Now carrying out the integral over $\mathbf{q}$ in both the numerator and denominator gives

$$ g[\lambda] = \exp \left( -\frac{D_0}{2} \left\{ \int_0^t \int_0^t ds \, ds' \min(s,s')\lambda(s) \cdot \lambda(s') - \frac{1}{t} \left[ \int_0^t ds \, s\lambda(s) \right]^2 \right\} \right). \quad (A11) $$

From this expression for the generating functional of the Gaussian process we immediately see that the temporal correlation function is given by Eq. (A7). Making now the change of variables $s = ut$ and use Brownian scaling leads to the proof of the Proposition in Eq. (A1).

The representation of the propagator in Eq. (A1) identifies explicitly the occurrence of the variable $t$. Let us now derive the small-$t$ expansion up to $O(t^3)$, and recover Eq. (242). To this purpose it is sufficient to Taylor expand the integral in the exponential on the right-hand side of Eq. (A1) to $O(t^3)$, giving

$$ \int_0^1 du \, V[\mathbf{x}(1-u) + \mathbf{y}u + \sqrt{D_0 t} \mathbf{B}_u] = A_0 + t^{\frac{1}{2}} A_1 + t A_2 + A_3 t^{\frac{3}{2}} + A_4 t^2 + o(t^2). \quad (A12) $$

The various terms are explicitly given by (using the Einstein summation convention)

$$ A_0 = \frac{1}{\hbar} \int_0^1 du \, V[\mathbf{X}(u)], \quad A_p = \frac{D_0^{p/2}}{p!\hbar} \int_0^1 du \, \nabla_{i_1} \dots \nabla_{i_p} V[\mathbf{X}(u)] B_{i_1 u} \dots B_{i_p u}, \quad \mathbf{X}(u) = \mathbf{x} + u(\mathbf{y} - \mathbf{x}), \quad (A13) $$

where $\mathbf{X}(u)$ denotes the straight-line path between $\mathbf{x}$ and $\mathbf{y}$ for $u \in [0,1]$. The resulting expansion can then be averaged using the cumulant expansion which, keeping only terms of $O(t^3)$, gives

$$ \left\langle \exp \left\{ -\frac{t}{\hbar} \int_0^1 du \, V[\mathbf{x}(1-u) + \mathbf{y}u + \sqrt{D_0 t} \mathbf{B}_u] \right\} \right\rangle_{\mathbf{B}} \\ \approx \exp \left( -tA_0 - \langle t^{\frac{3}{2}} A_1 + t^2 A_2 + t^{\frac{5}{2}} A_3 + t^3 A_4 \rangle_{\mathbf{B}} + \frac{1}{2} t^3 \left[ \langle A_1^2 \rangle_{\mathbf{B}} - \langle A_1 \rangle_{\mathbf{B}}^2 \right] \right), \quad (A14) $$

where we have used the fact that $A_0$ does not depend on $\mathbf{B}$. Now, we note that odd moments of $\mathbf{B}$ are zero so $\langle A_1 \rangle_{\mathbf{B}} = \langle A_3 \rangle_{\mathbf{B}} = 0$ and, consequently, we have

$$ \left\langle \exp \left\{ -\frac{t}{\hbar} \int_0^1 du \, V[\mathbf{x}(1-u) + \mathbf{y}u + \sqrt{D_0 t} \mathbf{B}_u] \right\} \right\rangle_{\mathbf{B}} \approx \exp \left( -tA_0 - t^2 \langle A_2 \rangle_{\mathbf{B}} + t^3 \left[ \frac{1}{2} \langle A_1^2 \rangle_{\mathbf{B}} - \langle A_4 \rangle_{\mathbf{B}} \right] \right). \quad (A15) $$

Explicit computation of the moments then leads to the result given in Eq. (242). It is interesting to note that the short-time expansion as formulated here is an expansion about the, constant velocity, direct straight-line path between $\mathbf{x}$ and $\mathbf{y}$ (in contrast to the semiclassical expansion in powers of $\hbar$ for the real-time path integral, which is given about classical paths obeying Newton's equation).

