\tilde{\sigma}_{np}&=(\sqrt{n-1/2}+\sqrt{p-1/2})\left(\frac{1}{\sqrt{n-1/2}}+\frac{1}{\sqrt{p-1/2}}\right)^{1/3}
\;.\end{aligned}$$ This leads to a very significant improvement in the quality of the Tracy-Widom approximation for our simulations. Simple manipulations (explained in section 2.2) show that we have some freedom in choosing the centering and scaling: if we replace $n$ by $n+a$ and $p$ by $p+b$ (where $a$ and $b$ are fixed real numbers) in the definitions of $\mu_{np}$ and $\sigma_{np}$, Theorem \[ThJ\] and Theorem \[ThJextended\] still hold. The particular choice used here is motivated by a careful theoretical analysis of the entries of $K_N$ mentioned in section 2.2.

Table 1 summarizes the “quantile" properties of the empirical distributions we obtained and compare them to the Tracy-Widom reference. We used the same reference points as @imj.

   TW Quantiles   TW    10$\times$1000   10$\times$ 4000   10$\times$ 10000   100$\times$4000   30$\times$5000
  -------------- ----- ---------------- ----------------- ------------------ ----------------- ----------------
       -3.9       .01       0.009             0.010             0.015              0.012            0.013
      -3.18       .05       0.047             0.050             0.060              0.053            0.055
      -2.78       .10       0.102             0.107             0.112              0.103            0.105
      -1.91       .30       0.303             0.308             0.316              0.304            0.303
      -1.27       .50       0.506             0.506             0.522              0.508            0.503
      -0.59       .70       0.705             0.704             0.723              0.706            0.702
       0.45       0.9       0.904             0.904             0.913              0.901            0.904
       0.98       .95       0.953             0.951             0.958              0.951            0.953
       2.02       .99       0.992             0.990             0.992              0.991            0.991

  : **Quality of 