For Gaussian matrices with arbitrary variance profiles, the expected ℓ_p to ℓ_q norm is comparable, up to constants depending only on p and q, to the sum of the largest row and column norms plus the expected maximum entry.
van Handel, On the spectral norm of Gaussian random matrices , Trans
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Operator $\ell_p\to\ell_q$ norms of Gaussian matrices
For Gaussian matrices with arbitrary variance profiles, the expected ℓ_p to ℓ_q norm is comparable, up to constants depending only on p and q, to the sum of the largest row and column norms plus the expected maximum entry.