  This manuscript goes through the fundamental connections between statistical
mechanics and estimation theory by focusing on the particular problem of
compressive sensing. We first show that the asymptotic analysis of a sparse
recovery algorithm is mathematically equivalent to the problem of calculating
the free energy of a spin glass in the thermodynamic limit. We then use the
replica method from statistical mechanics to evaluate the performance in the
asymptotic regime. The asymptotic results have several applications in
communications and signal processing. We briefly go through two instances of
these applications: Characterization of joint sparse recovery algorithms used
in distributed compressive sensing, and tuning of receivers employed for
detection of spatially modulated signals.