*An alternative derivation using diagrammatic techniques.* To address the calculation of the kernel in a more systematic way, we now develop a diagrammatic method. Let us consider the formula (168) for the kernel (for simplicity at $T=0$) and use the expression of the propagator in Eq. (A1). Let us further set $d=1$ and work in the natural units where $m=\hbar=1$, for simplicity. The generalization to higher dimension is straightforward. One has

$$ K_{\mu}(x, y) = \int_{\Gamma} \frac{dt}{(2\pi t)^{3/2}i} \exp \left[ -\frac{(x-y)^2}{2t} + \mu t - S(x, y; t) \right], \quad S(x, y; t) = -\ln \left\langle e^{-t \int_0^1 du \, V[x(1-u)+yu+\sqrt{t}B_u]} \right\rangle_{\mathbf{B}}. \quad (A16) $$

Making a standard cumulant expansion of (A16) and Wick contractions for the Gaussian Brownian bridge, one can calculate term by term using an elegant diagrammatic expansion, as shown below. Let us denote the vertex at position $u$, and the Brownian bridge correlator $g(u, u') = \langle B_u B_{u'} \rangle_{\mathbf{B}}$, respectively, by

$$ \bullet = -t V(x(1-u) + yu), \quad \qquad \qquad = t g(u, u'). \quad (A17) $$

Then, the function $-S(x, y; t)$ is the sum of all connected diagrams

$$ \begin{aligned} -S = & \bullet (t) + \bigcirc (t^2) + \bigcirc (t^3) + \cdots + \bullet \cdots \bullet (t^3) + \bullet \cdots \bullet (t^4) \\ & + \bigcirc \cdots \bigcirc (t^5) + \cdots + \bigcirc \cdots \bigcirc (t^5) + \cdots + \bigcirc \cdots \bigcirc (t^6) + \cdots \end{aligned} \quad (A18) $$

The first family of diagrams is $O(V)$, the second $O(V^2)$, the third $O(V^3)$, and so on. The power of $t$ is indicated next to the graph in parentheses, the rule being, from (A17), that each vertex and each propagator bring a power of $t$. There is one distinct
index $u_i$ per vertex, each line emerging from a vertex leads to a derivative acting on the vertex, and all the $u_i$'s are integrated at the end, each in $[0, 1]$. For instance (omitting the combinatorial factor),

$$ \bullet \text{---} \bullet = \int_{0}^{1} du_{1} \int_{0}^{1} du_{2} g(u_{1}, u_{2}) V'[x(1-u_{1}) + yu_{1}] V'[x(1-u_{2}) + yu_{2}] . \quad (A19) $$

Until now, all terms in this systematic expansion in powers of $t$ have been written exactly, and are valid for arbitrary potentials (sufficiently differentiable, although some extensions are possible). At this stage, it is so general that it is not clear what is the true expansion parameter of the above series (A18), in other words, for the series to be under control, $t$ must be small compared to what? The answer to that question will depend on the class of physical situations studied.

Now, we assume a soft potential and discuss the gradient expansion near the edge. By definition of the edge, $\mu = V(x_e)$ and here, for compactness, we denote $x_e = x_{\text{edge}}$. We set $x = x_e + w_N a$, $y = x_e + w_N b$ where $a, b$ are assumed to be of order $O(1)$ in the large-$N$ limit. Hence, $X(u) = x(1-u) + yu = x_e + w_N[a(1-u) + bu]$ and the scale factor $w_N$ is to be fixed later. For instance, (A19) becomes

$$ \bullet \text{---} \bullet = c_{00} V'(x_e)^2 + 2c_{12} V'(x_e) V''(x_e) w_N + \mathcal{O}(w_N^2) \quad (A20) $$

$$ c_{pq} = \int_{0}^{1} du_{1} \int_{0}^{1} g(u_{1}, u_{2}) [a(1 - u_{1}) + bu_{1}]^{p} [a(1 - u_{2}) + bu_{2}]^{q}. \quad (A21) $$

We can now enumerate the various terms, and keep the leading and subleading ones in each family. In the exponential in Eq. (A16), we have $(a-b)^{2} \frac{w_N^2}{2t} - S$, where the terms in $S$ are [after cancellation of $\mu$ and $V(x_e)$] (without showing the combinatorial factors)

$$ \bullet \quad V'(x_e)w_N t + V''(x_e)w_N^2 t + \dots, \quad \bigcirc \quad V''(x_e) t^2 + \dots \\ \bullet \quad V'(x_e)^2 t^3 + V'(x_e)V''(x_e)w_N t^3 + \dots, \quad \bigcirc \quad V''(x_e)^2 t^4 + V''(x_e)V'''(x_e)w_N t^4 + \dots \\ \bullet \quad V'(x_e)^2 V''(x_e)t^5 + V''(x_e)^3 w_N t^5 + \dots, \quad \bigcirc \quad V''(x_e)^3 t^6 + V''(x_e)^2 V'''(x_e)w_N t^6 + \dots \quad (A22) $$

