Non-crossing Brownian Paths and Dyson Brownian Motion Under a Moving Boundary

Abstract

We compute analytically the probability S(t) that a set of N Brownian paths do not cross each other and stay below a moving boundary $g(\tau )= W \sqrt{\tau }$ up to time t. We show that for large t it decays as a power law $S(t) \sim t^{- \beta (N,W)}$. The decay exponent $\beta (N,W)$ is obtained as the ground state energy of a quantum system of N non-interacting fermions in a harmonic well in the presence of an infinite hard wall at position W. Explicit expressions for $\beta (N,W)$ are obtained in various limits of N and W, in particular for large N and large W. We obtain the joint distribution of the positions of the walkers in the presence of the moving barrier $g(\tau ) =W \sqrt{\tau }$ at large time. We extend our results to the case of N Dyson Brownian motions (corresponding to the Gaussian Unitary Ensemble) in the presence of the same moving boundary $g(\tau )=W\sqrt{\tau }$. For $W=0$ we show that the system provides a realization of a Laguerre biorthogonal ensemble in random matrix theory. We obtain explicitly the average density near the barrier, as well as in the bulk far away from the barrier. Finally we apply our results to N non-crossing Brownian bridges on the interval [0, T] under a time-dependent barrier $g_B(\tau )= W \sqrt{\tau (1- \frac{\tau }{T})}$.

Appendices

Appendix A: Hermite Polynomials

In this Appendix, we recall some basic properties of the Hermite polynomials $H_n$ defined as

$$\begin{aligned} H_n(x) = (-1)^n e^{x^2} \frac{{d}^n}{{d}x^n}e^{-x^2} . \end{aligned}$$

(A1)

The first ones read

$$\begin{aligned} \begin{aligned} H_{0}(x)&=1 \\ H_{1}(x)&=2 x \\ H_{2}(x)&=4 x^{2}-2 \\ H_{3}(x)&=8 x^{3}-12 x .\end{aligned} \end{aligned}$$

(A2)

They solve the following differential equation

$$\begin{aligned} \frac{d}{d x}\left( e^{-x^{2}} \frac{{d} H_{n}(x)}{{d} x}\right) +2 n \ e^{-x^{2}} H_{n}(x)=0 . \end{aligned}$$

(A3)

They also satisfy the recurrence relation

$$\begin{aligned} H_{n+1}(x)=2 x H_{n}(x)-2 n H_{n-1}(x) \end{aligned}$$

(A4)

as well as the following relation for the derivative of the n-th Hermite polynomial

$$\begin{aligned} H_{n}^{\prime }(x)=2 n H_{n-1}(x) . \end{aligned}$$

(A5)

For our purpose, it is also interesting to write $H_n(\frac{X}{\sqrt{2}})$, from the definition (A1), as

$$\begin{aligned} H_n\left( \frac{X}{\sqrt{2}}\right) = (-1)^n 2^\frac{n}{2} e^\frac{X^2}{2} \frac{{d}^n}{{d}X^n}e^{-\frac{X^2}{2}} . \end{aligned}$$

(A6)

Furthermore, introducing the double factorial $(n-1) ! ! = (n-1)(n-3)\cdots 1$, the Hermite numbers are given by

$$\begin{aligned} H_{n}= H_n(0)=\left\{ \begin{array}{ll}{0 ,} &{} {{\text { if }} n {\text { is odd }}} \\ {(-1)^{n / 2} 2^{n / 2}(n-1) ! ! .} &{} {{\text { if }} n {\text { is even }}}\end{array}\right. \end{aligned}$$

(A7)

We will also use the explicit formula for the generating function of Hermite polynomials

$$\begin{aligned} e^{2 x t-t^{2}}=\sum _{n=0}^{\infty } H_{n}(x) \frac{t^{n}}{n !}. \end{aligned}$$

(A8)

In addition to these standard formulae for Hermite polynomials, we will need the following result

$$\begin{aligned} \int _{-\infty }^{0} H_{m}(x) H_{n}(x) e^{-x^{2}} d x=\frac{1}{2(m-n)}\left( H_{m}(0) H_{n}^{\prime }(0)-H_{n}(0) H_{m}^{\prime }(0)\right) . \end{aligned}$$

(A9)

This identity can be proved by multiplying (A3) by $H_m(x)$ and integrating over x to obtain

$$\begin{aligned} \int _{-\infty }^{0} H_{m}(x) \frac{{d}}{{d} x}\left( e^{-x^{2}} \frac{{d} H_{n}(x)}{{d} x}\right) {d}x=-2 n \int _{-\infty }^{0} e^{-x^{2}} H_{m}(x) H_{n}(x) {d} x . \end{aligned}$$

Then, integrating by parts yields

$$\begin{aligned} - \int _{-\infty }^{0} \frac{{d} H_{m}(x) }{{d} x} e^{-x^{2}} \frac{{d} H_{n}(x)}{{d} x}{d}x + H_m(0) H_n'(0) =-2 n \int _{-\infty }^{0} e^{-x^{2}} H_{m}(x) H_{n}(x) {d} x .\nonumber \\ \end{aligned}$$

(A10)

By permuting m and n and substracting to (A10) one obtains the relation in (A9).

Another useful property of Hermite polynomials is that the determinant of the matrix $\left( H_{i-1}(x_j) \right) _{1 \le i,j \le N}$ is a Vandermonde determinant. Indeed, as the Hermite polynomials form a sequence of orthogonal polynomials with successive degrees, the polynomials in the determinant simplify to their leading monomials and the determinant reads

$$\begin{aligned} \det _{1 \le i,j \le N} \left( H_{i-1}(x_j) \right)&= \det _{1 \le i,j \le N} \left( 2^{i-1} x_j^{i-1} \right) = 2^{\frac{N(N-1)}{2}} \det _{1 \le i,j \le N} ( x_j^{i-1} ) \nonumber \\&= 2^{\frac{N(N-1)}{2}} \prod _{1\le i < j \le N} (x_j - x_i) . \end{aligned}$$

(A11)

Appendix B: Harmonic Oscillator Wavefunctions

The harmonic oscillator without a wall (23) with $\hbar = m =1, \omega =\frac{1}{2}$ (and zero ground-state energy) is described by the Hamiltonian

$$\begin{aligned} \hat{H} = \ \ - \frac{ 1 }{ 2} \frac{ \partial ^ { 2 } }{ \partial X ^ { 2 } } + \frac{ 1 }{ 8 } X ^ { 2 } - \frac{ 1 }{4 } \end{aligned}$$

(B1)

The eigenfunctions are expressed in terms of Hermite polynomials (see e.g. [77])

$$\begin{aligned} \phi _k(X) = c_k \ H_k\left( \frac{X}{\sqrt{2}}\right) \ e^{-\frac{X^2}{4}} \end{aligned}$$

(B2)

with the normalization constant $c_k$ and energy level $E_k$ given by

$$\begin{aligned} {\left\{ \begin{array}{ll} c_k =(\sqrt{2 \pi } 2^{k} k !)^{-\frac{1}{2}} ,\\ E_k = \frac{k}{2} . \end{array}\right. } \end{aligned}$$

