In this paper, we study a particular Painlevé V (denoted PV) that arises from multi-input-multi-output wireless communication systems. Such PV appears through its intimate relation with the Hankel determinant that describes the moment generating function (MGF) of the Shannon capacity. This originates through the multiplication of the Laguerre weight or the gamma density xαe−x, x > 0, for α > −1 by (1 + x/t)λ with t > 0 a scaling parameter. Here the λ parameter “generates” the Shannon capacity; see Chen, Y. and McKay, M. R. [IEEE Trans. Inf. Theory 58, 4594–4634 (2012)]. It was found that the MGF has an integral representation as a functional of y(t) and y′(t), where y(t) satisfies the “classical form” of PV. In this paper, we consider the situation where n, the number of transmit antennas, (or the size of the random matrix), tends to infinity and the signal-to-noise ratio, P, tends to infinity such that s = 4n2/P is finite. Under such double scaling, the MGF, effectively an infinite determinant, has an integral representation in terms of a “lesser” PIII. We also consider the situations where and α ∈ {0, 1, 2, …}, λ ∈ {1, 2, …}, linking the relevant quantity to a solution of the two-dimensional sine-Gordon equation in radial coordinates and a certain discrete Painlevé-II. From the large n asymptotic of the orthogonal polynomials, which appears naturally, we obtain the double scaled MGF for small and large s, together with the constant term in the large s expansion. With the aid of these, we derive a number of cumulants and find that the capacity distribution function is non-Gaussian.
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December 2017
Research Article|
December 12 2017
Single-user MIMO system, Painlevé transcendents, and double scaling
Hongmei Chen
;
Hongmei Chen
Department of Mathematics, Faculty of Science and Technology, University of Macau
, Macau, China
and Department of Mathematics, Lancaster University
, Lancaster, United Kingdom
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Min Chen;
Min Chen
a)
Department of Mathematics, Faculty of Science and Technology, University of Macau
, Macau, China
and Department of Mathematics, Lancaster University
, Lancaster, United Kingdom
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Gordon Blower
;
Gordon Blower
b)
Department of Mathematics, Faculty of Science and Technology, University of Macau
, Macau, China
and Department of Mathematics, Lancaster University
, Lancaster, United Kingdom
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b)
Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 58, 123502 (2017)
Article history
Received:
March 03 2017
Accepted:
November 24 2017
Citation
Hongmei Chen, Min Chen, Gordon Blower, Yang Chen; Single-user MIMO system, Painlevé transcendents, and double scaling. J. Math. Phys. 1 December 2017; 58 (12): 123502. https://doi.org/10.1063/1.5017127
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