To classify the possible behaviors, we start with the diffusion term $(a-b)^2 \frac{w_N^2}{2t}$. Assume that the exponent $(a-b)^2 \frac{w_N^2}{2t} - S$ is dominated by some typical value of $t$ (called $t_N$ in the text). There are three possibilities: (i) the diffusion term blows up, i.e., $t_N \ll w_N^2$, (ii) it vanishes, i.e., $t_N \gg w_N^2$, or (iii) it is of order unity $t_N \sim w_N^2$. The cases (i) and (ii) being more exotic, we now consider the case (iii).

Now, let us first discuss the standard class (leading to the Airy kernel). It is such that the first term in Eq. (A22), using $t_N \sim w_N^2$, is $V'(x_e)w_N t_N \sim V'(x_e)w_N^3 = O(1)$ which means $w_N \sim |V'(x_e)|^{-1/3}$, as found in the text. Interestingly, the first term of the first diagram in the second line of (A22), $V'(x_e)^2 t_N^3 = [V'(x_e)w_N^3]^2 = O(1)$, is also of order unity, and that leads to the Airy kernel (37), if we can neglect all other terms. Let us examine now the conditions for neglecting all the other terms.

The diagrams of the first line [tadpoles, $O(V)$ diagrams] can all be written as $\sim t_N(w_N \partial_{x_e})^q (t_N \partial_{x_e}^2)^{\ell} V(x_e)$ with $\ell \ge 0$ number of loops, and $q \ge 0$ the degree in the gradient expansion, with $(\ell, q) = (0, 0)$ being excluded. Now, we want all these diagrams to be subdominant, excluding $(\ell, q) = (0, 1)$ which is $O(1)$. Hence, it requires $|V^{(2\ell+q)}(x_e)| \ll |V'(x_e)|^{(2\ell+2+q)/3}$ for $\ell = 0, q \ge 2$ and for $\ell \ge 1, q \ge 0$, which is equivalent to the set of conditions

$$ |V^{(2+n)}(x_e)| \ll |V'(x_e)|^{(4+n)/3} \quad n = 0, 1, 2, \dots \quad (A23) $$

with $n = 2\ell + q - 2$, the first of these conditions being

$$ |V''(x_e)| \ll |V'(x_e)|^{4/3}. \quad (A24) $$

Similarly, the diagrams on the second line [$O(V^2)$ diagrams] can all be written as $\sim t_N^{\ell+3} (w_N \partial_{x_e})^q [\partial_{x_e}^{\ell+1} V(x_e)]^2$ with $\ell \ge 0$ and $q \ge 0$. Now, we want all to be subdominant, excluding $(\ell, q) = (0, 0)$ which is $O(1)$. Hence, it requires

$$ |\partial_{x_e}^q [\partial_{x_e}^{\ell+1} V(x_e)]^2| \sim \left| \sum_{p=0}^{q} V^{(\ell+1+p)}(x_e) V^{(\ell+1+q-p)}(x_e) \right| \ll |V'(x_e)|^{(2\ell+6+q)/3}, \quad \ell = 0, q \ge 1 \quad \text{and} \quad \ell \ge 1, q \ge 0. \quad (A25) $$

Remarkably, it is a simple exercise to check that if the conditions (A23) are verified, then this new set of conditions is also verified.
It is a tedious exercise to continue this process, and check that if the conditions (A23) are verified then all conditions for all graphs $O(V^k)$, $k \ge 1$, are satisfied. As an illustration, let us point out for instance the conditions from the leading terms of the first two diagrams $O(V^3)$ on the next line. One finds simply (A24). Hence, it means that the set of conditions (A24) defines the exact basin of attraction of the Airy kernel edge statistics at $T = 0$.

