The $p,q$-binomial distribution applied to the Ising model

Monte Carlo simulations have shown that the $p,q$-binomial distribution closely fits the magnetisation distribution for the $d$-dimensional Ising model at all temperatures when $d>4$. It also fits well for some temperatures near $T_c$ for $d=2,3$ and especially so for $d=4$. At high and low temperatures, away from $T_c$, the $p,q$-distribution always fits extremely well. However, it appears very difficult to determine how the parameters $p$ and $q$ depend of the temperature. From high and low temperature series expansions we can get partial results on their temperature dependence. Near $T_c$ for $d=5$ we have approximately that $p=1-0.0736/L^5$ and $q=1-9.87/L^5$ whereas for $d<5$ the linear coefficient of $q$ grows logarithmically. We show numerically how the parameters behave near $T_c$ with increasing $d$.