33) $\displaystyle P\succeq 0\hskip
7.22743pt\mbox{satisfies a matrix Lyapunov Equation}$ (I.34)

and where
$\\{\cdot\\}^{+}\stackrel{{\scriptstyle\triangle}}{{=}}\max\\{1,\cdot\\}$, the
$({\bf C},{\bf D},K_{\overline{W}})$ are specific matrices, related to the
channel model and channel input matrices.

Problem #2. Under the conditions of Problem #1, we also show that the limit
and the supremum can be interchanged and

$\displaystyle C^{\infty}(\kappa,{\bf
P}_{Y_{1}})\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\sup_{\lim_{n\longrightarrow\infty}\frac{1}{n}{\bf
E}\big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\big{\\}}\leq\kappa}\lim_{n\longrightarrow\infty}\frac{1}{n}\Big{(}\sum_{t=1}^{n}H(I_{t})-H(\hat{I}_{t})\Big{)}$
(I.35) $\displaystyle=$ $\displaystyle C^{o}(\kappa,{\bf
P}_{Y_{1}})=C(\kappa)\in[0,\infty),\hskip 7.22743pt\forall{\bf P}_{Y}.$ (I.36)

Direct Coding Theorem. By (I.31) and (I.35), the convergence is uniform over
all ${\bf P}_{Y_{1}}$. Hence, the asymptotic equipartition (AEP) and the
information stability hold, from which follows directly that $C(\kappa)$ is
the nonfeedback capacity, even for unstable channels, similar to the feedback
capacity in [20, 21].

Notation.
${\mathbb{Z}}_{+}\stackrel{{\scriptstyle\triangle}}{{=}}\\{1,2,\ldots\\},{\mathbb{Z}}_{+}^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{1,2,\ldots,n\\}$,
where $n$ is a finite positive integer.
${\mathbb{R}}\stackrel{{\scriptstyle\triangle}}{{=}}(-\infty,\infty)$, and
${\mathbb{R}}^{m}$ is the vector space of tuples of the real numbers for an
integer $m\in{\mathbb{Z}}_{+}$. ${\mathbb{R}}^{n\times m}$ is the set of $n$
by $m$ matrices with entries from the set of real numbers for
$(n,m)\in{\mathbb{Z}}_{+}\times{\mathbb{Z}}_{+}$.
$I_{n}\in{\mathbb{R}}^{n\times n},n\in{\mathbb{Z}}_{+}$ denotes the identity
matrix, $tr\big{(}A\big{)}$ denotes the trace of any matrix
$A\in{\mathbb{R}}^{n\times n},n\in{\mathbb{Z}}_{+}$.
${\mathbb{C}}\stackrel{{\scriptstyle\triangle}}{{=}}\\{a+jb:(a,b)\in{\mathbb{R}}\times{\mathbb{R}}\\}$
is the space of complex numbers.
${\mathbb{D}}_{o}\stackrel{{\scriptstyle\triangle}}{{=}}\big{\\{}c\in{\mathbb{C}}:|c|<1\big{\\}}$
is the open unit disc of the space of complex number ${\mathbb{C}}$.
$spec(A)\subset{\mathbb{C}}$ is the spectrum of a matrix
$A\in{\mathbb{R}}^{q\times q},q\in{\mathbb{Z}}_{+}$ (the set of all its
eigenvalues). A matrix $A\in{\mathbb{R}}^{q\times q}$ is called exponentially
stable if all its eigenvalues are within the open unit disc, that is,
$spec(A)\subset{\mathbb{D}}_{o}$.
$X\in G(\mu_{X},K_{X}),K_{X}\succeq 0$ denotes a Gaussian distributed RV $X$,
with mean $\mu_{X}={\bf E}\\{X\\}$ and covariance $K_{X}\succeq 0$,
$K_{X}=cov(X,X)\stackrel{{\scriptstyle\triangle}}{{=}}{\bf
E}\big{\\{}\big{(}X-{\bf E}\big{\\{}X\big{\\}}\big{)}\big{(}X-{\bf
E}\big{\\{}X\big{\\}}\big{)}^{T}\big{\\}}$. Given another Gaussian RV
$Y:\Omega\rightarrow{\mathbb{R}}^{n_{y}}$, which is jointly Gaussian
distributed with $X$, i.e., with joint distribution ${\bf P}_{X,Y}$, the
conditional covariance of $X$ given $Y$ is (by properties of Gaussian RVs)

$\displaystyle
K_{X|Y}=cov(X,X\Big{|}Y)\stackrel{{\scriptstyle\triangle}}{{=}}{\bf
E}\big{\\{}\big{(}X-{\bf E}\big{\\{}X\Big{|}Y\big{\\}}\big{)}\big{(}X-{\bf
E}\big{\\{}X\Big{|}Y\big{\\}}\big{)}^{T}\big{\\}}.$

##  II Asymptotic Characterization of Capacity

###  II-A Sequential Characterizations of $n-$FTwFI Capacity

We recall the characterization of $C_{n}(\kappa,{\bf P}_{Y_{1}})$ of (I.18),
which shows its dependence on two DREs and a Lyapunov equation.

######  Theorem II.1.

[19, Theorem III.2] Sequential characterization of ${C}_{n}(\kappa,{\bf
P}_{Y_{1}})$. Consider the MIMO AGN channel (I.1), the noise of Definition I.1
and the input (I.20)-(I.25). Define

$\displaystyle\Theta_{t}\stackrel{{\scriptstyle\triangle}}{{=}}\left(\begin{array}[]{cc}\Xi_{t}^{T}&S_{t}^{T}\end{array}\right)^{T},\hskip
7.22743pt\overline{W}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}\left(\begin{array}[]{cc}Z_{t}^{T}&W_{t}^{T}\end{array}\right)^{T},\hskip
7.22743pt\overline{W}_{t}\in G(0,K_{\overline{W}_{t}}),$
$\displaystyle\Pi_{t}\stackrel{{\scriptstyle\triangle}}{{=}}cov\big{(}\Theta_{t},\Theta_{t}\Big{|}Y^{t-1}\big{)}={\bf
E}\Big{\\{}\Big{(}\Theta-\widehat{\Theta}_{t}\Big{)}\Big{(}\Theta_{t}-\widehat{\Theta}_{t}\Big{)}^{T}\Big{\\}},$
$\displaystyle\widehat{\Theta}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}{\bf
E}\Big{\\{}\Theta_{t}\Big{|}Y^{t-1}\Big{\\}},\;t=2,\ldots,n,\;\widehat{\Theta}_{1}\stackrel{{\scriptstyle\triangle}}{{=}}\mu_{\Theta_{1}},\;\Pi_{1}\stackrel{{\scriptstyle\triangle}}{{=}}K_{\Theta_{1}},$
$\displaystyle
P_{t}\stackrel{{\scriptstyle\triangle}}{{=}}cov\big{(}\Xi_{t},\Xi_{t})={\bf
E}\Big{\\{}\Big{(}\Xi_{t}-{\bf
E}\Big{\\{}\Xi_{t}\Big{\\}}\Big{)}\Big{(}\Xi_{t}-{\bf
E}\Big{\\{}\Xi_{t}\Big{\\}}\Big{)}^{T}\Big{\\}}.$