At this stage we have specified very little about $V(x)$, apart from being smooth. All we know is that

$$V(x_e) = \mu, \quad N = \frac{\sqrt{2}}{\pi} \int dx \sqrt{\mu - V(x)} \theta[\mu - V(x)]. \qquad (A26)$$

The only large parameter being $N$, this equation determines $\mu$ and $x_e$. From there the conditions (A24) can be checked, the width of the edge region being $w_N \sim |V'(x_e)|^{-1/3}$. There are many possible cases to consider, especially if one also considers $N$-dependent trap potentials. There are, however, two large classes, with either $x_e \to \pm\infty$ or $x_e \to x_e^*$ as $N \to +\infty$.

For the (N-independent) power-law potentials $V(x) \sim |x|^p$, the condition (A23) becomes

$$|x_e|^{p-2-n} \ll |x_e|^{(p-1)\frac{4+n}{3}} \Rightarrow 1 \ll |x_e|^{(p+2)\frac{1+n}{3}}. \qquad (A27)$$

The conditions are thus verified for all $p > 0$ since in all these cases $x_e \to +\infty$. This case was studied in the text. It also includes limiting cases such as $V(x) \propto \ln|x|$. Note that the width is $w_N \sim x_e^{(1-p)/3}$.

The case $p < 0$ occurs in two situations. The first one is a (bounded from below) confining potential with a power-law tail at large $|x|$

$$V(x) \approx_{|x| \to +\infty} V_0 - c|x|^{-\alpha} \qquad (A28)$$

with $\alpha = -p > 0$. It is known that such a potential has an infinite number of bound states if $\alpha < 2$ [96]. Since then $|x_e| \to +\infty$, the conditions (A27) are satisfied which indicates that it belongs to the Airy class at $T = 0$ with a width scaling as $w_N \sim x_e^{(1+\alpha)/3} \ll x_e$. In the case $\alpha > 2$, there are only a finite number of bound states, which do not enter in the class studied here (since the limit of large $N$ then cannot be studied). The case $\alpha = 2$ is marginal and deserves a separate discussion, but there clearly the conditions (A27) fail.

The second situation with $p = -\alpha < 0$ corresponds to wall potentials $V(x) \sim 1/x^\alpha$ near $x = 0$. In this case, the average density will have a left edge $x_e \to 0^+$ as $N \to +\infty$ and one finds that the condition (A23) is valid only if $p < -2$, i.e., $\alpha > 2$. In addition, the width now scales as $w_N \sim x_e^{(1+\alpha)/3}$. Hence, for $\alpha > 2$ one has $w_N \ll x_e$, which is completely physical, while for $2 < \alpha < 0$ one finds $w_N \gg x_e$ which is in contradiction with the assumptions (because of the wall). Hence, both criteria are valid for $\alpha > 2$, and break down for $0 < \alpha \le 2$.

The above analysis concerns the Airy universality at $T = 0$. While the class $V(x) \sim |x|^p$ leads to the universal finite-temperature kernel studied in this paper (same as the harmonic oscillator), the class of potential of type (A28) is much more sensitive to temperature since its spectrum has a continuum part and the fermions can unbind. Hence, we expect there a different behavior which remains to be studied. Note that for potentials $V(x) \sim |x|^p$ with $p < 1$ the temperature scale $T_{\text{edge}}$ defined in Eq. (285) decreases with increasing $N$.

# APPENDIX B: LONG-TIME-LOW-TEMPERATURE EXPANSIONS FOR KPZ OR FERMIONS AROUND THE TRACY-WIDOM DISTRIBUTION

We start from the solution for droplet initial conditions recalled in the text, expressed as

$$\mathrm{Pr}(\xi < s) = \mathrm{Pr} \left[ h(0,t) + \frac{t}{12} + \gamma < st^{1/3} \right] = \mathrm{Det} \left[ I - P_s K_t^{\mathrm{KPZ}} P_s \right], \qquad (B1)$$

where $\gamma$ is a Gumbel variable, independent of $h(0,t)$. In this form it describes both the fermions, via the variable $\xi$ defined in the text, and the KPZ height, and we recall that

$$K_t^{\mathrm{KPZ}}(r,r') = K_{b=t^{1/3}}^{\mathrm{edge}}(r,r') = \int_{-\infty}^{+\infty} du \frac{\mathrm{Ai}(r+u)\mathrm{Ai}(r'+u)}{1+e^{-t^{1/3}u}}. \qquad (B2)$$

Using that $\frac{1}{1+e^{-u}} = 1 - \frac{1}{1+e^u}$ and the same manipulations as to derive the Sommerfeld expansion of the Fermi factor (see, e.g., [97]) we can rewrite

$$K_t^{\mathrm{KPZ}}(r,r') = K_{\mathrm{Ai}}(r,r') - t^{-1/3} \int_{-\infty}^{+\infty} du \frac{\mathrm{Ai}\left(r + \frac{u}{t^{1/3}}\right)\mathrm{Ai}\left(r' + \frac{u}{t^{1/3}}\right)\mathrm{sgn}(u)}{1 + e^{|u|}}, \qquad (B3)$$

which, until now, is an exact expression for all $t$.

