# What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?

The Tracy–Widom distributions admit many interpretations.

One of them 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 $$T\rightarrow 0$$?

• I don't quite understand the question; you ask about finite temperatures, but even at zero temperature it is only the $\beta=2$ GUE that has the quantum mechanical interpretation in terms of the many-body wave function of free fermions; the orthogonal or symplectic ensembles have no such interpretation, so why would going to nonzero temperature change that? Commented Jul 15, 2022 at 19:00
• @CarloBeenakker The quantum-mechanical interpretation is my justification of the word "temperature"; I believe there are other interpretations of $F_2$ with $F_1$/$F_4$ counterparts that can be deformed into nonzero temperature. Commented Jul 15, 2022 at 19:11
• Your macro \Ai is undefined. Commented Jul 15, 2022 at 20:43

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 which contains formulas you can look at (start with formula (64) there). Hope this helps.