(B3)

The quantum propagator in imaginary time is then defined as

$$\begin{aligned} G(X,T \mid X_0, T_0) = \langle {X}| e^{-(T-T_0) \hat{H}} |{X_0}\rangle = \sum \limits _{k=0} ^{\infty } \phi _k(X) \phi _k^*(X_0) e^{- \frac{k}{2} (T-T_0)} \end{aligned}$$

(B4)

and it can be computed explicitly (using Mehler’s formula), leading to

$$\begin{aligned} G(X,T \mid X_0, T_0) =\frac{1}{\sqrt{2\pi \left( 1-e^{-\left( T-T_{0}\right) }\right) }} \exp \left( - \frac{1}{2} \frac{\left( X-X_{0} e^{-\frac{1}{2} \left( T-T_{0}\right) }\right) ^{2}}{1-e^{-\left( T-T_{0}\right) }}+ \frac{X^2 - X_0^2}{4}\right) .\nonumber \\ \end{aligned}$$

(B5)

In the presence of a hard wall at position $W=0$, the wall imposes a zero of the wavefunction at $X=0$. Thus, the wavefunctions of this system are the odd wavefunctions of the harmonic oscillator, with an extra $\sqrt{2}$ factor due to the normalization

$$\begin{aligned} \phi _k(X,W=0) = \sqrt{2} \ \phi _{2k+1}(X) = \sqrt{2} \ c_{2k+1} \ H_{2k+1}\left( \frac{X}{\sqrt{2}} \right) \ e^{-\frac{X^2}{4}} . \end{aligned}$$

(B6)

Let us compute the half-space integrals for the system with a hard wall in $W=0$, which are needed for the perturbative expansion around $W=0$ in Eq. (74) in the text, namely

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\Delta E _{2k+1}^{(1)} = \langle {2k+1}| \Delta \hat{H} |{2k+1}\rangle \\ &{} \\ &{} \Delta E _{2k+1}^{(2)} = \sum \limits _{k'\ne k} \frac{|{ \langle 2k'+1 |\Delta \hat{H}| {2k+1}\rangle }|^2 }{E_{2k+1} -E_{2k'+1} } \end{array}\right. } , \end{aligned}$$

(B7)

where $\Delta {\hat{H}} = X/4$.

• The first matrix element reads

  $$\begin{aligned} \langle {2k+1}| \Delta \hat{H} |{2k+1}\rangle&= \int _{-\infty }^{0} 2 \;\phi _{2k+1}(X)^{2} \frac{X}{4} d X \nonumber \\&= \frac{1}{2} c_{2k+1}^2 \int _{-\infty }^{0} X \ H_{2 k+1}^{2}\left( \frac{X}{\sqrt{2}}\right) e^{-\frac{X^{2}}{2}} {d}X , \end{aligned}$$

  (B8)

  where we have used the explicit expression of $\phi _k(X,W=0)$ given in Eq. (B6). Performing the change of variable $X \rightarrow \sqrt{2} X$ and using the recurrence relation for Hermite polynomials (A4) one gets

  $$\begin{aligned}&\int _{-\infty }^{0} X \ H_{2 k+1}^{2}\left( \frac{X}{\sqrt{2}}\right) e^{-\frac{X^{2}}{2}} {d}X = \int _{-\infty }^{0} 2 X \ H_{2 k+1}^{2} ( X ) e^{- X^{2} } {d}X \end{aligned}$$

  (B9)
  $$\begin{aligned}&\quad = \int _{-\infty }^{0} ( H_{2k+2}(X) + 2(2k+1) H_{2k}(X) ) H_{2k+1}(X) e^{- X^{2} } {d}X . \end{aligned}$$

  (B10)

  These integrals can then be evaluated using the identities in (A5), (A7) and (A9), namely

  $$\begin{aligned} {\left\{ \begin{array}{ll} &{}\int _{-\infty }^{0} H_{2 k+1}(X) H_{2 k}(X) e^{-X^{2}} d X =-(2 k+1) \dfrac{(2 k) !^{2}}{k !^{2}}, \\ &{}\\ &{}\int _{-\infty }^{0} H_{2 k+2}(X) H_{2 k+1}(X) e^{-X^{2}} d X= -\dfrac{(2 k+2) !(2 k+1) !}{k !(k+1) !} . \end{array}\right. } \end{aligned}$$

  (B11)

  Finally injecting these results in Eqs. (B8) and (B10) and using the explicit expression of the coefficients $c_k$ (113), one obtains

  $$\begin{aligned} \Delta E _{2k+1}^{(1)} = \langle {2k+1}| \Delta \hat{H} |{2k+1}\rangle = -\frac{1}{\sqrt{2 \pi } 2^{2 k}} \frac{(2 k+1) !}{k !^{2}} , \end{aligned}$$

  (B12)

  which is the result given in Eq. (75) in the text.

• The more general matrix element needed for the computation of $\Delta E _{2k+1}^{(2)}$ in Eq. (B7) can be computed similarly and it yields

  $$\begin{aligned}&\langle {2k'+1}|\Delta \hat{H} |{2k+1}\rangle \nonumber \\&\quad = \frac{1}{2} c_{2k+1} c_{2k'+1} \int _{-\infty }^{0} X \ H_{2 k+1}\left( \frac{X}{\sqrt{2}}\right) H_{2 k'+1}\left( \frac{X}{\sqrt{2}}\right) e^{-\frac{X^{2}}{2}} {d}X \end{aligned}$$

  (B13)
  $$\begin{aligned}&\quad = \frac{1}{2} c_{2k+1} c_{2k'+1} \int _{-\infty }^{0} H_{2 k' +1 }(X)\left( \frac{1}{2} H_{2 k+2}(X)+(2 k+1) H_{2 k}(X)\right) e^{-X^{2} } {d}X\nonumber \\ \end{aligned}$$

  (B14)
  $$\begin{aligned}&\quad = \frac{(-1)^{k+k'}}{\sqrt{2 \pi } \, 2^{k+k'}\left( 4(k-k')^{2}-1\right) } \frac{\sqrt{(2 k+1) !(2 k'+1) !}}{k ! k' !} , \end{aligned}$$

  (B15)

  where, again, we have used the recurrence relation (A4) together with the identities in (A5), (A7) and (A9) as well as the explicit expression of $c_k$ in (B3). Finally, one obtains $\Delta E _{2k+1}^{(2)}$ by injecting this expression in the second line of Eq. (B7), which yields the expression (B7) given in the text. With these expressions, $\beta (N,W)$ can be evaluated up to order $W^2$, as given in the text in Eq. (78).

