Looking through the paper of Sommer et al. (Sommers, HJ, Crisanti, A., Sompolinsky, H., Y., Stein (1988), Spectrum of Large Random Asymmetric Arrays, Physical Review Letters, 60 (19), 1895.) They Perform the Next Integral on Variables :

begin {equation}

left langle int left ( prod_j frac { mathrm {d} z_ {j} mathrm {d} overline {z} _ {j}} { pi} right) exp left {- epsilon sum_ {j} left | z_ {j} right | ^ {2} – sum_ {j, k, l} overline {z} _ {j} left ( overline { omega} delta_ {jl} -J_ {jl} ^ {T} right) left ( omega delta_ {lk} -J_ {lk} right) z_ {k} right } right rangle_ mathbf {J}

end {equation}

Or $ langle cdot rangle $ denotes an overall average on $ mathbf {J} $, a non-symmetric random matrix with real elements, using a Gaussian measure (mean zero and $ 1 / N $ variance): $$ P ( mathbf {J}) prop exp left {- frac {N} {2} sum_ {ij} J_ {ij} ^ 2 right } $$

To average on $ mathbf {J} $, they decouple the quadratic terms into $ J_ {ij} $ with a complex Gaussian transformation. However, I do not know how they did it. Usually, I can identify a $ b $ such as:

$$ exp left { frac { overline {b}} { alpha} right } = alpha int frac { mathrm {d} v mathrm {d} overline {vb} } {2 mathrm {i} pi} exp left {- overline {vb} v alpha + overline {b} v + b overline {vb} right } $$

however $ text {Re} ( alpha)> 0 $ is necessary. Which is not the case here.

$$ exp left {- sum_ {j, k, l} overline {z} _jJ_ {lj} J_ {lk} z_k right } = exp left {- sum_ {l} left ( overline { sum_j (z_jJ_ {lj})} right) left ( sum_kJ_ {lk} z_k right) right } = exp left {- sum_ {l} overline {b } _lb_l right } $$

How could I linearize the terms in $ J_ {ij} $ before assessing their average? In the end, after unbundling the terms, I should be able to average the $ J_ {ij} $ and get the following result:

$$

int left ( prod_j frac { mathrm {d} z_ {d} mathrm {d} overline {z} _ {j}} { pi} right) exp left {- N left ( epsilon r + ln (1 + r) + frac {r | omega | ^ {2}} {1 + r} right) right }

$$

or $ r = 1 / N sum_i z_i overline {z_i} $