In this paper, we introduce the \(\omega \) -transform on the fermionic operators and apply it to study the uniform measure for the self-conjugate partitions. We first derive the q-difference equation which is satisfied by the n-point correlation function related to the uniform measure. As applications, we give explicit formulas for the one-point and two-point functions via Eisenstein series. Motivated by this, we prove the quasimodularity of the general n-point function. We also derive the limit shape of self-conjugate partitions under the Gibbs uniform measure and compare it to the leading asymptotics of the one-point function.