We close this section by presenting the asymptotic analysis of $\Delta E _{2k+1}^{(2)}$, yielding the result (77) given in the text. We start with the formula (B7) given in the text

$$\begin{aligned} \Delta E_{2k+1}^{(2)} = \frac{(2k+1)!}{\pi \, 2^{2k+1}\, (k!)^2}\sum _{k'\ge 0, k' \ne k} \frac{1}{k-k'} \frac{(2k'+1)!}{2^{2k'} (k'!)^2} \frac{1}{(4(k-k')^2 - 1)^2} . \end{aligned}$$

(B16)

In the sum, we perform the change of variable $q=k'-k$, which yields

$$\begin{aligned} \Delta E_{2k+1}^{(2)} = -\frac{(2k+1)!}{ \pi \, 2^{4k+1}\, (k!)^2}\sum _{q\ge -k, q \ne 0} \frac{1}{q} \frac{(2k+2q+1)!}{2^{2q} ((k+q)!)^2} \frac{1}{(4q^2 - 1)^2} . \end{aligned}$$

(B17)

We now use the large k expansions, obtained from Stirling’s formula

$$\begin{aligned} \frac{(2k+2q+1)!}{((k+q)!)^2} = \frac{2^{2k} 2^{2q+1}}{\sqrt{\pi }} \sqrt{k} + \frac{2^{2k} 2^{2q-2}(3+4q)}{\sqrt{\pi }} \frac{1}{\sqrt{k}} + \mathcal{O}(k^{-3/2}) . \end{aligned}$$

(B18)

Inserting this expansion (B18) into Eq. (B17) and using the large k expansion (obtained again from Stirling’s formula)

$$\begin{aligned} \frac{(2k+1)!}{2^{2k} (k!)^2} = \frac{2}{\sqrt{\pi }} \sqrt{k} + \mathcal{O}(1) , \end{aligned}$$

(B19)

one obtains that

$$\begin{aligned} \Delta E_{2k+1}^{(2)} = - \frac{1}{4\pi ^2} \left( 8 k \sum _{q\ge -k, q \ne 0} \frac{1}{q(4q^2-1)^2} + \sum _{q\ge -k, q \ne 0} \frac{3+4q}{q(4q^2-1)^2} + \mathcal{O}(k^{-1})\right) .\nonumber \\ \end{aligned}$$

(B20)

In the first sum, one notices that the summand $1/(q(4q^2-1)^2)$ is an odd function of q and, therefore, for large k, it is easy to see that this first term is actually of order $\mathcal{O}(k^{-3})$. The leading term, for large k, is thus the second sum in Eq. (B20), which yields

$$\begin{aligned} \lim _{k \rightarrow \infty } \Delta E_{2k+1}^{(2)} = -\frac{1}{4\pi ^2} \sum _{q \in \mathbb {Z}^*} \frac{3+4q}{q(4q^2-1)^2} = \frac{1}{\pi ^2} - \frac{1}{8} , \end{aligned}$$

(B21)

as announced in Eq. (77).

Appendix C: Derivation of the Constrained Propagator via the Karlin–McGregor Formula

In this section, we give another derivation of the probability for N vicious Brownian motions to have survived and be at $\vec {x}$ at time t, starting from $\vec {x}_0$ at time $t_0$ which was obtained in (117). From the Karlin McGregor formula, we can write this propagator for N non-crossing particles as an $N \times N$ determinant involving only the single particle propagators

$$\begin{aligned} P_N^{ B r } ( \vec {x} , t | {\vec {x_ {0}}} , t _ { 0 } ; + \infty ) = \det _{ 1 \le i,j \le N} \left( P^{ B r }( x_j , t | x _ { 0i }, t _ { 0 } ) \right) \end{aligned}$$

(C1)

where $P^{ B r }( x_j , t | x _ { 0i }, t _ { 0 } )$ is the propagator of the free Brownian motion

$$\begin{aligned} P^{ B r }( x_j , t | x _ { 0i }, t _ { 0 } ) = \frac{1}{\sqrt{2 \pi (t-t_0) }} e^{ - \frac{ (x_j - x_{0i})^2 }{2(t-t_0)} } . \end{aligned}$$

(C2)

therefore, by injecting (C2) in (C1) and factoring out the common factors of the determinants, one finds

$$\begin{aligned} P_N^{ B r }( \vec {x} , t | {\vec {x_ {0}}} , t _ { 0 } ; + \infty ) = \frac{1}{\sqrt{2 \pi (t-t_0)}^N} e^{ - \frac{1}{2(t-t_0)} \sum \nolimits _{i=1}^N (x_i^2 + x_{0i}^2 ) } \det _{ 1 \le i,j \le N} \left( e^{\frac{x_j x_{0i} }{t-t_0}} \right) \end{aligned}$$

(C3)

As in the main text, we analyse this propagator in the limit $t \rightarrow \infty$, $x \rightarrow \infty$ keeping $x/ \sqrt{t}$ fixed, while $x_0$ and $t_0$ are fixed and of order $\mathcal{O}(1)$. In this limit, we truncate the exponential factor, such that the determinant, to leading order for large t, can be computed by using the identity

$$\begin{aligned} \bigg \vert \sum _{k=0}^{N-1} x_j^k \frac{(x_{0i}/t)^k }{k! } \bigg \vert _{1\le i,j \le N}= & {} \begin{vmatrix} 1&x_1&\cdots&x_1^{N-1} \\ 1&\cdots&\cdots&\cdots \\ 1&x_N&\cdots&x_N^{N-1} \end{vmatrix} \times \begin{vmatrix} 1&1&1 \\ (x_{01}/t)&\cdots&(x_{0N}/t) \\ \cdots&\cdots&\cdots \\ \frac{(x_{01}/t)^{N-1}}{(N-1)!}&\cdots&\frac{(x_{0N}/t)^{N-1}}{(N-1)!} \end{vmatrix} \end{aligned}$$

(C4)

$$\begin{aligned}= & {} \ \Delta (\vec {x}) \ \frac{1}{t^{N(N-1)/2} \prod \nolimits _{k=0}^{N-1} k! } \ \Delta (\vec {x}_0) , \end{aligned}$$

(C5)

where $\Delta (\vec {x}) = \prod _{1\le i < j \le N} ( x_j - x_i )$ is the Vandermonde determinant. Finally, the large time limit of the Karlin–McGregor formula for N non-crossing Brownian motions yields

$$\begin{aligned} P_N^{ B r } ( \vec {x} , t | {\vec {x_ {0}}} , t _ { 0 } ; + \infty ) \simeq \frac{1 }{ (2\pi ) ^{\frac{N}{2}} G(N+1)} \ e^{-\frac{1}{2t} \sum \limits _{i=1}^{N} x_i^2 } \ \prod \limits _{1\le i < j \le N} (x_j - x_i) (x_{0j}-x_{0i}) \quad t^{-\frac{N^2}{2}} \end{aligned}$$

(C6)

in agreement with (117) given in the text.

Appendix D: Computation of the Survival Amplitude in Terms of Pfaffians

We detail here the computations of the survival probability prefactors in the special cases $W \rightarrow \infty$ and $W = 0$ (see Eq. (125) and below in the main text).