(i) The joint Gaussian process $(X^{n},Y^{n},V^{n})$ is represented by

$\displaystyle\Theta_{t+1}={\bf A}_{t}\Theta_{t}+{\bf
B}_{t}\overline{W}_{t},\hskip 7.22743ptt=1,\ldots,n-1,$ (II.37) $\displaystyle
Y_{t}={\bf C}_{t}\Theta_{t}+{\bf D}_{t}\overline{W}_{t},\hskip
7.22743ptt=1,\ldots,n$ (II.38) $\displaystyle{\bf
A}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}\left(\begin{array}[]{cc}F_{t}&0\\\
0&A_{t}\end{array}\right),\hskip 7.22743pt{\bf
B}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}\left(\begin{array}[]{cc}G_{t}&0\\\
0&B_{t}\end{array}\right),$ (II.43) $\displaystyle{\bf
C}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}\left(\begin{array}[]{cc}H_{t}\Gamma_{t}&C_{t}\end{array}\right),\hskip
7.22743pt{\bf
D}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}\left(\begin{array}[]{cc}H_{t}D_{t}&N_{t}\end{array}\right).$
(II.46)

where ${\bf A}_{t},{\bf B}_{t},{\bf C}_{t},{\bf D}_{t}$ are appropriate
matrices.
(ii) The covariance of the error,
$\Pi_{t}\stackrel{{\scriptstyle\triangle}}{{=}}{\bf
E}\big{\\{}\widehat{E}_{t}\widehat{E}_{t}^{T}\big{\\}}$,$\widehat{E}_{t}=\Theta_{t}-\widehat{\Theta}_{t}$,
satisfies the generalized matrix DRE

$\displaystyle\Pi_{t+1}={\bf A}_{t}\Pi_{t}{\bf A}_{t}^{T}+{\bf
B}_{t}K_{\overline{W}_{t}}{\bf B}_{t}^{T}-\Big{(}{\bf A}_{t}\Pi_{t}{\bf
C}_{t}^{T}+{\bf B}_{t}K_{\overline{W}_{t}}{\bf D}_{t}^{T}\Big{)}$
$\displaystyle\hskip 2.84544pt.\Big{(}{\bf D}_{t}K_{\overline{W}_{t}}{\bf
D}_{t}^{T}+{\bf C}_{t}\Pi_{t}{\bf C}_{t}^{T}\Big{)}^{-1}\Big{(}{\bf
A}_{t}\Pi_{t}{\bf C}_{t}^{T}+{\bf B}_{t}K_{\overline{W}_{t}}{\bf
D}_{t}^{T}\Big{)}^{T},$ $\displaystyle\hskip 2.84544pt\hskip
7.22743pt\Pi_{t}\succeq 0,\hskip 7.22743pt\Pi_{1}=K_{\Theta_{1}}\succeq
0,\hskip 7.22743ptt=1,\ldots,n.$ (II.47)

Moreover, the error
$\widehat{E}_{t}\stackrel{{\scriptstyle\triangle}}{{=}}{\Theta}_{t}-\widehat{\Theta}_{t}$
satisfies the recursion

$\displaystyle\widehat{E}_{t+1}={\bf
F}^{CL}_{t}(\Pi_{t})\widehat{E}_{t}+\big{(}{\bf B}_{t}-{\bf
F}_{t}(\Pi_{t}){\bf D}_{t}\big{)}\overline{W}_{t},\;t=1,\dots,n,$ (II.48)
$\displaystyle{\bf F}^{CL}_{t}(\Pi_{t})={\bf A}_{t}-{\bf F}_{t}(\Pi_{t}){\bf
C}_{t},$ (II.49) $\displaystyle{\bf F}_{t}(\Pi_{t})=\big{(}{\bf
A}_{t}\Pi_{t}{\bf C}_{t}^{T}+{\bf B}_{t}K_{\overline{W}_{t}}{\bf
D}_{t}^{T}\big{)}\big{(}{\bf D}_{t}K_{\overline{W}_{t}}{\bf D}_{t}^{T}+{\bf
C}_{t}\Pi_{t}{\bf C}_{t}^{T}\big{)}^{-1}.$

(iii) The innovations process $I_{t}$ of $Y^{n}$ for $t=1,\ldots,n$, is

$\displaystyle I_{t}\stackrel{{\scriptstyle\triangle}}{{=}}Y_{t}-{\bf
E}\Big{\\{}Y_{t}\Big{|}Y^{t-1}\Big{\\}}={\bf
C}_{t}\big{(}\Theta_{t}-\widehat{\Theta}_{t}\big{)}+{\bf
D}_{t}\overline{W}_{t},$ (II.50) $\displaystyle I_{t}\in G(0,K_{I_{t}}),\hskip
7.22743ptK_{I_{t}}={\bf C}_{t}\Pi_{t}{\bf C}_{t}^{T}+{\bf
D}_{t}K_{\overline{W}_{t}}{\bf D}_{t}^{T}.$ (II.51)

(iv) The matrix $P_{t}=cov\big{(}\Xi_{t},\Xi_{t})$ satisfies Lyapunov
recursion,

$\displaystyle P_{t+1}=F_{t}P_{t}F_{t}^{T}+G_{t}K_{Z_{t}}G_{t}^{T},\hskip
7.22743ptP_{t}\succeq 0,\hskip 7.22743ptP_{1}=K_{\Xi_{1}}.$ (II.52)

(v) The average power constraint is

$\displaystyle\frac{1}{n}{\bf
E}\Big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\Big{\\}}=\frac{1}{n}\sum_{t=1}^{n}tr\Big{(}\Gamma_{t}P_{t}\Gamma_{t}^{T}+D_{t}K_{Z_{t}}D_{t}^{T}\Big{)}.$
(II.53)

(vii) The entropy of $Y^{n}$ is $H(Y^{n})=\sum_{t=1}^{n}H({I}_{t})$, is given
by

$\displaystyle H(Y^{n})=\frac{1}{2}\sum_{t=1}^{n}\ln\big{(}(2\pi
e)^{n_{y}}\det\big{(}{\bf C}_{t}\Pi_{t}{\bf C}_{t}^{T}+{\bf
D}_{t}K_{\overline{W}_{t}}{\bf D}_{t}^{T}\big{)}\big{)}$ (II.54)

and the entropy of $V^{n}$ is $H(V^{n})=\sum_{t=1}^{n}H(\hat{I}_{t})$, is
given by

$\displaystyle H(V^{n})=\frac{1}{2}\sum_{t=1}^{n}\ln\big{(}(2\pi
e)^{n_{y}}\det\big{(}C_{t}\Sigma_{t}C_{t}^{T}+N_{t}K_{W_{t}}N_{t}^{T}\big{)}\big{)}$
(II.55) $\displaystyle\hat{I}_{t}\in G(0,K_{\hat{I}_{t}})\hskip
7.22743pt\mbox{an orth. innov. proc. indep. of $V^{t-1}$},$ (II.56)
$\displaystyle
K_{\hat{I}_{t}}\stackrel{{\scriptstyle\triangle}}{{=}}cov(\hat{I}_{t},\hat{I}_{t})=C_{t}\Sigma_{t}C_{t}^{T}+N_{t}K_{W_{t}}N_{t}^{T}=K_{V_{t}|V^{t-1}}.$
(II.57)

where $\Sigma_{t}$ satisfies the generalized matrix DRE

$\displaystyle\Sigma_{t+1}=$ $\displaystyle
A_{t}\Sigma_{t}A_{t}^{T}+B_{t}K_{W_{t}}B_{t}^{T}-\Big{(}A_{t}\Sigma_{t}C_{t}^{T}+B_{t}K_{W_{t}}N^{T}\Big{)}$
$\displaystyle\hskip
2.84544pt.\Big{(}N_{t}K_{W_{t}}N_{t}^{T}+C_{t}\Sigma_{t}C_{t}^{T}\Big{)}^{-1}\Big{(}A_{t}\Sigma_{t}C_{t}^{T}+B_{t}K_{W_{t}}N_{t}^{T}\Big{)}^{T},$
$\displaystyle\hskip 2.84544pt\hskip 7.22743pt\Sigma_{t}\succeq 0,\hskip
7.22743ptt=1,\ldots,n,\hskip 7.22743pt\Sigma_{1}=K_{S_{1}}\succeq 0.$ (II.58)

(v) An equivalent characterization of ${C}_{n}(\kappa,{\bf P}_{Y_{1}})$ is

$\displaystyle{C}_{n}(\kappa,{\bf P}_{Y_{1}})=\sup_{\frac{1}{n}{\bf
E}\big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\big{\\}}\leq\kappa}\frac{1}{2}\sum_{t=1}^{n}\ln\Big{\\{}\frac{\det(K_{I_{t}})}{\det(K_{\hat{I}_{t}})}\Big{\\}}^{+}$
(II.59)

where the supremum is over
$(F_{t},G_{t},\Gamma_{t},D_{t},K_{Z_{t}}),t=1,\ldots,n$.

