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 $T\rightarrow 0$?

  • $\begingroup$ 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? $\endgroup$ Jul 15, 2022 at 19:00
  • $\begingroup$ @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. $\endgroup$ Jul 15, 2022 at 19:11
  • $\begingroup$ Your macro \Ai is undefined. $\endgroup$
    – LSpice
    Jul 15, 2022 at 20:43

1 Answer 1


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.


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