Let us now perform an expansion for large $t$, by first expanding the kernel. Expanding in powers of the factor $\frac{u}{t^{1/3}}$, and performing the integrals over $u$, one finds that only odd terms $\sim u^{1+2q}$ survive, leading to

$$K_t^{\mathrm{KPZ}}(r,r') = K_{\mathrm{Ai}}(r,r') - t^{-2/3} \sum_{q \ge 0} t^{-\frac{2q}{3}} 2(1-2^{-1-2q})\zeta(2+2q) \partial_s^{1+2q} [\mathrm{Ai}(r+s)\mathrm{Ai}(r'+s)]|_{s=0}. \qquad (B4)$$
Replacing $t = b^3$, where $b = N^{1/3}\hbar\omega/T$, we simultaneously obtain the low-$T$ expansion of the edge kernel for the fermions. For instance, let us give the complete expansion for the universal scaling function of the density near the edge

$$F_{1,b}(r) = K_b^{\text{edge}}(r,r) = F_1(r) - b^{-2} \sum_{q \ge 0} b^{-2q} 2(1 - 2^{-1-2q}) \zeta(2+2q) \partial_r^{1+2q} [\text{Ai}(r)^2]. \quad (B5)$$

Now, we would like to obtain the large-$t$ expansion of the PDF of the KPZ height, equivalently the low-$T$ expansion of the PDF of the position of the rightmost fermion, around the Tracy-Widom distribution. For this we need to expand the Fredholm determinant in Eq. (B1). Let us rewrite the FD as follows:

$$\text{Det}[I - P_s K_t^{\text{KPZ}} P_s] = \text{Det}[I - P_0 K_{t,s} P_0], \quad K_{t,s}(r,r') = K_t^{\text{KPZ}}(r+s,r'+s). \quad (B6)$$

Let us write the two leading corrections

$$K_{t,s}(r,r') = K_{\text{Ai},s}(r,r') - \frac{\pi^2}{6t^{2/3}} \partial_s [\text{Ai}(r+s)\text{Ai}(r'+s)] - \frac{7\pi^4}{360t^{4/3}} \partial_s^3 [\text{Ai}(r+s)\text{Ai}(r'+s)] + O(t^{-2}), \quad (B7)$$

where we denote $K_{\text{Ai},s}(r,r') = K_{\text{Airy}}(r+s,r'+s)$. Recalling that the CDF of the Tracy-Widom distribution is $F_2(s) = \text{Det}[I - P_0 K_{\text{Ai},s} P_0]$, we can expand the FD to second order as

$$\text{Det}[I - P_0 K_{t,s} P_0] = F_2(s) + t^{-2/3} \frac{\pi^2}{3} \text{Tr}[P_0(I - P_0 K_{\text{Ai},s} P_0)^{-1} \text{Ai}_s' \text{Ai}_s^\dagger] F_2(s) + Q_4(s) t^{-4/3} + O(t^{-2}), \quad (B8)$$

where $\text{Ai}_s' \text{Ai}_s^\dagger(r,r') = \text{Ai}'(r+s)\text{Ai}'(r'+s)$ and $Q_4(s)$ can in principle be computed. We can now use the identity (see a derivation in Ref. [98])

$$F_2''(s) = 2F_2(s)\text{Tr}[P_0(I - P_0 K_{\text{Ai},s} P_0)^{-1} \text{Ai}_s' \text{Ai}_s^\dagger] \quad (B9)$$

to simplify the expression in Eq. (B8) and obtain

$$\text{Det}[I - P_0 K_{t,s} P_0] = F_2(s) + t^{-2/3} \frac{\pi^2}{6} F_2''(s) + Q_4(s) t^{-4/3} + O(t^{-2}). \quad (B10)$$