###  II-B Asymptotic Analysis

To address the limit
$\lim_{n\longrightarrow\infty}\frac{1}{n}C_{n}(\kappa,{\bf P}_{V_{1}})$, under
Case 1 and Case 2, we investigate the convergence properties of generalized
matrix DREs (II.47), (II.58) and Lyapunov matrix difference equation (II.52)
to their limits. Such properties are summarized in [21, Theorem A.1] and [14,
Theorem III.2].

We present sufficient conditions for convergence in Corollary II.1, Theorem
II.2, Theorem II.3, irrespectively of whether the noise is stable, or
unstable.

######  Corollary II.1.

Consider Case 1 or Case 2
Let $\Sigma_{t},t=1,2,\ldots$ denote the solution of the matrix DRE (II.58).
Let $\Sigma=\Sigma^{T}\succeq 0$ be a solution of the corresponding ARE

$\displaystyle\Sigma=$ $\displaystyle A\Sigma A^{T}+BK_{W}B^{T}-\Big{(}A\Sigma
C^{T}+BK_{W}N^{T}\Big{)}$ $\displaystyle\hskip
2.84544pt.\Big{(}NK_{W}N^{T}+C\Sigma C^{T}\Big{)}^{-1}\Big{(}A\Sigma
C^{T}+BK_{W}N^{T}\Big{)}^{T}.$ (II.60)

Define the matrices

$\displaystyle
A^{*}\stackrel{{\scriptstyle\triangle}}{{=}}A-BK_{W}N^{T}\big{(}NK_{W}N^{T}\big{)}^{-1}C,\hskip
7.22743ptG\stackrel{{\scriptstyle\triangle}}{{=}}B,$ $\displaystyle
B^{*}\stackrel{{\scriptstyle\triangle}}{{=}}K_{W}-K_{W}N^{T}\Big{(}NK_{W}N^{T}\Big{)}^{-1}\Big{(}K_{W}N^{T}\Big{)}^{T}.$
(II.61)

Suppose (see [22, 23, 24] for definitions)

$\\{A,C\\}$ is detectable, and $\\{A^{*},GB^{*,\frac{1}{2}}\\}$ is
stabilizable. (II.62)

Any solution $\Sigma_{t},t=1,2,\ldots,n$ to the generalized matrix DRE (II.58)
with arbitrary initial condition $\Sigma_{1}\succeq 0$, is such that
$\lim_{n\longrightarrow\infty}\Sigma_{n}=\Sigma$, where $\Sigma\succeq 0$ is
the unique solution of the generalized matrix ARE (II.60) with
$spec\big{(}M^{CL}(\Sigma)\big{)}\in{\mathbb{D}}_{o}$.

###### Proof.

For Case 1, the convergence of $\Sigma_{n},n=1,2,\ldots$, follows from the
detectability and stabilizability conditions. For Case 2, the statements of
convergence of $\Sigma_{n},n=1,2,\ldots$ hold, due to continuity property of
solutions of generalized difference Riccati equations, with respect to its
coefficients. ∎

######  Theorem II.2.

Consider Case 1 or Case 2.
Let $\Pi_{t},t=1,\ldots,$ denote the solution of the DRE (II.47).
Let $\Pi=\Pi^{T}\succeq 0$ be a solution of the corresponding ARE

$\displaystyle\Pi={\bf A}\Pi{\bf A}^{T}+{\bf B}K_{\overline{W}}{\bf
B}^{T}-\Big{(}{\bf A}\Pi{\bf C}^{T}+{\bf B}K_{\overline{W}}{\bf D}^{T}\Big{)}$
$\displaystyle\hskip 2.84544pt.\Big{(}{\bf D}K_{\overline{W}}{\bf D}^{T}+{\bf
C}\Pi{\bf C}^{T}\Big{)}^{-1}\Big{(}{\bf A}\Pi{\bf C}^{T}+{\bf
B}K_{\overline{W}}{\bf D}^{T}\Big{)}^{T}.$ (II.63)

Define the matrices [22, 23, 24]

$\displaystyle{\bf A}^{*}\stackrel{{\scriptstyle\triangle}}{{=}}{\bf A}-{\bf
B}K_{\overline{W}}{\bf D}^{T}\big{(}{\bf D}K_{\overline{W}}{\bf
D}^{T}\big{)}^{-1}{\bf C},\hskip 7.22743pt{\bf
G}\stackrel{{\scriptstyle\triangle}}{{=}}{\bf B},$ $\displaystyle{\bf
B}^{*}\stackrel{{\scriptstyle\triangle}}{{=}}K_{\overline{W}}-K_{\overline{W}}{\bf
D}^{T}\Big{(}{\bf D}K_{\overline{W}}{\bf
D}^{T}\Big{)}^{-1}\Big{(}K_{\overline{W}}{\bf D}^{T}\Big{)}^{T}.$ (II.64)

Suppose [22, 23, 24]

$\\{{\bf A},{\bf C}\\}$ is detectable and $\\{{\bf A}^{*},{\bf G}{\bf
B}^{*,\frac{1}{2}}\\}$ is stabilizable. (II.65)

Any solution $\Pi_{t},t=1,2,\ldots,n$ to the generalized matrix DRE (II.47)
with arbitrary initial condition $\Pi_{1}\succeq 0$, is such that
$\lim_{n\longrightarrow\infty}\Pi_{n}=\Pi$, where $\Pi\succeq 0$ is the unique
solution of the generalized matrix ARE (II.63), with $spec\big{(}{\bf
F}^{CL}(\Pi)\big{)}\in{\mathbb{D}}_{o}$.

###### Proof.

Similar to Corollary II.1. ∎

Theorem II.3 identifies conditions for the average power (II.53) to converge,
using $P_{t}=cov\big{(}\Xi_{t},\Xi_{t})$, which satisfies (II.52).

######  Theorem II.3.

Convergence of average power
Consider the average power of Thm II.1, for Cases 1 or 2.
Let $P_{t},t=1\ldots,n$ be a solution of Lyapunov recursion (II.52).
Let $P\succeq 0$ be a solution of

$\displaystyle P=FPF^{T}+GK_{Z}G^{T}.$ (II.66)

Suppose $F$ is an exponentially stable matrix. Any solution
$P_{t},t=1,2,\ldots,n$ to the Lyapunov recursion DRE (II.52), with arbitrary
initial condition $P_{1}\succeq 0$, is such that
$\lim_{n\longrightarrow\infty}P_{n}=P$, where $P\succeq 0$ is the unique
solution of (II.66). Moreover,

$\displaystyle\lim_{n\longrightarrow\infty}$ $\displaystyle\frac{1}{n}{\bf
E}\Big{\\{}\sum_{t=1}^{n}||X_{t}||_{{\mathbb{R}}^{n_{x}}}^{2}\Big{\\}}=\lim_{n\longrightarrow\infty}\frac{1}{n}\sum_{t=1}^{n}tr\Big{(}\Gamma
P_{t}\Gamma^{T}+DK_{Z}D^{T}\Big{)}$ $\displaystyle=$ $\displaystyle
tr\Big{(}\Gamma P\Gamma^{T}+DK_{Z}D^{T}\Big{)},\;\forall P_{1}\succeq 0.$
(II.67)

###### Proof.

For Case 1, these are known[24]. For Case 2, the statements are due to
continuity property of solutions of Lyapunov equations, with respect to their
coefficients. ∎

###  II-C Asymptotic Characterizations of Nonfeedback Capacity

######  Theorem II.4.

Characterization of $C^{\infty}(\kappa,{\bf P}_{Y_{1}})$ for Case 1
Consider the time-invariant noise and channel input strategies of Case 1,
i.e., (I.27) and (I.28) hold.
Define the per unit time limit and supremum by222If at any time $t$, the
information $H(Y_{t}|Y^{t-1})-H(V_{t}|V^{t-1})=+\infty$ then it is removed, as
it is usually the case [25].