1.1 1. $W\rightarrow \infty$

Equation (125) can be computed in the case where there is no wall. In this case, the eigenfunctions are simply those of the harmonic oscillator. The (i, j) term in the Pfaffian is then:

$$\begin{aligned} A^\infty _{i,j}= & {} \int \limits _{\mathbb {R}^2} c_i c_j \ e^{-\frac{X^2 + \tilde{X}^2}{2}} \ H_i\left( \frac{X}{\sqrt{2}}\right) H_j\left( \frac{ \tilde{X} }{\sqrt{2}}\right) \ {\mathrm {sgn}}(X-\tilde{X}) \ {d}X {d}\tilde{X} \end{aligned}$$

(D1)

$$\begin{aligned}= & {} (-1)^{i+j} \ c_i c_j \ 2^\frac{i+j}{2} \int \limits _{\mathbb {R}^2} \ \left( \frac{{d}^i}{{d}X^i} e^{-\frac{X^2}{2}} \right) \left( \frac{{d}^j}{{d}\tilde{X}^j} e^{-\frac{\tilde{X}^2}{2}} \right) \ {\mathrm {sgn}}(X-\tilde{X}) \ {d}X {d} \tilde{X}\nonumber \\ \end{aligned}$$

(D2)

where we have used the definition of Hermite polynomials given in (A1). Integrating by parts with respect to the X variable:

$$\begin{aligned} A^\infty _{i,j}= & {} (-1)^{i+j+1} \ c_i c_j 2^\frac{i+j}{2} \int \limits _{\mathbb {R}^2} \ \left( \frac{d^{i-1}}{{d} X^{i-1}} e^{-\frac{X^2}{2}} \right) \left( \frac{d^j}{d\tilde{X}^j} e^{-\frac{\tilde{X}^2}{2}} \right) \ 2 \delta (X-\tilde{X}) \ {d}X {d} \tilde{X}\nonumber \\ \end{aligned}$$

(D3)

$$\begin{aligned}= & {} (-1)^{i+j+1} \ c_i c_j 2^\frac{i+j+2}{2} \int \limits _{\mathbb {R}} \ \left( \frac{d^{i-1}}{{d} X^{i-1}} e^{-\frac{X^2}{2}} \right) \left( \frac{d^j}{ {d} X^j} e^{-\frac{X^2}{2}} \right) \ {d} X \end{aligned}$$

(D4)

Integrating by parts again $i-1$ times:

$$\begin{aligned} A^\infty _{i,j}= & {} (-1)^{j} \ c_i c_j 2^\frac{i+j+2}{2} \int \limits _{\mathbb {R}} \ e^{-\frac{X^2}{2}} \left( \frac{{d}^{i+j-1} }{{d}X^{i+j-1} } e^{-\frac{X^2}{2}} \right) \ {d}X \end{aligned}$$

(D5)

$$\begin{aligned}= & {} (-1)^{i-1} 2^{\frac{3}{2}} \ c_i c_j \int \limits _{\mathbb {R}} \ e^{-X^2} H_{i+j-1}\left( \frac{X}{\sqrt{2}}\right) \ {d}X \end{aligned}$$

(D6)

$$\begin{aligned} A^\infty _{i,j}= & {} (-1)^{i-1} 4 \ c_i c_j \int \limits _{\mathbb {R}} \ e^{-2 X^2} H_{i+j-1}(X) {d}X \end{aligned}$$

(D7)

The term is nonzero only if i and j have opposite parity, as can be read from (D7). This ensures the antisymmetry of $A^\infty _{i,j}$ upon exchanging i and j, as expected from (D1). This integral can be evaluated starting from the identity for the generating function of Hemite polynomials (A8), as explained in Appendix E, yielding

$$\begin{aligned} \int \limits _{\mathbb {R}} \ e^{-2X^2} H_{2m}(X) \ {d} X = \sqrt{\frac{\pi }{2}} \left( -\frac{1}{2} \right) ^m \frac{(2m)!}{m!} . \end{aligned}$$

(D8)

The Pfaffian term of the prefactor in the survival probability is then

$$\begin{aligned} \underset{ 0 \le i,j \le N-1 }{\mathrm {Pf}} \left( A^\infty _{i,j} \right) = \underset{ 0 \le i,j \le N-1 }{\mathrm {Pf}} \left( \frac{(-1)^{i + \frac{i+j-3}{2}} }{2^{i+j-3/2}} \frac{ (i+j-1)! }{ \sqrt{i! j! } \left( \frac{i+j-1}{2}\right) ! } \ \mathbf {1}_{ i+j \notin 2\mathbb {N}} \right) . \end{aligned}$$

(D9)

And the survival amplitude is, in the case $W \rightarrow \infty$, given by

$$\begin{aligned} A(\vec {x_0},t_0;W \rightarrow \infty )&= t_0^{ \frac{N(N-1)}{4} } \underset{ \begin{array}{c} 0 \le i,j \le N-1 \\ i+j \notin 2\mathbb {N} \end{array} }{\mathrm {Pf}} \left( \frac{(-1)^{i + \frac{i+j-3}{2}} }{2^{i+j-3/2}} \frac{ (i+j-1)! }{ \sqrt{i! j! } \left( \frac{i+j-1}{2}\right) ! } \right) \nonumber \\&\quad \times \det _{1 \le i,j \le N} \left( \phi _{i-1} ^* ({X_0}_j) e^{\frac{{X_0}_j^2}{4} } \right) , \end{aligned}$$

(D10)

as given in Eq. (126) in the text.

1.2 2. $W = 0$

As explained in Appendix B, in the $W=0$ case, the k-th wavefunction is the $(2k+1)$-th wavefunction of the harmonic oscillator:

$$\begin{aligned} \phi _k(X) = \sqrt{2} \ c_{2k+1} \ H_{2k+1} \left( \frac{X}{\sqrt{2}} \right) \ e^{-\frac{X^2}{4}} \end{aligned}$$

(D11)

The generic term $A_{i,j}^0$ in the Pfaffian in Eq. (127) of the main text reads

$$\begin{aligned} A_{i,j}^0= & {} 2 \ c_{2i+1} c_{2j+1} \ \int \limits _{({\mathbb {R}^-})^2} e^{-\frac{X^2 + \tilde{X}^2}{2}} \ H_{2i+1}\left( \frac{X}{\sqrt{2}}\right) H_{2j+1}\left( \frac{ \tilde{X} }{\sqrt{2}}\right) \nonumber \\&\times {\mathrm {sgn}}(X-\tilde{X}) \ {d} X {d} \tilde{X} \end{aligned}$$

(D12)

$$\begin{aligned}= & {} 2^{i+j+2} \ c_{2i+1} c_{2j+1} \int \limits _{({\mathbb {R}^-})^2} \left( \frac{d^{2i+1}}{{d}X^{2i+1}} e^{-\frac{X^2}{2}}\right) \left( \frac{d^{2j+1}}{d\tilde{X}^{2j+1} } e^{-\frac{\tilde{X}^2}{2}} \right) \nonumber \\&\times {\mathrm {sgn}}(X-\tilde{X}) \ {d}X {d}\tilde{X} \end{aligned}$$

(D13)

$$\begin{aligned}= & {} 2^{i+j+2} \ c_{2i+1} c_{2j+1} \left( - 2 \int \limits _{-\infty }^0 \left( \frac{d^{2i}}{{d}X^{2i}} e^{-\frac{X^2}{2}} \right) \left( \frac{d^{2j+1}}{{d}X^{2j+1}} e^{-\frac{X^2}{2}} \right) {d} X \right. \nonumber \\&\left. + \frac{H_{2i}(0) H_{2j}(0)}{2^{i+j}} \right) \end{aligned}$$

