Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent

Abstract

I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $\varPi _n=M_nM_{n-1}\ldots M_1$, where $M_i$’s are i.i.d. Following Tutubalin (Theor Probab Appl 10(1):15–27, 1965), the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the group $\mathrm {SL}(2,\mathbb {R})$ where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrödinger equation where the random potential is a Lévy noise (derivative of a Lévy process). In this case, I obtain a general formula for the variance of $\ln ||\varPi _n||$ and for the variance of $\ln |\psi (x)|$, where $\psi (x)$ is the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.

Appendices

A Representation with Multipliers: Another Convention

In this appendix, the connection is made with the convention of Ref. [36] for the construction of the representation with multipliers. This second convention, used in the mathematical literature, is probably more appropriate for generalization to groups of matrices of larger dimensions.

The starting point is to introduce the adjoint of the transfer operator, with same definition as in the paper,

$$\begin{aligned} \big [ \mathscr {T}_M^\dagger (q) \psi \big ](z) {\mathop {=}\limits ^{{\mathrm{def}}}}J^{-q/2}(M,z) \, \psi \left( \mathcal {M}(z) \right) . \end{aligned}$$

(A.1)

We now define the scalar product as

$$\begin{aligned} \left( \, \psi \,|\, \tilde{f} \, \right) {\mathop {=}\limits ^{{\mathrm{def}}}}\int {\mathrm{d}}z \,\rho (z)\, \psi (z)^*\, \tilde{f}(z) . \end{aligned}$$

(A.2)

The relation with the scalar product (2.61) can be simply established by writing

$$\begin{aligned} \left( \, \psi \,|\, \tilde{f} \, \right) = \langle \, \psi \,|\, f \, \rangle \qquad \text{ with } f(z)= \rho (z)\, \tilde{f}(z). \end{aligned}$$

(A.3)

With this new scalar product in hand, we can obtain the new transfer operator as follows

$$\begin{aligned}&\left( \, \mathscr {T}_M^\dagger (q)\psi \,|\, \tilde{f} \, \right) = \int {\mathrm{d}}y\, \rho (y) \, J^{-q/2}(M,y)\, \psi \left( \mathcal {M}(y) \right) ^*\, \tilde{f}(y) \nonumber \\&\qquad = \int {\mathrm{d}}z\, \rho \left( \mathcal {M}^{-1}(z)\right) \, \frac{{\mathrm {d}}\mathcal {M}^{-1}(z)}{{\mathrm {d}}z}\, J^{-q/2}(M,\mathcal {M}^{-1}(z))\, \psi (z)^*\, \tilde{f}\left( \mathcal {M}^{-1}(z) \right) \nonumber \\&\qquad = \left( \, \psi \,|\, \widetilde{\mathscr {T}}_M(q)\tilde{f} \, \right) \end{aligned}$$

(A.4)

with

$$\begin{aligned} \left[ \widetilde{\mathscr {T}}_M(q) f \right] (z) = J^{1+q/2}(M^{-1},z) \, f\left( \mathcal {M}^{-1}(z) \right) . \end{aligned}$$

(A.5)

From (A.3), we see that the two representations (2.41,A.5) are related by

$$\begin{aligned} \frac{1}{\rho (z)}\, \mathscr {T}_M(q)\, \rho (z)= \widetilde{\mathscr {T}}_M(q) . \end{aligned}$$

(A.6)

Contrary to $\mathscr {T}_M={\mathscr {T}}_M(q=0)$, which preserves the norm $\int {\mathrm{d}}z\,f(z)$, the operator $\widetilde{\mathscr {T}}_M(q=0)$ conserves $\int {\mathrm{d}}z\,\rho (z)\,\tilde{f}(z)$: $\int {\mathrm{d}}z\, \rho (z)\,\big [ \widetilde{\mathscr {T}}_M(0) \tilde{f} \big ](z) = \int {\mathrm{d}}z\, \rho (z)\,\tilde{f}(z) .$ This is one of the reasons why we have preferred the representation (2.41) in the paper.

Finally, note that the transfer operators (A.5) are related to infinitesimal generators of the form

$$\begin{aligned} \widetilde{\mathscr {D}}_i(q) = \frac{1}{\rho (z)}\, \mathscr {D}_i(q)\, \rho (z) = g_i(z)\,\frac{{\mathrm {d}}}{{\mathrm {d}}z} + (q+2)\, h_i(z) \qquad \text{ for } i\in \{ K, \, A, \, N \} . \end{aligned}$$

(A.7)

It is also easy to check that they realise the Lie algebra of $\mathrm {SL}(2,\mathbb {R})$ and that they are adjoint to the infinitesimal generators (2.69), with respect with the scalar product (A.2).