$\displaystyle C^{\infty}(\kappa,{\bf
P}_{Y_{1}})\stackrel{{\scriptstyle\triangle}}{{=}}$ $\displaystyle\sup_{{\cal
P}_{\infty}(\kappa)}\lim_{n\longrightarrow\infty}\frac{1}{2n}\sum_{t=1}^{n}\ln\Big{\\{}\frac{\det\big{(}{\bf
C}\Pi_{t}{\bf C}^{T}+{\bf D}K_{\overline{W}}{\bf
D}^{T}\big{)}}{\det\big{(}C\Sigma_{t}C^{T}+NK_{W}N^{T}\big{)}}\Big{\\}}^{+}$
(II.68)

where the average power constraint is defined by

$\displaystyle{\cal
P}_{\infty}(\kappa)\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}(F,G,\Gamma,D,K_{Z})\in{\cal
P}^{\infty}\Big{|}$ (II.69) $\displaystyle\hskip
28.45274pt\lim_{n\longrightarrow\infty}\frac{1}{n}\sum_{t=1}^{n}tr(\Gamma
P_{t}{\Gamma}^{T}+DK_{Z}{D}^{T})\leq\kappa\Big{\\}},$ (II.70)
$\displaystyle{\cal
P}^{\infty}\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}(F,G,\Gamma,D,K_{Z}),\mbox{
such that the following hold}$ $\displaystyle\mbox{(i) the detectability and
stabilizability of (\ref{st})},$ (II.71) (ii) the detectability and
stabilizability of (II.65) (II.72) $\displaystyle\mbox{(iii) $F$ is
exponentially stable}\Big{\\}}.$ (II.73)

Then, $C^{\infty}(\kappa,{\bf P}_{Y_{1}})$ is given by

$\displaystyle C^{\infty}(\kappa,{\bf P}_{Y_{1}})$ $\displaystyle=\sup_{{\cal
P}^{\infty}(\kappa)}\frac{1}{2}\ln\Big{\\{}\frac{\det\big{(}{\bf C}\Pi{\bf
C}^{T}+{\bf D}K_{\overline{W}}{\bf D}^{T}\big{)}}{\det\big{(}C\Sigma
C^{T}+NK_{W}N^{T}\big{)}}\Big{\\}}^{+}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}C^{\infty}(\kappa),\hskip
7.22743pt\forall{\bf P}_{Y_{1}}$ (II.74)

where ${\cal P}^{\infty}(\kappa)$ is defined by

$\displaystyle{\cal P}^{\infty}(\kappa)$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}(F,G,\Gamma,D,K_{Z})\in{\cal
P}^{\infty}\Big{|}$ $\displaystyle K_{Z}\succeq 0,\hskip 7.22743pttr(\Gamma
P{\Gamma}^{T}+DK_{Z}D^{T})\leq\kappa\Big{\\}}$ (II.75)

and $\Sigma\succeq 0$ and $\Pi\succeq 0$ are the unique and stabilizable
solutions, i.e., $spec\big{(}M^{CL}(\Sigma)\big{)}\in{\mathbb{D}}_{o}$ and
$spec\big{(}{\bf F}^{CL}(\Pi)\big{)}\in{\mathbb{D}}_{o}$ of the generalized
matrix AREs (II.60) and (II.63) respectively, $P\succeq 0$ is the unique
solution of the matrix Lyapunov equation (II.66), provided there exists
$\kappa\in[0,\infty)$, such that ${\cal P}^{\infty}(\kappa)$ is non-empty.
Moreover, the optimal $(F,G,\Gamma,D,K_{Z})\in{\cal P}^{\infty}(\kappa)$, is
such that,
(i) if the noise is stable, then the input and the output processes
$(X_{t},Y_{t}),t=1,\ldots$ are asymptotic stationary and
(ii) if the noise is unstable, then the input and the innovations processes
$(X_{t},I_{t}),t=1,\ldots$ are asymptotic stationary.

###### Proof.

By the definition of the set ${\cal P}^{\infty}$, then Corollary II.1, Theorem
II.2 and Theorem II.3 hold. Hence, the following summands converge in
$[0,\infty)$ uniformly, $\forall{\bf P}_{Y_{1}}$:

$\displaystyle\lim_{n\longrightarrow\infty}\frac{1}{n}\sum_{t=1}^{n}tr(\Gamma
P_{t}\Gamma^{T}+DK_{Z}D^{T})=tr(\Gamma P\Gamma^{T}+DK_{Z}D^{T}),$ (II.76)
$\displaystyle\lim_{n\longrightarrow\infty}\frac{1}{2n}\Big{\\{}\sum_{t=1}^{n}\ln\big{(}\frac{\det\big{(}{\bf
C}\Pi_{t}{\bf C}^{T}+{\bf D}K_{\overline{W}}{\bf
D}^{T}\big{)}}{\det\big{(}C\Sigma_{t}C^{T}+NK_{W}N^{T}\big{)}}\big{)}\Big{\\}}$
$\displaystyle=\frac{1}{2}\ln\big{(}\frac{\det\big{(}{\bf C}\Pi{\bf
C}^{T}+{\bf D}K_{\overline{W}}{\bf D}^{T}\big{)}}{\det\big{(}C\Sigma
C^{T}+NK_{W}N^{T}\big{)}}\big{)},\hskip 7.22743pt\forall{\bf P}_{Y_{1}}.$
(II.77)

The last part of the theorem follows from the asymptotic properties of the
Kalman-filter, as follows. For (i). $\Xi_{t},t=1,\ldots$ is asymptotically
stationary, which implies
$X_{t}=\Gamma\Xi_{t}+DZ_{t},X_{1}=\Gamma{\Xi}_{1}+DZ_{1},t=2,\ldots,n,Z_{t}\in
G(0,K_{Z}),\hskip 7.22743ptK_{Z}\succeq 0$, $I_{t},t=1,\ldots$ and
$Y_{t}=HX_{t}+V_{t},t=1,\ldots$ are asymptotically stationary. Similarly for
(ii), with the exception that $Y_{t}=HX_{t}+V_{t},t=1,\ldots$ is not
asymptotically stationary, because $V_{t},t=1,\ldots$ is unstable. ∎

Next, we show that Theorem II.4 remains valid for Case 2.