(D14)

where we have used (A1) and integrated by parts with respect to X. Integrating by parts 2i times, we compute the integral as:

$$\begin{aligned}&\int \limits _{-\infty }^0 \left( \frac{{d}^{2i}}{{d}^{2i}X} e^{-\frac{X^2}{2}} \right) \left( \frac{{d}^{2j+1}}{{d}^{2j+1}X} e^{-\frac{X^2}{2}} \right) {d} X \nonumber \\&\quad = \int \limits _{-\infty }^0 e^{-\frac{X^2}{2}} \left( \frac{d^{2i+2j+1}}{{d}X^{2i+ 2j+1}} e^{-\frac{X^2}{2}} \right) {d} X - \sum _{m=1}^{i} \frac{H_{2i-2m}(0) H_{2j+2m}(0) }{ 2^{i+j} } \end{aligned}$$

(D15)

$$\begin{aligned}&\quad = -\frac{1}{2^{i+j}\sqrt{2} } \int \limits _{-\infty }^0 e^{-X^2} H_{2i+2j+1}\left( \frac{X}{\sqrt{2}} \right) {d} X - (-1)^{i+j}\nonumber \\&\qquad \times \sum _{m=1}^{i} (2i-2m-1)!! (2j+2m-1)!! \end{aligned}$$

(D16)

$$\begin{aligned}&\quad = -\frac{1}{2^{i+j} } \int \limits _{-\infty }^0 e^{-2 X^2} H_{2i+2j+1}(X) {d} X - (-1)^{i+j} \nonumber \\&\qquad \times \sum _{m=1}^{i} (2i-2m-1)!! (2j+2m-1)!! \end{aligned}$$

(D17)

The remaining integral and the discrete sum can be evaluated explicitly (see Eq. (E6) below for the computation of the integral)

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}\int \limits _{-\infty }^0 e^{-2 X^2} H_{2n + 1}(X) {d} X \quad = \quad \frac{(-1)^{n+1} (2n+1)! }{2^{n+1} n! } \ {}_2F_1\left( \frac{1}{2}, -n , \frac{3}{2}, -1\right) \\ &{}\overset{i}{\underset{m=1}{\sum }} (2i-2m-1)!! (2j+2m-1)!! \\ &{}= 2^{i+j} \left( \frac{2 \Gamma \left( \frac{3}{2}+i+j \right) }{\sqrt{\pi }} {}_2F_1\left( 1,\frac{3}{2}+i+j;\frac{3}{2}, -1\right) -(-1)^{i} \frac{\Gamma \left( \frac{3}{2}+j\right) }{\Gamma \left( \frac{3}{2}-i\right) } {}_2F_1\left( 1,\frac{3}{2}+j;\frac{3}{2}-i , -1\right) \right) , \end{array}\right. } \end{aligned}$$

(D18)

where ${}_2F_1(a,b; c, z)$ is the standard hypergeometric function. Finally, the generic term $A_{i,j}^0$ in the Pfaffian in Eq. (127) reads

$$\begin{aligned} A_{i,j}^0 =&\frac{(-1)^{i+j+1}}{\sqrt{ \frac{\pi }{2} (2i+1)! (2j+1)! }} \left[ \left( \frac{1}{4}\right) ^{i+j} \frac{(2i+2j+1)!}{(i+j)!} {}_2F_1 \left( \frac{1}{2},-i-j;\frac{3}{2}, -1 \right) \right. \nonumber \\&- (2i-1)!!(2j-1)!! \nonumber \\&\left. - (2)^{i+j+1} \left( \frac{2 \Gamma \left( \frac{3}{2}+i+j \right) }{\sqrt{\pi }} {}_2F_1 \left( 1,\frac{3}{2}+i+j;\frac{3}{2}, -1 \right) \right. \right. \nonumber \\&\left. \left. + (-1)^{1+i} \frac{\Gamma \left( \frac{3}{2}+j\right) }{\Gamma \left( \frac{3}{2}-i\right) } {}_2F_1 \left( 1,\frac{3}{2}+j;\frac{3}{2}-i , -1 \right) \right) \right] , \end{aligned}$$

(D19)

as given in Eq. (128) in the main text.

Appendix E: Computation of Some Integrals

The generating function of the Hermite polynomial (A8) enables us to compute some integrals which are useful for the computations presented in Appendix D.

• By multiplying both sides of Eq. (A8) by $e^{-2x^2}$ and integrating over $x \in {\mathbb {R}}$ one obtains

  $$\begin{aligned}&\sum _{n=0}^{\infty } \frac{t^n}{n!} \int \limits _{\mathbb {R}} \ e^{-2x^2} H_{n}(x) dx = \int \limits _{\mathbb {R}} \ e^{-2x^2} e^{2 x t-t^{2}} {d}x = \sqrt{\frac{\pi }{2}} e^{-\frac{t^2}{2} } \nonumber \\&\quad = \sum _{m=0}^{\infty } \sqrt{\frac{\pi }{2}} \left( -\frac{1}{2}\right) ^m \frac{t^{2m}}{m!} . \end{aligned}$$

  (E1)

  By identifying the powers of t, one obtains the identity given in Eq. (D8).

• By evaluating the same integral on $\mathbb {R}^-$ one gets

  $$\begin{aligned} \sum _{n=0}^{\infty } \frac{t^n}{n!} \int _{-\infty }^{0} e^{-2x^2}H_n(x){d}x = e^{-\frac{t^2}{2}} \int _{-\infty }^{-t/2} e^{-2x^2} {d}x = \frac{\sqrt{\pi }}{2 \sqrt{2}} e^{-\frac{t^2}{2}} \text {erfc}\left( \frac{t}{\sqrt{2}}\right) . \end{aligned}$$

  (E2)

  Using the series expansion $\text {erfc}(x) = 1 - \text {erf}(x) = 1 -\frac{2}{\sqrt{\pi }} \sum _n \frac{(-1)^n x^{2n+1} }{ n! (2n+1) }$:

  $$\begin{aligned}&\sum _{n=0}^{\infty } \frac{t^n}{n!} \int _{-\infty }^{0} e^{-2x^2}H_n(x){d}x =\frac{\sqrt{\pi }}{2 \sqrt{2}} \left( \sum _{n=0}^{\infty } \frac{(-1)^n t^{2n}}{2^n n!} \right) \nonumber \\&\quad \times \left( 1 - \frac{2}{\sqrt{\pi }} \sum _{n=0}^{\infty } \frac{(-1)^n t^{2n+1}}{ \sqrt{2} \, 2^n n! (2n+1)} \right) . \end{aligned}$$

  (E3)

  Identifying the coefficient of the term $\propto t^{2n+1}$ on both sides of this identity yields

  $$\begin{aligned} \frac{1 }{ (2n+1)!} \int _{-\infty }^{0} e^{-2x^2}H_{2n+1}(x) {d} x= & {} - \sum _{m=0}^{n} \frac{(-1)^{n-m} }{(n-m)! 2^{n-m} } \frac{(-1)^m }{ 2^{m+1} (2m+1) m!}\nonumber \\ \end{aligned}$$

  (E4)
  $$\begin{aligned}= & {} \frac{(-1)^{n+1} }{2^{n+1} } \sum _{m=0}^{n} \frac{1}{ (2m+1) \ m! (n-m)! } \end{aligned}$$

  (E5)

  Finally the sum over m can be expressed in terms of a hypergeometric function, which gives finally

  $$\begin{aligned} \int _{-\infty }^{0} e^{-2x^2}H_{2n+1}(x) {d}x = \frac{(-1)^{n+1} (2n+1)! }{2^{n+1} n! } \ {}_2F_1\left( \frac{1}{2}, -n , \frac{3}{2}, -1\right) , \end{aligned}$$

  (E6)

as given in the first line of Eq. (D18).