B Irreducible Representations of $\mathrm {SL}(2,\mathbb {R})$

We recall basic properties of the representation theory of real unimodular matrices, taken from the monograph [53]. For $M\in \mathrm {SL}(2,\mathbb {R})$, let us start with the group action in the space $\mathscr {F}(\mathbb {R}^2)$ of infinitely differentiable functions defined in the plane

$$\begin{aligned} (T_M F)(\varvec{x}) {\mathop {=}\limits ^{{\mathrm{def}}}}F(M^{-1}\cdot \varvec{x}) \end{aligned}$$

(B.1)

Classification of the irreducible representations of the group requires to identify the smallest proper invariant subspaces of functions. For this purpose, homogeneous functions play an important role. A function satisfying the property

$$\begin{aligned} F(\lambda \,\varvec{x}) = \left( {{\mathrm {sign}}}(\lambda ) \right) ^{1-\epsilon } \,|\lambda |^{-\eta }\,F(\varvec{x}) \qquad \forall \, \lambda \in \mathbb {R} \end{aligned}$$

(B.2)

is called a homogeneous function of degree $\eta \in \mathbb {C}$, where $\epsilon \in \{+1,\,-1\}$ defines even and odd sectors. We denote by $\mathscr {E}_{\eta }^+$ the space of homogeneous functions of degree $\eta$ with even parity, and $\mathscr {E}_{\eta }^-$ with odd parity. Clearly, the property (B.2) is preserved by the transformation (B.1), hence we have identified subspaces of $\mathscr {F}(\mathbb {R}^2)$ invariant under the action of the group. The space $\mathscr {E}_{\eta }^\pm$ for noninteger $\eta$ is known to generate an irreducible representation of $\mathrm {SL}(2,\mathbb {R})$ [53].

A key observation is that each homogeneous function of given parity is uniquely determined by its value on the projective line: for any $F\in \mathscr {E}_{\eta }^+$, we have $F(x,y)=|y|^{-\eta }F(x/y,1)=|x|^{-\eta }F(1,y/x)$, hence each homogeneous function of degree $\eta$ can be represented by an infinitely differentiable function defined on the projective line

$$\begin{aligned} f(z) = F(z,1) = |z|^{-\eta }\, F(1,1/z) . \end{aligned}$$

(B.3)

This shows that, on the projective line, the elements of $\mathscr {E}_{\eta }^+$ are represented by functions such that the two limits

$$\begin{aligned} \lim _{z\rightarrow -\infty } (-z)^{\eta }\,f(z) \,\,\,\text{ and }\,\,\, \lim _{z\rightarrow +\infty } z^{\eta }\,f(z) \,\,\,\text{ exist } \text{ and } \text{ are } \text{ equal. } \end{aligned}$$

(B.4)

In order to apply these considerations to our case we must relate the degree of the representation to the parameter q. Given a matrix (2.1), we consider the transformation $\tilde{f}=\mathscr {T}_M(q)f$ defined by (2.41). It is straighforward to get the asymptotic behaviour

$$\begin{aligned} \tilde{f}(z) \underset{z\rightarrow \pm \infty }{\simeq } {\left\{ \begin{array}{ll} \displaystyle \left( \frac{\sqrt{\rho (-d/c)}}{|c|}\right) ^q\frac{f(-d/c)}{c^2}\,\frac{1}{z^2|z\sqrt{\rho (z)}|^q} &{} \text{ for } c\ne 0 \\ \displaystyle |a|^{-2-q}\left( \frac{\rho (z/a^2)}{\rho (z)}\right) ^{q/2}\, f(z/a^2) &{} \text{ for } c=0 \end{array}\right. } \end{aligned}$$

(B.5)

In the paper, we have considered Jacobians involving symmetric densities $\rho (z)=\rho (-z)$ with power law tail $\rho (z)\sim |z|^{-2\omega }$ where

$$\begin{aligned} {\left\{ \begin{array}{ll} \omega =0 &{} \text{ for } \rho _N \\ \omega =1/2 &{} \text{ for } \rho _A \\ \omega =1 &{} \text{ for } \rho _K \end{array}\right. } \end{aligned}$$

(B.6)

It is now clear from (B.5) that the operators $\mathscr {T}_M(q)$ preserve the property (B.4), hence these operators form an irreducible representation of the unimodular group of degree

$$\begin{aligned} \eta =2+(1-\omega )\,q . \end{aligned}$$

(B.7)

For example, for the choice of Jacobian $J=J_N$, we have simply $\eta =2+q$ (Sect. 3.5). For $J=J_A$, we have $\eta =2+q/2$ (Sect. 3.6).

A remark:

• the case $c=0$ corresponds to the subgroup of matrices AN. As it is clear from (B.5), the exponent $\eta$ cannot be related to q in this case.