######  Corollary II.2.

Characterization of $C^{\infty}(\kappa,{\bf P}_{Y_{1}})$ for Case 2
Consider the asymptotically time-invariant noise and channel input strategies
of Case 2, i.e., (I.29) and (I.30) hold.
Define the per unit time limit and supremum by

$\displaystyle C^{\infty,+}(\kappa,{\bf
P}_{Y_{1}})\stackrel{{\scriptstyle\triangle}}{{=}}\sup_{{\cal
P}_{\infty}^{+}(\kappa)}\lim_{n\longrightarrow\infty}\frac{1}{2n}\Big{\\{}$
$\displaystyle\hskip 28.45274pt\sum_{t=1}^{n}\ln\Big{\\{}\frac{\det\big{(}{\bf
C}_{t}\Pi_{t}{\bf C}_{t}^{T}+{\bf D}_{t}K_{\overline{W}_{t}}{\bf
D}_{t}^{T}\big{)}}{\det\big{(}C_{t}\Sigma_{t}C_{t}^{T}+N_{t}K_{W_{t}}N_{t}^{T}\big{)}}\Big{\\}}^{+}\Big{\\}}$
(II.78) $\displaystyle{\cal
P}_{\infty}^{+}(\kappa)\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}\\{(F_{n},G_{n},\Gamma_{n},D_{n}K_{Z_{n}})|n=1,2,\ldots\\}\in{\cal
P}_{\infty}^{+}\Big{|}$ $\displaystyle\hskip
28.45274pt\lim_{n\longrightarrow\infty}\frac{1}{n}\sum_{t=1}^{n}tr\big{(}\Gamma_{t}P_{t}\Gamma_{t}^{T}+D_{t}K_{Z_{t}}D_{t}^{T}\big{)}\big{\\}}\leq\kappa\Big{\\}},$
(II.79) $\displaystyle{\cal
P}_{\infty}^{+}\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}\\{(F_{n},G_{n},\Gamma_{n},D_{n},K_{Z_{n}})|n=1,2,\dots\\}\Big{|}$
$\displaystyle\lim_{n\longrightarrow\infty}(F_{n},G_{n},\Gamma_{n},D_{n}K_{Z_{n}})=(F,G,\Gamma,D,K_{Z})\in{\cal
P}^{\infty}\Big{\\}}.$ (II.80)

Then,

$\displaystyle C^{\infty,+}(\kappa,{\bf P}_{Y_{1}})=C^{\infty}(\kappa,{\bf
P}_{Y_{1}})=C^{\infty}(\kappa)=\mbox{(\ref{ll_3})},\hskip 7.22743pt\forall{\bf
P}_{Y_{1}}.$ (II.81)

and the statements of Theorem II.4.(i), (ii), remain valid.

###### Proof.

The solutions of the DREs and the Lyapunov equation are,
$\Sigma_{n+1}=\Sigma_{n+1}(\Sigma_{n},A_{n},B_{n},C_{n},N_{n},K_{W_{n}})$,
$\Pi_{n+1}=\Pi_{n+1}(\Pi_{n},\Sigma_{n},{\bf A}_{n},{\bf B}_{n},{\bf
C}_{n},{\bf D}_{n},K_{\overline{W}_{n}}),$
$P_{n+1}=P_{n+1}(P_{n},F_{n},G_{n},\Gamma_{n},D_{n},K_{Z_{n}}),\;n=1,2,\ldots$
and these are continuous with respect to their coefficients. Moreover, for all
elements of the set ${\cal P}^{\infty}$, by (I.29), then

$\displaystyle\lim_{n\longrightarrow\infty}\frac{1}{n}\sum_{t=1}^{n}tr(\Gamma_{t}P_{t}{\Gamma_{t}}^{T}+D_{t}K_{Z_{t}}{D_{t}}^{T})=tr(\Gamma
P\Gamma^{T}+DK_{Z}D^{T}),\;\forall{\bf P}_{Y_{1}},$ (II.82)
$\displaystyle\lim_{n\longrightarrow\infty}\frac{1}{2n}\sum_{t=1}^{n}\ln\Big{\\{}\frac{\det\big{(}{\bf
C}_{t}\Pi_{t}{\bf C}_{t}^{T}+{\bf D}_{t}K_{\overline{W}_{t}}{\bf
D}_{t}^{T}\big{)}}{\det\big{(}C_{t}\Sigma_{t}C_{t}^{T}+N_{t}K_{W_{t}}N_{t}^{T}\big{)}}\Big{\\}}^{+}$
$\displaystyle=\frac{1}{2}\ln\big{(}\frac{\det\big{(}{\bf C}\Pi{\bf
C}^{T}+{\bf D}K_{\overline{W}}{\bf D}^{T}\big{)}}{\det\big{(}C\Sigma
C^{T}+NK_{W}N^{T}\big{)}}\big{)},\hskip 7.22743pt\forall{\bf P}_{Y_{1}}.$
(II.83)

The rest follows by repeating the proof of Theorem II.4. ∎

Identity $C^{o}(\kappa,{\bf P}_{Y_{1}})=C^{\infty}(\kappa,{\bf
P}_{Y_{1}})=C^{\infty}(\kappa),\forall{\bf P}_{Y_{1}}$ for Case 2, follows
from the uniform convergence of Theorem II.4 and Corollary II.2; the
derivation is omitted due to space limitation.

######  Theorem II.5.

Characterization of $C^{o}(\kappa,{\bf P}_{Y_{1}})$ for Case 2
Consider the asymptotically time-invariant noise and channel input strategies
of Case 2, i.e., (I.29) and (I.30) hold.
Define the per unit time limit and supremum by

$\displaystyle C^{o}(\kappa,{\bf
P}_{Y_{1}})\stackrel{{\scriptstyle\triangle}}{{=}}\lim_{n\longrightarrow\infty}\sup_{{\cal
P}_{n}^{o,+}(\kappa)}\frac{1}{2n}\Big{\\{}$ $\displaystyle\hskip
28.45274pt\sum_{t=1}^{n}\ln\Big{\\{}\frac{\det\big{(}{\bf C}_{t}\Pi_{t}{\bf
C}_{t}^{T}+{\bf D}_{t}K_{\overline{W}_{t}}{\bf
D}_{t}^{T}\big{)}}{\det\big{(}C_{t}\Sigma_{t}C_{t}^{T}+N_{t}K_{W_{t}}N_{t}^{T}\big{)}}\Big{\\}}^{+}\Big{\\}}$
(II.84) $\displaystyle{\cal
P}_{\infty}^{+}(\kappa)\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}\\{(F_{n},G_{n},\Gamma_{n},D_{n}K_{Z_{n}})|n=1,2,\ldots\\}\in{\cal
P}_{\infty}^{+}\Big{|}$ $\displaystyle\hskip
28.45274pt\frac{1}{n}\sum_{t=1}^{n}tr(\Gamma_{t}P_{t}\Gamma_{t}^{T}+D_{t}K_{Z_{t}}D_{t}^{T})\big{\\}}\leq\kappa\Big{\\}}.$
(II.85)

Then,

$\displaystyle C^{o}(\kappa,{\bf P}_{Y_{1}})=C^{\infty}(\kappa,{\bf
P}_{Y_{1}})=C^{\infty}(\kappa)=\mbox{(\ref{ll_3})},\>\forall{\bf P}_{Y_{1}}$
(II.86)

and the statements of Theorem II.4.(1), (ii), remain valid.

###### Proof.

The derivation uses the uniform limits (II.82) and (II.83), Theorem II.4 and
Corollary II.2. ∎

######  Remark II.1.

If the stabilizability condition is replaced by the unit circle
controllability (see [22] for definition and [14, 16, 17, 15] for specific
examples), then Theorem II.4 and Theorem II.5 remain valid with the
fundamental difference that all limits are not uniform for all ${\bf
P}_{V_{1}}$. For such a relaxation, the limits depend on ${\bf
P}_{V_{1}},\Sigma_{1},P_{1},\Pi_{1}$ and the asymptotic optimization problem
$C^{\infty}(\kappa)$ may not be convex. Specific examples are found in [16].

##  III Conclusion

This paper presents new asymptotic characterizations of nonfeedback capacity
of MIMO additive Gaussian noise (AGN) channels, when the noise is
nonstationary and unstable. The asymptotic characterizations of nonfeedback
capacity, involve two generalized matrix algebraic Riccati equations (AREs) of
filtering theory and a Lyapunov matrix equation of stability theory of
Gaussian systems. Identified, are conditions for uniform convergence of the
asymptotic limits, which imply that the nonfeedback capacity is independent of
the initial states.