Appendix F: Dyson Brownian Motion and Non-crossing Brownian Paths

In this appendix we derive the relation given in Eq. (133) of the text.

1.1 Relation Between Propagators

As in the main text, we call ${ P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 )$ the propagator of the Dyson Brownian motion with Dyson index $\beta =2$, and ${ P }^{Br}_N ( \vec {x} , t | {\vec {x}}_0 , t_0 )$ the propagator for independent Brownian Motions with boundary condition ${ P }^{Br}_N ( \vec {x} , t | {\vec {x}}_0 , t_0 ) = 0$ whenever $x_i = x_j$. Let us first show the following relation

$$\begin{aligned} { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ) = \frac{ \prod \nolimits _ { i< j } \left( x _ { j } - x_ { i } \right) }{ \prod \nolimits _ { i < j } \left( x_ { 0j } - x_ { 0i } \right) } \ { P }^{Br}_N ( \vec {x} , t | {\vec {x}}_0 , t_0 ) . \end{aligned}$$

(F1)

We follow the argument of [59], and consider a general $\beta$ for now. The propagator ${ P }^{Br}_N$ satisfies the diffusion equation

$$\begin{aligned} \frac{ \partial }{ \partial t } { P }^{Br}_N \left( \vec {x} , t | {\vec {x}}_0 , t_0 \right) = \frac{ 1 }{ 2 } \sum _ { i = 1 } ^ { N } \frac{ \partial ^ { 2 } }{ \partial x _ { i } ^ { 2 } } { P }^{Br}_N \left( \vec {x} , t | {\vec {x}}_0 , t_0 \right) \end{aligned}$$

(F2)

together with the non-crossing condition, i.e. ${ P }^{Br}_N \left( \vec {x} , t | {\vec {x}}_0 , t_0 \right) = 0 \text { if } x _ { i } = x _ { j }$.

On the other hand, the Dyson Brownian motion propagator ${ P }_N^{\mathrm {DBM}}$ satisfies (see e.g. [78])

$$\begin{aligned} \frac{ \partial }{ \partial t } { P }_N^{\mathrm {DBM}} = \frac{ 1 }{ 2 } \sum _ { i = 1 } ^ { N } \frac{ \partial ^ { 2 } }{ \partial x _ { i } ^ { 2 } } { P }_N^{\mathrm {DBM}} - \frac{ \beta }{ 2 } \sum _ { i = 1 } ^ { N } \frac{ \partial }{ \partial x _ { i } } \left[ \sum _ { 1 \le j \ne i \le N } \frac{ 1 }{ x _ { i } - x _ { j } } { P }_N^{\mathrm {DBM}} \right] \end{aligned}$$

(F3)

Applying the transform:

$$\begin{aligned} { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ) = \frac{ \exp \left[ \frac{ \beta }{ 2 } \sum \nolimits _ { 1 \le i< j \le N } \log \left( x _ { j } - x _ { i } \right) \right] }{ \exp \left[ \frac{ \beta }{ 2 } \sum \nolimits _ { 1 \le i < j \le N } \log \left( x _ {0 j } - x _ { 0 i } \right) \right] } \times { W }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ) \end{aligned}$$

(F4)

we obtain:

$$\begin{aligned} \frac{ \partial }{ \partial t } { W }_N^{\mathrm {DBM}} = \frac{ 1 }{ 2 } \sum _ { i = 1 } ^ { N } \frac{ \partial ^ { 2 } }{ \partial x _ { i } ^ { 2 } } { W }_N^{\mathrm {DBM}} - \frac{ \beta }{ 8 } ( \beta - 2 ) \sum _ { i = 1 } ^ { N } \sum _ { 1 \le j \ne i \le N } \frac{ 1 }{ \left( x _ { j } - x _ { i } \right) ^ { 2 } } { W }_N^{\mathrm {DBM}} \end{aligned}$$

(F5)

For $\beta = 2$, ${ W }_N^{\mathrm {DBM}}$ verifies the same equation as ${ P }^{Br}_N$. It also verifies the annihilating condition, since ${ P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ) \sim \left( x _ { j } - x _ { i } \right) ^ { \beta } , x _ { i } \rightarrow x _ { j }$. We conclude ${ W }_N^{\mathrm {DBM}} = { P }^{Br}_N$ by unicity of the solution of this linear PDE, and thus we obtain Eq. (F1).

1.2 Equivalence of the Two Processes

Assuming known final positions $x_i$ at time t and initial positions $\vec {x}_0$ at time $t_0$, the probability to be in $\vec {y}$ at some intermediate time $\tau$ is the same in the DBM and Brownian cases, by telescoping the extra factor:

$$\begin{aligned} \frac{ { P }_N^{\mathrm {DBM}} ( \vec {y} ,\tau | \vec {x}_0 , t_0 ) \ { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {y} , \tau ) }{ { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ) } = \frac{ { P }^{Br}_N ( \vec {y} ,\tau | \vec {x}_0 , t_0 ) \ { P }^{Br}_N ( \vec {x} , t | \vec {y} , \tau ) }{ { P }^{Br}_N ( \vec {x} , t | \vec {x}_0 , t_0 ) } \end{aligned}$$

(F6)

More generally, the finite-dimensional distributions are equal for the two processes. Assuming fixed final and initial positions, the probability to be in $(y_1, \ldots , y_m)$ at times $t_0< \tau _1< \cdots< \tau _m < t$:

$$\begin{aligned} \frac{ { P }_N^{\mathrm {DBM}} ( \vec {y}_1 ,\tau _1| \vec {x}_0 , t_0 ) \cdots { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {y}_m , \tau _m ) }{ { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ) } = \frac{ { P }^{Br}_N ( \vec {y}_1 ,\tau _1 | \vec {x}_0 , t_0 ) \cdots { P }^{Br}_N ( \vec {x} , t | \vec {y}_m , \tau _m ) }{ { P }^{Br}_N ( \vec {x} , t | \vec {x}_0 , t_0 ) } \end{aligned}$$

(F7)

From this equivalence we obtain that, conditioning on fixed final positions, the probability to stay below a deterministic moving barrier g(t) is the same for the two processes. The relation between the propagators is thus still correct when adding a moving barrier:

$$\begin{aligned} { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ; g(t) )&= { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 )\nonumber \\&\quad \times {\mathrm{Pr}}(\text {DBM remains below the barrier}~ g(t)| \vec {x} , t ; \vec {x}_0 , t_0) \nonumber \\&= { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 )\nonumber \\&\times {\mathrm{Pr}}(\text {Brownian motions remains below the barrier}~ g(t)~ \text {and do not cross}| \vec {x} , t ; \vec {x}_0 , t_0) \nonumber \\&= \frac{ \prod \nolimits _ { i< j } \left( x _ { j } - x_ { i } \right) }{ \prod \nolimits _ { i < j } \left( x_ { 0j } - x_ { 0i } \right) } \ { P }^{Br}_N ( \vec {x} , t | {\vec {x}}_0 , t_0 ; g(t) ) , \end{aligned}$$

(F8)

which shows the Eq. (133) of the text.

