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In this paper we study a particular Painlev\'e V (denoted ${\rm P_{V}}$) thatarises from Multi-Input-Multi-Output (MIMO) wireless communication systems.Such a $P_V$ appears through its intimate relation with the Hankel determinantthat describes the moment generating function (MGF) of the Shannon capacity.This originates through the multiplication of the Laguerre weight or the Gammadensity $x^{\alpha} {\rm e}^{-x},\;x> 0,$ for $\alpha>-1$ by$(1+x/t)^{\lambda}$ with $t>0$ a scaling parameter. Here the $\lambda$parameter "generates" the Shannon capacity, see Yang Chen and Matthew McKay,IEEE Trans. IT, 58 (2012) 4594--4634. It was found that the MGF has an integralrepresentation as a functional of $y(t)$ and $y'(t)$, where $y(t)$ satisfiesthe "classical form" of $P_V$. In this paper, we consider the situation where$n,$ the number of transmit antennas, (or the size of the random matrix), tendsto infinity, and the signal-to-noise ratio (SNR) $P$ tends to infinity, suchthat $s={4n^{2}}/{P}$ is finite. Under such double scaling the MGF, effectivelyan infinite determinant, has an integral representation in terms of a "lesser"$P_{III}$. We also consider the situations where $\alpha=k+1/2,\;\;k\in\mathbb{N},$ and $\alpha\in\{0,1,2,\dots\}$ $\lambda\in\{1,2,\dots\},$ linkingthe relevant quantity to a solution of the two dimensional sine-Gordon equationin radial coordinates and a certain discrete Painlev\'e-II. From the large $n$asymptotic of the orthogonal polynomials, that appears naturally, we obtain thedouble scaled MGF for small and large $s$, together with the constant term inthe large $s$ expansion. With the aid of these, we derive a number of cumulantsand find that the capacity distribution function is non-Gaussian.

Single-use MIMO system, Painlev\'e transcendents and double scaling