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# Non-Gaussianity of the topological charge distribution in $\mathrm{SU}(3)$
Yang–Mills theory

Scuola Normale Superiore, Pisa & INFN - Sezione di Pisa
E-mail

###### Abstract:

In Yang–Mills theory, the cumulants of the naïve lattice discretization of the
topological charge evolved with the Yang–Mills gradient flow coincide, in the
continuum limit, with those of the universal definition. We sketch in these
proceedings the main points of the proof. By implementing the gradient-flow
definition in numerical simulations, we report the results of a precise
computation of the second and the fourth cumulant of the $\mathrm{SU}(3)$
Yang–Mills theory topological charge distribution, in order to measure the
deviation from Gaussianity. A range of high-statistics Monte Carlo simulations
with different lattice volumes and spacings is used to extrapolate the results
to the continuum limit with confidence by keeping finite-volume effects
negligible with respect to the statistical errors. Our best result for the
topological susceptibility is $t_{0}^{2}\chi=$6.67(7)\text{\times}{10}^{-4}$$,
while for the ratio between the fourth and the second cumulant we obtain
$R=$0.233(45)$$.

##  1 Introduction

The Witten–Veneziano (WV) mechanism [1, 2] relates the mass of the
$\eta^{\prime}$ meson to the topological susceptibility in Yang–Mills (YM)
theory, a quantity that can be measured on the lattice. However, the naïve
lattice discretization of the topological charge density needs an unambiguous
renormalization condition to be correctly defined. When considering its
cumulants, such as the susceptibility, short-distance singularities contribute
also to an additive renormalization. On the contrary, if defined using
Ginsparg–Wilson (GW) fermions [3, 4, 5], the topological charge and its
cumulants have finite and unambiguous continuum limits without
renormalization, which satisfy the anomalous chiral Ward identities behind the
WV mechanism [6, 7, 8].

The recent proposal of evolving the naïve discretization with the YM gradient
flow [9] is an alternative definition with a finite and unambiguous continuum
limit [9, 10], and is particularly appealing because its numerical evaluation
is significantly cheaper than the definition using Neuberger’s GW satisfying
fermions.

In these proceedings we discuss our recent results [11]. In Section 2 we show
that in the YM theory the cumulants of the topological charge with the naïve
discretization at positive flow time coincide, in the continuum limit, with
those of the universal definition. In the following sections, by implementing
the gradient-flow definition we compute the topological susceptibility with a
precision $5$ times better than the reference computation with Neuberger’s
definition [12] and the ratio $R$ between the fourth and the second cumulant
with all systematics negligible with respect to the statistical error.

##  2 Cumulants of the topological charge on the lattice

In the continuum, given the topological nature of the topological charge

$Q=\int\differential[4]{x}q(x)\;,\qquad
q(x)=\frac{1}{32\pi^{2}}\epsilon_{\mu\nu\rho\sigma}\tr{F_{\mu\nu}F_{\rho\sigma}}\;,$
(1)

it’s easy to see that $Q$ is invariant under continuous deformation of the
fields induced by the YM gradient flow. Indeed, denoting with a superscript
$t$ the flow-time dependence111If not explicitly indicated, we assume $t>0$.,

$\partial_{t}q^{t}=\partial_{\rho}w_{\rho}^{t}\;,\qquad
w_{\rho}^{t}=\frac{1}{8\pi^{2}}\epsilon_{\mu\nu\rho\sigma}\tr{F^{t}_{\mu\nu}D_{\alpha}F^{t}_{\alpha\sigma}}\;,$
(2)

where $w_{\rho}^{t}$ is a local dimension-$5$ gauge-invariant pseudovector
field, thus $\partial_{t}Q^{t}=0$.

Eq. (2) implies, for any finite (multi)local field $O(y)$ at a physical
distance from $x$,

$\expectationvalue{q^{t}(x)O(y)}=\expectationvalue{q^{t=0}(x)O(y)}+\partial_{\rho}\int_{0}^{t}\differential{t^{\prime}}\expectationvalue{w_{\rho}^{t^{\prime}}(x)O(y)}\;,\qquad
x\neq y\;.$ (3)

The $t=0$ limit of this equation is taken as the definition of the
renormalized $q(x)$. For $x\neq y$, there are no local fields of $d<5$ that
$w_{\rho}^{t}$ can mix with, thus

$\expectationvalue{q^{t=0}(x)O(y)}=\lim_{t\to
0}\expectationvalue{q^{t}(x)O(y)}.$ (4)

We are interested in cumulants of the topological charge

$C_{n}^{t}=\int\differential[4]{z_{1}}\dots\differential[4]{z_{2n-1}}\expectationvalue{q^{t}(0)q^{t}(z_{1})\dots
q^{t}(z_{2n-1})}\;.$ (5)

To study the short-distance singularities at $t=0$, we supplement the theory
with degenerate valence quarks and substitute a single $q^{t=0}(z_{i})$ with a
chain of renormalized (pseudo)scalar densities [8]

$\int\differential[4]{z}q^{t=0}(z)\to-m^{5}\int\left(\prod\differential[4]{x_{i}}\right)\expectationvalue{P_{51}(x_{1})S_{12}(x_{2})S_{23}(x_{3})S_{34}(x_{4})S_{45}(x_{5})}_{\text{F}}\;.$
(6)

When one of the $x_{j}$ is close to $0$, according to the operator product
expansion

$q^{t=0}(0)S_{ij}(x)\xrightarrow{\absolutevalue{x}\to
0}c(x)P_{ij}(0)+\dots\;,$ (7)

where $c(x)$ is a function that diverges as $\absolutevalue{x}^{-4}$ for
$\absolutevalue{x}\to 0$ and the dots indicate subleading contributions. The
Wilson coefficient of the leading short-distance singularity can be computed
in perturbation theory. From Eq. (2) we have

$\expectationvalue{q^{t}(0)S_{ij}(x)O(y)}=\expectationvalue{q^{t=0}(0)S_{ij}(x)O(y)}+\partial_{\rho}\int_{0}^{t}\differential{t^{\prime}}\expectationvalue{w_{\rho}^{t^{\prime}}(0)S_{ij}(x)O(y)}\;,$
(8)

where again $O(y)$ is any finite (multi)local field at a physical distance
from $0$ and $x$. When $t>0$, the l.h.s. has no singularities for
$\absolutevalue{x}\to 0$, thus singularities must cancel in the r.h.s. and
$c(x)$ must be of the form

$c(x)=\partial_{\rho}u_{\rho}(x)$ (9)

which does not contribute when integrated over spacetime, and leads to
$\partial_{t}C_{n}^{t}=0$ for every $t\geq 0$.

###  2.1 Ginsparg–Wilson discretization of cumulants

On the lattice, the gauge field evolved with the YM gradient flow is smooth on
the scale of the cut-off. Gauge-invariant local fields constructed with the
gauge field at positive flow time are finite as they stand, with the
universality class determined only by their asymptotic behaviour in the
classical continuum limit [9, 10]. Short-distance singularities cannot arise
because of the exponential damping at high frequencies enforced with the flow.
Therefore, to show that cumulants of the topological charge at $t>0$ satisfy
the proper singlet chiral Ward identities it is sufficient to work with a
particular discretization of $q$. The GW definition $q_{\text{GW}}(x)$ [3, 4,
5] has a privileged rôle222A concrete example is the definition with
Neuberger’s fermions [3]:
$a^{4}q_{\text{N}}(x)=-(\ifrac{\bar{a}}{2})\tr\gamma_{5}D_{\text{N}}(x,x)$.
since the bare lattice cumulants with the GW lattice discretization,

$C_{\text{GW},n}^{t}=a^{4(2n-1)}\sum_{x_{i}}\expectationvalue{q_{\text{GW}}^{t}(0)q_{\text{GW}}^{t}(x_{1})\dots
q_{\text{GW}}^{t}(x_{2n-1})}\;,$ (10)

are finite and satisfy the anomalous chiral Ward identity for $t=0$ [6, 7, 8].