1.3 Alternative Derivation of Eq. (F8)

We present here another derivation of the relation between the propagators in the presence of a moving barrier.

1.3.1 Constant Barrier $g(t) = 0$

If the barrier is fixed at $g(t)=0$, the propagator of the DBM in the presence of the barrier is the solution of Eq. (F3) which vanishes at coinciding arguments and furthermore satisfies the additional condition

$$\begin{aligned} { P }_N^{\mathrm {DBM}} ( \vec {x} , t | \vec {x}_0 , t_0 ;g(t)=0) = 0 , \quad \text {if any} \, \, x_i=0 \end{aligned}$$

(F9)

Since $P_N^{\mathrm {Br}} ( \vec {x} , t | \vec {x}_0 , t_0 ;g(t)=0)$ satisfies the same additional condition, the relation (F8) is still valid in this case.

1.3.2 Moving Barrier g(t)

For a moving barrier g(t), we consider the processes $z_i(t)=x_i(t) - g(t)$. For these processes the absorbing boundary condition is fixed at $z_i=0$. The Langevin equation satisfied by a shifted processes $z_i(t)$ reads

$$\begin{aligned} \frac{dz_i(t)}{dt} = \frac{dx_i(t)}{dt} - \frac{dg(t)}{dt} = \frac{\beta }{2} \sum _{j \ne i} \frac{1}{z_i(t) - z_j(t)} - g'(t) + \xi _i(t) \end{aligned}$$

(F10)

This Langevin equation is identical to the original one up to an additional drift term $- g'(t)$. The corresponding Fokker–Planck equation is identical to Eq. (F3) with $x_i \rightarrow z_i$ together with the additional term from the drift (the arguments of all the functions are now the $z_i$)

$$\begin{aligned}&- \sum _i \frac{\partial }{\partial z_i} (- g'(t) { P }_N^{\mathrm {DBM}}) \nonumber \\&\quad = g'(t) \sum _i \frac{\partial { P }_N^{\mathrm {DBM}}}{\partial z_i} \end{aligned}$$

(F11)

$$\begin{aligned}&\quad = \frac{ \prod \nolimits _ { i< j } \left( z_ { j } - z_ { i } \right) }{ \prod \nolimits _ { i < j } \left( z_ { 0j } - z_ { 0i } \right) } g'(t) \sum _i \left( \frac{\partial { W }_N^{\mathrm {DBM}}}{ \partial z_i} + \frac{\beta }{2} \sum _{j\ne i} \frac{1}{z_i - z_j} { W }_N^{\mathrm {DBM}}\right) \end{aligned}$$

(F12)

However, we note that, by symmetry:

$$\begin{aligned} \sum _i \sum _{j \ne i} \frac{1}{z_i - z_j} = 0 \end{aligned}$$

(F13)

Such that the additional term is:

$$\begin{aligned} - \sum _i \frac{\partial }{\partial z_i} (- g'(t) { P }_N^{\mathrm {DBM}}) =- \frac{ \prod \nolimits _ { i< j } \left( z_ { j } - z_ { i } \right) }{ \prod \nolimits _ { i < j } \left( z_ { 0j } - z_ { 0i } \right) } \sum _i \frac{\partial }{ \partial z_i} ( - g'(t) { W}_N^{\mathrm {DBM}} ) \end{aligned}$$

(F14)

We see that, in the translated frame, the PDE verified by the translated ${ W}_N^{\mathrm {DBM}}$ is exactly the same as that of ${ P}_N^{ Br}$. As a consequence, the relation (F8) between the two propagators still holds for an arbitrary g(t).

Appendix G: Bulk Density of the Dyson Brownian Motion with a Boundary at $W=0$

In this section, we derive the large N limit of the density for the Dyson Brownian motion with a boundary at $W=0$. The starting point of our computations is the joint PDF of the positions $x_i<0$ given, at large time t, by (138), which we write here with an overall prefactor $K_N$ including all terms that do not depend on $\vec {x}$:

$$\begin{aligned} P_N^{DBM}(\vec {x},t|\vec {x_0},t_0;0)\approx & {} K_N \prod _{i=1}^N x_i \prod _{i<j} \left[ (x_j - x_i) ( x_j^2 - x_i^2)\right] \ e^{- \frac{1}{2t} \sum \nolimits _{i=1}^N x_i^2} \end{aligned}$$

(G1)

$$\begin{aligned}\approx & {} K_N \prod _{i=1}^N x_i \prod _{i<j} \left[ (x_j - x_i)^2 ( x_j + x_j)\right] \ e^{- \frac{1}{2t} \sum \nolimits _{i=1}^N x_i^2} . \end{aligned}$$

(G2)

Note that this joint PDF is very similar to the one encountered in the so called O(n) matrix model [64, 65], with the value $n=-1$ in this case. To compute the density in the limit of large N, we will follow the method exposed in [41, 67], which is based on a method developed by Bueckner [79]. We first perform a change of variables

$$\begin{aligned} y_i = - \frac{x_i}{\sqrt{2Nt}} , \end{aligned}$$

(G3)

such that the joint PDF of the $y_i$’s reads,

$$\begin{aligned} P_{\mathrm{joint}}(y_1, \cdots , y_N) = \frac{1}{Z'_N} e^{-E_N ( \vec {y} ) } \end{aligned}$$

(G4)

where

$$\begin{aligned} E_N ( \vec {y} ) = N \sum _i y_i^2 - \frac{1}{2} \sum _{i \ne j} \ln |{y_i + y_j }| - \sum _{i \ne j} \ln |{ y_j - y_i}| - \frac{1}{2} \sum _i \ln |{y_i}| . \end{aligned}$$

(G5)

Let us introduce the average bulk density ${\tilde{r}}(y)$

$$\begin{aligned} {\tilde{r}}(y) = \frac{1}{N} \sum _i \langle \delta (y - y_i) \rangle , \end{aligned}$$

(G6)

where the average is computed with respect to the joint PDF in (G4). In the limit of large N, the density can be computed using a standard Coulomb gas method and one finds that ${\tilde{r}}(y)$ is given by the solution of the following integral equation

(G7)

which holds for y inside the support of ${\tilde{r}}$, together with the normalisation condition $\int _0^\infty {\tilde{r}}(y) dy = 1$. It turns out that ${\tilde{r}}(y)$ has a finite support [0, L], and the solution of (G7) can be obtained explicitly along the lines explained in Ref. [67] (see Sect. 6.3).