Taking a derivative, we thus obtain the PDF, noted $p_t(s)$ of the variable $\xi$ for large $t$, as

$$p_t(s) = f_2(s) + t^{-2/3} \frac{\pi^2}{6} f_2''(s) + q_4(s) t^{-4/3} + O(t^{-2}), \quad (B11)$$

where $f_2(s) = F_2'(s)$ and $q_4(s) = Q_4'(s)$. Let us denote $m_p = \langle \chi^p \rangle_{\text{GUE}}$ and $\kappa_p = \langle \chi^p \rangle_{\text{GUE}}^c$ the moments and cumulants of the TW distribution for GUE, and $a_p$ the moments associated to $q_4(s)$. From (B11) we obtain

$$\langle \xi^p \rangle = m_p + t^{-2/3} \frac{\pi^2}{6} p(p-1) m_{p-2} + a_p t^{-4/3} + O(t^{-2}). \quad (B12)$$

Let us give the first three cumulants of the variable $\xi$ for the fermions

$$\langle \xi \rangle = m_1 + a_1 t^{-4/3}, \quad (B13)$$

$$\langle \xi^2 \rangle^c = \kappa_2 + \frac{\pi^2}{3} t^{-2/3} + (a_2 - 2a_1 m_1) t^{-4/3}, \quad (B14)$$

$$\langle \xi^3 \rangle^c = \langle \xi^3 \rangle - 3 \langle \xi^2 \rangle \langle \xi \rangle + 2 \langle \xi \rangle^3 = \kappa_3 + (a_3 - 3a_2 m_1 - 3a_1 m_2 + 6a_1) t^{-4/3}. \quad (B15)$$

Since the cumulants are additive for uncorrelated variables, one immediately obtains the large-time expansion of the cumulants for the scaled KPZ field $\tilde{h} \equiv \tilde{h}(0,t)$ as

$$\langle \tilde{h}^p \rangle^c = \langle \xi^p \rangle^c - \gamma_p, \quad (B16)$$

where the cumulants of the Gumbel distribution are $\gamma_p = \langle \gamma^p \rangle_\gamma^c = (p-1)!\zeta(p)$ for $p \ge 2$ and $\gamma_1 = \gamma_E$. Note that for the first three cumulants the leading correction to TW is equal to the corresponding cumulant of minus a Gumbel variable

$$\langle \tilde{h}^p \rangle^c = \kappa_p + (-1)^p \gamma_p t^{-p/3} + O(t^{-4/3}), \quad p = 1, 2, 3 \quad (B17)$$

the leading correction to all higher cumulants being a priori of order $O(t^{-4/3})$. This fact is actually known [41]. It can be obtained equivalently from (310) and (311), showing that $p_t(u) = f_2(u) + O(t^{-4/3})$. Indeed, manipulations of the integrals similar as above allow to show that

$$B_t(r,r') = K_{\text{Airy}}(r,r') + t^{-1/3} \int_{-\infty}^{+\infty} dv \frac{\text{sgn}(v)}{e^{|v|} - 1} \left[ \text{Ai}\left(r + \frac{v}{t^{1/3}}\right) \text{Ai}\left(r' + \frac{v}{t^{1/3}}\right) - \text{Ai}(r)\text{Ai}(r') \right] \quad (B18)$$

$$= K_{\text{Airy}}(r,r') + t^{-2/3} \frac{\pi^2}{3} [\text{Ai}(r)\text{Ai}'(r) + \text{Ai}'(r)\text{Ai}(r)] + O(t^{-4/3}). \quad (B19)$$
Now, since AiAi† is a rank-one projector, one can rewrite (311) as

$$p_t(u) = \frac{1}{2}\text{Det}[I - P_u(B_t - \text{AiAi}^\dagger)P_u] - \frac{1}{2}\text{Det}[I - P_u(B_t + \text{AiAi}^\dagger)P_u], \quad (B20)$$

and inserting (B19), it is clear that the $O(t^{-2/3})$ term cancels between the two terms. Thus, the (nontrivial) leading correction to $p_t(u)$ around the TW distribution is $O(t^{-4/3})$, hence, the first three cumulants are corrected first by (minus) the Gumbel variable in Eq. (310). The method that we used here, however, allows one to obtain the cumulants for the fermion problem in a more direct manner.

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