This is not obvious at $t>0$. To prove it, first consider the susceptibility
$C_{\text{GW},1}^{t=0}$ and apply the flow to one of the two densities. While
no short-distance singularities are present and $q_{\text{GW}}^{t}$ is finite
as it stands, $q_{\text{GW}}^{t=0}$ can have a renormalization constant
$Z_{q}$333$Z_{q}$ is at most logarithmically divergent, since there are no
other pseudoscalar gauge-invariant operators with $d\leq 4$.. Using the
lattice equivalent of Eq. (6), it’s possible to show that
$q_{\text{GW}}^{t=0}$ with $Z_{q}=1$ converges to the correct continuum limit
with rate $a^{2}$.

Then, consider again the susceptibility $C_{\text{GW},1}^{t=0}$ and replace
one of the two $q_{\text{GW}}^{t=0}$ with its lattice density-chain
expression. The discussion in the continuum case applies here and Eq. (9)
guarantees that there are no short-distance singularities. This result and
$Z_{q}=1$ imply that

$\lim_{t\to 0}\lim_{a\to
0}a^{4}\sum_{x}\expectationvalue{q_{\text{GW}}^{t}(0)q_{\text{GW}}^{t=0}(x)}=\lim_{a\to
0}a^{4}\sum_{x}\expectationvalue{q_{\text{GW}}^{t=0}(0)q_{\text{GW}}^{t=0}(x)}\;.$
(11)

By replacing the $t=0$ density on the l.h.s. with the evolved one, no further
singularities arise. The same argument applies to $C_{\text{GW},n}^{t}$
replacing $2n-1$ charges with their density-chain expressions, and we arrive
to the result

$\lim_{t\to 0}\lim_{a\to 0}C_{\text{GW},n}^{t}=\lim_{a\to
0}C_{\text{GW},n}^{t=0}\;.$ (12)

###  2.2 Universality at positive flow time

The naïve discretization of the topological charge density $q_{\text{L}}(x)$
is the lattice version of Eq. (1) in which a discretized field strength tensor
is used

$F_{\mu\nu}^{a}(x)=-\frac{\mathrm{i}}{4a^{2}}\tr{[Q_{\mu\nu}(x)-Q_{\nu\mu}(x)]T^{a}}\;,$
(13)

with

$\begin{split}Q_{\mu\nu}(x)=&\,U_{\mu}(x)U_{\nu}(x+a\hat{\mu})U^{\dagger}_{\mu}(x+a\hat{\nu})U^{\dagger}_{\nu}(x)\\\
+&\,U_{\nu}(x)U^{\dagger}_{\mu}(x-a\hat{\mu}+a\hat{\nu})U^{\dagger}_{\nu}(x-a\hat{\mu})U_{\mu}(x-a\hat{\mu})\\\
+&\,U^{\dagger}_{\mu}(x-a\hat{\mu})U^{\dagger}_{\nu}(x-a\hat{\mu}-a\hat{\nu})U_{\mu}(x-a\hat{\mu}-a\hat{\nu})U_{\nu}(x-a\hat{\nu})\\\
+&\,U^{\dagger}_{\nu}(x-a\hat{\nu})U_{\mu}(x-a\hat{\nu})U_{\nu}(x+a\hat{\mu}-a\hat{\nu})U^{\dagger}_{\mu}(x)\;.\end{split}$
(14)

At $t=0$, $q_{\text{L}}^{t=0}(x)$ require a multiplicative renormalization
constant [13]. The corresponding cumulants $Q_{\text{L}}^{t=0}$ have
additional ultraviolet power-divergent singularities and they do not have a
well defined continuum limit. Conversely, at positive flow time the density
$q_{\text{L}}^{t}(x)$ shares with $q_{\text{GW}}^{t}(x)$ the same asymptotic
behaviour in the classical continuum limit. The universality at positive flow
time together with Eq. (12) imply that the topological charge cumulants
discretized with the naïve definition at positive flow time

$\lim_{t\to 0}\lim_{a\to 0}C_{\text{L},n}^{t}=\lim_{a\to
0}C_{\text{GW},n}^{t=0}$ (15)

and satisfies the anomalous chiral Ward identities. It is interesting to note,
however, that at fixed lattice spacing there can be differences444For
instance, the topological susceptibility with the naïve definition at $t>0$ is
not guaranteed to go to zero in the chiral limit at finite lattice spacing in
the presence of fermions..

##  3 Numerical setup

Lattice | $\beta$ | $\ifrac{L}{a}$ |  $L$ [$\mathrm{fm}$] |  $a$ [$\mathrm{fm}$] | $N_{\text{conf}}$ | $N_{\text{it}}$ | $\ifrac{t_{0}}{a^{2}}$ | $L^{4}\chi$ | $R$
---|---|---|---|---|---|---|---|---|---
$A_{1}$ | $5.96$ | $10$ | $1.0$ | $0.102$ | $36\,000$ | $30$ | $2.995(4)$ | $0.701(6)$ | $0.39(3)$
$B_{1}$ |  | $12$ | $1.2$ |  | $144\,000$ |  | $2.7984(9)$ | $1.617(6)$ | $0.187(24)$
$C_{1}$ |  | $13$ | $1.3$ |  | $280\,000$ |  | $2.7908(5)$ | $2.244(6)$ | $0.177(23)$
$D_{1}$ |  | $14$ | $1.4$ |  | $505\,000$ |  | $2.7889(3)$ | $3.028(6)$ | $0.209(23)$
$E_{1}$ |  | $15$ | $1.5$ |  | $880\,000$ |  | $2.788\,92(23)$ | $3.982(6)$ | $0.202(23)$
$F_{1}$ |  | $16$ | $1.6$ |  | $1\,440\,000$ |  | $2.788\,67(16)$ | $5.167(6)$ | $0.157(22)$
$B_{2}$ | $6.05$ | $14$ | $1.2$ | $0.087$ | $144\,000$ | $60$ | $3.7960(12)$ | $1.699(7)$ | $0.24(3)$
$D_{2}$ |  | $17$ | $1.5$ |  | $144\,000$ |  | $3.7825(8)$ | $3.686(14)$ | $0.22(5)$
$B_{3}$ | $6.13$ | $16$ | $1.2$ | $0.077$ | $144\,000$ | $90$ | $4.8855(15)$ | $1.750(7)$ | $0.22(3)$
$D_{3}$ |  | $19$ | $1.5$ |  | $144\,000$ |  | $4.8722(11)$ | $3.523(13)$ | $0.16(5)$
$B_{4}$ | $6.21$ | $18$ | $1.2$ | $0.068$ | $144\,000$ | $250$ | $6.2191(20)$ | $1.741(7)$ | $0.20(3)$
$D_{4}$ |  | $21$ | $1.4$ |  | $144\,000$ |  | $6.1957(14)$ | $3.266(12)$ | $0.21(5)$
Table 1: Ensembles and statistics used in this study together with measured
observables. For each lattice we give the label, $\beta=\ifrac{6}{g_{0}^{2}}$,
the spatial extent of the lattice, the lattice spacing, the number
$N_{\text{conf}}$ of independent configurations generated, the number of
sweeps $N_{\text{it}}$ required to space them, the measured reference scale
$\ifrac{t_{0}}{a^{2}}$, the susceptibility $L^{4}\chi$ and the ratio $R$.

