A recurrence relation for the average singular value of a complex Gaussian random matrix

Area of Science:

• Mathematics
• Random Matrix Theory

Background:

• The study of average singular values of large random matrices is crucial in various scientific fields.
• A 2016 conjecture by Bandeira, Kennedy, and Singer proposed that the average singular value of complex Gaussian matrices decreases monotonically with matrix dimension.
• The limiting behavior of these values is related to the Marchenko-Pastur distribution.

Purpose of the Study:

• To derive a recurrence relation for the average singular value, denoted as α(d), of complex Gaussian matrices.
• To investigate the conjecture regarding the monotonic decrease of α(d) as the matrix dimension d increases.
• To analyze the properties of the remainder term R(d) in the recurrence relation.

Main Methods:

• Derivation of a recurrence relation: α_C(d+1) = α_C(d) + R(d).
• Expressing the remainder term R(d) as a linear combination of two integrals involving Laguerre polynomials.
• Analyzing the non-summable hypergeometric nature of the integrals in R(d).

Main Results:

• A recurrence relation for the average singular value α(d) of complex Gaussian matrices was obtained.
• The remainder term R(d) was expressed in terms of integrals of Laguerre polynomials.
• The analysis confirmed that showing R(d) < 0 is a non-trivial analytical problem, essential for proving the conjecture.

Conclusions:

• The derived recurrence relation offers a potential pathway towards resolving the conjecture on the monotonic decrease of average singular values.
• The conjecture posits that α(d) decreases from √π/4 to 8/(3π) as d approaches infinity.
• Further analytical work is required to demonstrate R(d) < 0 and definitively prove the conjecture.