The [Tracy–Widom distributions](https://en.wikipedia.org/wiki/Tracy%E2%80%93Widom_distribution) admit many interpretations.


[One of them](https://www.ggi.infn.it/sft/SFT_2018/LectureNotes/Majumdar_slides.pdf) is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ground state, the position of the rightmost fermion (approximately) has probability density $P=\sqrt{2N} + \frac{N^{-1/6}}{\sqrt 2} F\_2$ where $F\_2$ is the PDF of the unitary Tracy-Widom distribution.


In the ground state, the wave-function $\Phi(x\_1, \dots , x\_N ) = \frac{1}{\sqrt{N!}} \det[\phi\_i(x\_j)]$ where $\phi \_i$ runs over the first $N$ Hermitian polynomials and $j$ runs over $1 \dots N$. The probability density of observing the particles, $\Phi^2$, can be written as a determinant $\frac{1}{N!}\det[K\_N(x\_i,x\_j)]$ where $K\_N$ is a kernel depending on $N$. When $N\rightarrow \infty$, the kernel $K\_N$ converges to the Airy kernel $K\_A(u,v) = \int\_0^{+\infty} Ai(u+x) Ai(v+x)dx$ at the edge location $\sqrt{2N}$ after scaling. This is why we have the Fredholm determinant representation $F\_2=\det(1-K\_A)\_{L^2([s,+\infty])}$.


When the temperature is above zero, there is no determinantal structure for $\Phi^2$ when $N$ is fixed. However, if we consider the grand canonical potential, then there is a similar edge kernel $K\_b(u,v)=\int\_{-\infty}^{+\infty}\frac{ Ai(u+x) Ai(v+x)}{1+e^{-b x}}dx$ where $b$ is related to the temperature. Thus we call the Fredholm determinant $F\_{2,b}=\det(1-K\_b)\_{L^2([s,+\infty])}$ the **Finite-temperature Tracy-Widom distribution** function.


**Question:** How can we find a similar distribution that degenerates to the orthogonal/symplectic Tracy-Widom distribution when $T\rightarrow 0$?



What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?
This is not a complete answer, but more of an approach and an invitation to look at the relevant literature. As you write, you would like to insert the so-called Fermi factor into the Fredholm Pfaffian expression for $F\_1$ or $F\_4$ (by analogy with the Fredholm determinant for $F\_2$). Such Fermi factors are also present in exact formulas for solutions of the KPZ equations. On the line, this is due to Amir-Corwin-Quastel and Sasamoto-Spohn (both 2010). For $F\_1$ or $F\_4$, you want to change the geometry. Some particle systems, in particular, the Facilitated/Open TASEP evolving on the half line, have $F\_4$ fluctuations. Therefore, you want to consider exact Pfaffian formulas for the KPZ equation on the half line. There is a growing literature on this topic, both rigorous and physics level. I found [this paper](https://arxiv.org/pdf/1905.05718.pdf) which contains formulas you can look at (start with formula (64) there). Hope this helps.