Let us introduce the resolvent

$$\begin{aligned} W(z) = \int _0^\infty dy \frac{{\tilde{r}}(y)}{z-y}, \end{aligned}$$

(G8)

which is defined on the complex plane with a cut on [0, L]. Equation (G7) gives the following constraint on the resolvent, for $y \in [0,L]$:

$$\begin{aligned} 2 y = W(y+i 0^+) + W(y-i 0^+) - W(-y) . \end{aligned}$$

(G9)

Hence, W is the solution of the following Riemann-Hilbert problem [41, 67]:

• 1. W is analytic everywhere except on the cut [0, L],

• 2. $W(z) \sim \frac{1}{z}$ as $|{z}| \rightarrow \infty$, which follows from its definition (G8) together with the normalization of ${\tilde{r}}$,

• 3. $W(z) \in \mathbb {R}$ for $z \in [L,+\infty ]$,

• 4. W satisfies (G9) .

Note the last condition can also be written as $y = {\mathrm {Re}} [W(y)] - \frac{1}{2} W(-y)$, see equation (6.60) in [67], and W has a jump as it approaches the cut, i.e. $W(y \pm i 0^+)={\text {Re}}[W(y)] \mp i \pi {\tilde{r}}(y)$.

The solution of this Riemann-Hilbert problem can be found as $W(z) = h(z) + \widetilde{W}(z)$ with a particular solution of (G9) given by $h(z) = \frac{2}{3}z$ (see Eq. (6.62) of [67]) while the homogeneous solution $\widetilde{W}(z)$ reads

$$\begin{aligned} \widetilde{W}(z) = P_0(z^2) \phi _0(z) + P_1(z^2) \phi _1(z) \end{aligned}$$

(G10)

where $P_{0,1}(x)$ are polynomials while the functions $\phi _{0,1}(z)$ are given by

$$\begin{aligned} \left\{ \begin{array}{ll} \phi _0(z) = 2 \frac{\cos \left( \frac{\omega }{3} + \frac{\pi }{6} \right) }{ \sqrt{3} } \\ \\ \phi _1(z) = \frac{\sin \left( \frac{\omega }{3} + \frac{\pi }{6} \right) }{ \tan ( \omega ) } \end{array} \right. , \quad \quad {\mathrm{with}} \quad \frac{L}{z}=\sin \omega . \end{aligned}$$

(G11)

The polynomials $P_{0,1}$ as well as the edge of the support L are then obtained by imposing that $W(z) \sim \frac{1}{z}$ as $|{z}| \rightarrow \infty$. Using the asymptotic behaviours for large z [67]

$$\begin{aligned} \phi _{0}(z)= & {} 1-\frac{1}{3} \tan \left[ \frac{\pi }{6}\right] \frac{L}{z} + \mathcal {O}\left( \frac{1}{z^2}\right) = 1 - \frac{1}{3 \sqrt{3}}\frac{L}{z} + \mathcal {O}\left( \frac{1}{z^2}\right) , \end{aligned}$$

(G12)

$$\begin{aligned} \phi _{1}(z)= & {} \sin \left[ \frac{\pi }{6}\right] \frac{z}{L}+\frac{1}{3} \cos \left[ \frac{\pi }{6}\right] -\left( \frac{1/9+1}{2}\right) \sin \left[ \frac{\pi }{6}\right] \frac{L}{z} + \mathcal {O}\left( \frac{1}{z^2}\right) \end{aligned}$$

(G13)

$$\begin{aligned}= & {} \frac{z}{2L} + \frac{1}{2\sqrt{3}} - \frac{5}{18} \frac{L}{z}+ \mathcal {O}\left( \frac{1}{z^2}\right) , \end{aligned}$$

(G14)

one obtains

$$\begin{aligned} \left\{ \begin{array}{ll} A(x)= A = \frac{2}{3 \sqrt{3}} L = \frac{1}{\sqrt{2}} , \\ \\ B(x) = B = - \frac{4}{3} L = - \sqrt{6} , \\ \\ L = \left( \frac{3}{2} \right) ^{3/2} . \end{array} \right. \end{aligned}$$

(G15)

Finally, the resolvent is given by (we recall that $\frac{L}{z}=\sin \omega$)

$$\begin{aligned} W(z) = \frac{2}{3} z + \sqrt{\frac{2}{3}} \cos \left( \frac{\omega }{3} + \frac{\pi }{6} \right) - \sqrt{6} \frac{\sin \left( \frac{\omega }{3} + \frac{\pi }{6} \right) }{ \tan ( \omega ) } , \end{aligned}$$

(G16)

from which one obtains the density ${\tilde{r}}(y)$ using the relation

$$\begin{aligned} {\tilde{r}}( y) = - \frac{1}{\pi } {\mathrm {Im}} ( W(y +i 0^+) ) . \end{aligned}$$

(G17)

Since $z=y \in [0,L]$ corresponds to $\omega =\frac{\pi }{2}-i \eta$ with $\eta >0$ such that $\frac{L}{y}=\sin \omega =\cosh \eta$ the density is given by [67]

$$\begin{aligned} {\tilde{r}}(y) = \frac{1}{2 \pi \sqrt{2} } \left( (e^{-\eta /3} - e^{ \eta /3}) + 3 \frac{y}{2 L}( e^{ \eta } - e^{ - \eta } ) ( e^{ \eta /3} + e^{ - \eta /3} ) \right) , \end{aligned}$$

(G18)

with $e^{\pm \eta } = \frac{L}{y} \pm \sqrt{\frac{L^2}{y^2} -1 }$ and $L = \left( \frac{3}{2} \right) ^{3/2}$, which eventually yields the expression given in Eq. (152).

As stated in the text, this is in accordance with Proposition 2.5 of [68]. Because of the change of variables we have applied in this paper, the relation between the variable s from this work and our variable y is the following

$$\begin{aligned} s=y^2 , \end{aligned}$$

(G19)

such that the density $\frac{d \mu _{V, \frac{1}{2}}^{*}(s)}{d s}$ obtained in [68] is related to $\tilde{r}(y)$ through :

$$\begin{aligned} \frac{d \mu _{V, \frac{1}{2}}^{*}}{d s} (y^2) = \frac{\tilde{r}(y)}{2y} . \end{aligned}$$

(G20)

This relation between the two formulas can be proved by changing variables to $\omega = \frac{1-\sqrt{1-\frac{y^{2}}{L^{2}}}}{1+\sqrt{1-\frac{y^{2}}{L^{2}}}}$. Indeed, replacing y by $\omega$ in both sides of (G20) through $y =L \sqrt{1-\left( \frac{1-\omega }{1+\omega }\right) ^{2}}$, one shows that :

$$\begin{aligned} \frac{d \mu _{V, \frac{1}{2}}^{*}}{d s} (y^2) = \frac{\tilde{r}(y)}{2y} = \frac{\left( 1- w^{1/3}\right) \left( w^{1/3}+1\right) ^3}{6 \sqrt{3} \pi w^{2/3}} . \end{aligned}$$

(G21)