$J$ are constant.

\[def:largeSystemLimit\] The signal length $N$ scales to infinity, and the number of measurements $M=M(N)$ depends on $N$ and also scales to infinity, where the ratio approaches a positive constant $R$ for practical problems, $$\label{eq:measurementRate}
\lim_{N\rightarrow\infty} \frac{M(N)}{N} = R>0.$$

We call $R$ the measurement rate.

Replica Analyses for MMV Settings {#sec:main}
=================================

Section \[sec:model\] discussed two MMV settings. Both settings have applications in real-world problems such as magnetic resonance imaging [@JuYeKi07; @JuSuNaKiYe09] and sensor networks [@Pottie2000]. Although numerous algorithms for MMV signal estimation have been proposed [@tropp2006ass; @chen2006trs; @malioutov2005ssr; @tropp2006ass2; @cotter2005ssl; @Mishali08rembo; @ZinielSchniter2011], what is missing is an information theoretic analysis of the best possible mean squared error (MSE) performance. Throughout this paper, we only consider the MSE as our performance metric.

Statistical physics background and replica method {#sec:set1}
-------------------------------------------------

![Illustration of MMV channel  with $J=3$ signal vectors (left), and one of its possible SMV forms (right). Different background patterns differentiate entries from different channels, and blank space denotes zeros.[]{data-label="fig:channel"}](MMV_channel_v2.png){width="8.5cm"}

In order to express  using a single channel, we transform it to an SMV form. One possible way to do so is illustrated in Fig. \[fig:channel\]. The equivalent SMV problem is $$\label{eq:MMVchannel}
  \y=\F\s+\z,$$ where $\F\in\mathbb{R}^{MJ\times NJ}$ is the 