To compute the topological susceptibility $\chi^{t}\equiv C_{\text{L},1}^{t}$
and the fourth cumulant $C_{\text{L},2}^{t}$, we discretize $\mathrm{SU}(3)$
YM theory using the standard Wilson plaquette action on a finite lattice of
size $\ifrac{L}{a}$ in all four spacetime directions and periodic boundary
conditions. The Monte Carlo updates the gauge field implementing the
Cabibbo–Marinari scheme, with a heat bath sweep of all lattice links followed
by $\ifrac{L}{(2a)}$ over-relaxation sweeps. We simulate the lattices listed
with their parameters in Table 1.

We use the $\\{B_{1}\dots B_{4}\\}$ series, with physical volume of
$\sim($1.2\text{\,}\mathrm{fm}$)^{4}$, to extrapolate the continuum limit of
the fourth cumulant. The choice of a very small volume is forced by the fact
that statistical error on $C_{\text{L},2}$ are $\order{V}$. Given the higher
statistical accuracy, we extrapolate the continuum limit of the susceptibility
from the $\\{D_{1}\dots D_{4}\\}$ series, with physical volume of
$\sim($1.4\text{\,}\mathrm{fm}$)^{4}$. The other lattices are used to check
that the finite-volume systematics is under control and negligible with
respect to the final statistical error.

High-precision scale setting is obtained measuring the energy density $E^{t}$
at positive flow time to compute the reference flow time $t_{0}$ [9] as the
solution of

$\left.t^{2}\expectationvalue{E^{t}}\right|_{t=t_{0}}=0.3\;,\qquad
E^{t}=\frac{1}{4V}\sum_{x}F_{\mu\nu}^{a,t}(x)F_{\mu\nu}^{a,t}(x)\;.$ (16)

The YM gradient flow equation is integrated numerically implementing a fourth
order structure-preserving Runge–Kutta–Munthe-Kaas method555See Appendix B of
Ref. [11] for the details.. On each lattice, the gauge field is evolved
approximatively up to $\sim 1.2t_{0}$. Two different RK step size are used to
check the systematics from the numerical integration and a smaller step size
is used on per-configuration basis to have negligible systematic errors with
respect to the statistical error.

Dedicated simulations are performed to measure the integrated autocorrelation
time $\tau_{\text{int}}$ of $Q$, its moments and $t_{0}$. In the range of
$\beta$ considered, $Q$ has the largest $\tau_{\text{int}}$ which increases
rapidly toward the continuum limit. The number of Cabibbo–Marinari updates
between measurements in the main runs, $N_{\text{it}}$ in Table 1, is chosen
to have statistically independent measurements of $Q$.

Observables are measured at $t=t_{0}$ and given in terms of the two
dimensionless quantities $t_{0}^{2}\chi\equiv t_{0}^{2}C_{\text{L},1}^{t_{0}}$
and ratio $R\equiv\ifrac{C_{\text{L},2}^{t_{0}}}{C_{\text{L},1}^{t_{0}}}$
between the fourth and second cumulant. Bare lattice results are reported in
Table 1.

##  4 Results

$\displaystyle 0.00$$\displaystyle 0.05$$\displaystyle 0.10$$\displaystyle
0.15$$\displaystyle 0.20$$\displaystyle 0.25$$\displaystyle
0.30$$\displaystyle 0.35$$\displaystyle a^{2}/t_{0}$$\displaystyle
6.0$$\displaystyle 6.2$$\displaystyle 6.4$$\displaystyle 6.6$$\displaystyle
6.8$$\displaystyle 7.0$$\displaystyle t_{0}^{2}\chi$$\displaystyle\times
10^{-4}$ (a)

$\displaystyle 0.00$$\displaystyle 0.05$$\displaystyle 0.10$$\displaystyle
0.15$$\displaystyle 0.20$$\displaystyle 0.25$$\displaystyle
0.30$$\displaystyle 0.35$$\displaystyle a^{2}/t_{0}$$\displaystyle
0.65$$\displaystyle 0.70$$\displaystyle 0.75$$\displaystyle
0.80$$\displaystyle 0.85$$\displaystyle 0.90$$\displaystyle
0.95$$\displaystyle 1.00$$\displaystyle
1.05$$\displaystyle\chi^{t}/\chi$$\displaystyle 0.25t_{0}$$\displaystyle
0.5t_{0}$$\displaystyle 0.75t_{0}$$\displaystyle 1.125t_{0}$ (b)

Figure 1: (a) Topological susceptibility $t_{0}^{2}\chi$ with extrapolation to
the continuum limit. (b) The ratio $\ifrac{\chi^{t}}{\chi}$ (errors are
smaller than symbols) for several values of $t$, and its extrapolation to the
continuum limit. $\displaystyle 0.00$$\displaystyle 0.05$$\displaystyle
0.10$$\displaystyle 0.15$$\displaystyle 0.20$$\displaystyle
0.25$$\displaystyle 0.30$$\displaystyle 0.35$$\displaystyle
a^{2}/t_{0}$$\displaystyle 0.0$$\displaystyle 0.2$$\displaystyle
0.4$$\displaystyle 0.6$$\displaystyle 0.8$$\displaystyle 1.0$$\displaystyle R$
Figure 2: The ratio $R$ with continuum extrapolation. In blue the dilute
instanton gas model prediction $R=1$.

At the coarsest lattice spacing, no systematic volume effects are observed in
the value of $t_{0}^{2}\chi$ starting from
$V\sim($1.4\text{\,}\mathrm{fm}$)^{4}$. Thus, the continuum value of
$t_{0}^{2}\chi$ is extrapolated in Figure 1a from the values measured on the
lattices $\\{D_{1},\dots,D_{4}\\}$, fitting with $\order{a^{2}}$
discretization errors according to Symanzik effective theory. The best fit is
obtained with a linear fit restricted to the three finest lattices and gives

$t_{0}^{2}\chi=$6.67(7)\text{\times}{10}^{-4}\text{\,}\;$.$ (17)

A linear fit including the coarsest lattice and a quadratic fit of all points
give compatible results.

The cumulants of the topological charge are expected to be $t$-independent in
the continuum limit. In Figure 1b we show the topological susceptibility
computed at various $t$ normalized to its value at $t_{0}$. The data points
have statistical errors in the range $0.1$–$1$ per mille, due to the
correlation between numerators and denominators. $t$-dependent discretization
effects are fitted with a quadratic function in $\ifrac{a^{2}}{t_{0}}$,
obtaining for each $t$ an intercept compatible with $1$ within statistical
errors in the range $0.5$–$5$ per mille. This represent a high-precision test
of universality in the continuum limit of the gradient-flow discretization of
the topological charge.

The continuum value of $R$ is extrapolated in Figure 2 from the results from
the $V\sim($1.2\text{\,}\mathrm{fm}$)^{4}$ lattices $\\{B_{1},\dots,B_{4}\\}$
without visible finite-volume systematics. The result of a linear fit in
$a^{2}$ is

$R=$0.233(45)$\;,$ (18)

with a compatible result obtained from a constant fit.

##  5 Conclusions

In Ref. [11] we prove that, in the continuum limit, the cumulants of the
topological charge defined by the Yang–Mills gradient flow coincide with those
of the universal definition appearing in the chiral Ward identities. We use
this definition to study the topological charge distribution with an
unprecedented precision. Our result in Eq. (18) is the first one for the ratio
$R$ with systematic and statistical errors under control. The value in Eq.
(18) agrees with the result of Ref. [14], it is inconsistent with the dilute
instanton gas model prediction of $R=1$, and it is compatible with the
large-$N_{c}$ prediction of being of order $\ifrac{1}{N_{c}^{2}}$. We also
measure the topological susceptibility with unprecedented precision. The value
in Eq. (17) is compatible with other lattice results [12, 15]. As a by-
product, we obtain a sub-percent numerical test of universality of the
gradient-flow definition.

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