1.1 B.1 Eigenvectors of the Casimir Operator and $\mathscr {D}_K(q)$

Irreducible representations of $\mathrm {SL}(2,\mathbb {R})$ can also be analysed by the algebraic method, as it well known for $\mathrm {SU}(2)$. For this purpose, we find more convenient to consider the decomposition of the group in terms of matrices of the subgroups $\mathrm {K}$, $\widetilde{\mathrm {K}}$ and $\mathrm {A}$, i.e. as $M=K(\theta )\widetilde{K}(\varphi )A(w)$, according to the notations of Sect. 2. With this choice, the three infinitesimal generators obey the algebra

$$\begin{aligned} \left[ \varGamma _K\,\,,\varGamma _{\widetilde{K}}\right] = 2\varGamma _A \,, \,\,\, \left[ \varGamma _K\,\,,\varGamma _A\right] = -2\varGamma _{\widetilde{K}} \,, \,\,\, \left[ \varGamma _{\widetilde{K}}\,\,,\varGamma _A\right] = -2\varGamma _K \end{aligned}$$

(B.8)

which is the usual form for the Lie algebra of $\mathrm {SO}(2,1)$, the Lorentz group in $2+1$ dimensions, studied by Bargmann [10] (this is also the Lie algebra for $\mathrm {SU}(1,1)$ as expected, cf. [32]).

The connection to the Lorentz group makes easy to identify the Casimir operator as $-\varGamma _K^2+\varGamma _{\widetilde{K}}^2+\varGamma _A^2=3\,\mathbf {1}_2$, and, for the representation of interest in the paper:

$$\begin{aligned} {\mathscr {C}}= -\mathscr {D}_{K}(q)^2 + \mathscr {D}_{\widetilde{K}}(q)^2 + \mathscr {D}_{A}(q)^2 = q\,(q+2) \,, \end{aligned}$$

(B.9)

where we have used the expressions (2.46) derived in Sect. 2.5.1. This makes clear that irreducible representations are classified by the index q (this is similar to the Bargmann index).¹⁷ This explains the origin of the symmetry $q\leftrightarrow -q-2$ discussed in Sect. 8.

Irreducible representations can be further characterised by diagonalization of one of the generator. We consider the operator $\mathscr {D}_K(q)=\frac{{\mathrm {d}}}{{\mathrm {d}}z}(1+z^2)+q\,z$, acting on functions $\phi (z)$ defined on the projective line. We have chosen here the measure $\rho =\rho _N$.

In a first step, it is more clear to map the projective line on the half circle, $z={{\mathrm {cotan}}}\theta$. Correspondingly, we introduce the transformed operator

$$\begin{aligned} \mathscr {B}_K(q) = \frac{{\mathrm {d}}z}{{\mathrm {d}}\theta } \, \mathscr {D}_K(q) \, \frac{{\mathrm {d}}\theta }{{\mathrm {d}}z} = -\frac{{\mathrm {d}}}{{\mathrm {d}}\theta } + q\, {{\mathrm {cotan}}}\theta \end{aligned}$$

(B.10)

acting on periodic functions $\psi (\theta )=\psi (\theta +\pi )$ (the projective space is the space of directions, hence the period $\pi$). Here, $\phi (z)$ and $\psi (\theta )$ play the same role as densities, related by $\phi (z)\,{\mathrm{d}}z=\psi (\theta )\,{\mathrm{d}}\theta$. It is now straightforward to get the eigenvector of $\mathscr {B}_K(q)$:

$$\begin{aligned} \psi _n^\mathrm {R}(\theta ) = \frac{1}{\sqrt{\pi }}\,{\mathrm {e}}^{2{\mathrm{i}}n\theta }\,(\sin \theta )^q \qquad \text{ for } \text{ eigenvalue } \qquad \lambda _n=-2{\mathrm{i}}n \,, \,\,\,\text{ with } n\in \mathbb {Z} . \end{aligned}$$

(B.11)

Similarly, it is instructive to derive the eigenvector of the adjoint operator $\mathscr {B}^\dagger _K(q) =\frac{{\mathrm {d}}}{{\mathrm {d}}\theta } + q^*\, {{\mathrm {cotan}}}\theta$ (we consider $q\in \mathbb {C}$ in this paragraph):

$$\begin{aligned} \psi _n^\mathrm {L}(\theta ) = \frac{1}{\sqrt{\pi }}\,{\mathrm {e}}^{2{\mathrm{i}}n\theta }\,(\sin \theta )^{-q^*} . \end{aligned}$$

(B.12)

The orthonormalisation reads $\int _0^\pi {\mathrm{d}}\theta \,\psi _n^\mathrm {L}(\theta )^*\psi _m^\mathrm {R}(\theta )=\delta _{n,m}$.

We now come back on the projective line. The right and left eigenvectors of $\mathscr {D}_K(q)$ are $\phi _n^\mathrm {R}(z)=\psi _n^\mathrm {R}(\theta )\frac{{\mathrm {d}}\theta }{{\mathrm {d}}z}$ and $\phi _n^\mathrm {L}(z)=\psi _n^\mathrm {L}(\theta )$:

$$\begin{aligned} \phi _n^\mathrm {R}(z) = \frac{1}{\sqrt{\pi }}\,\left( \frac{z+{\mathrm{i}}}{z-{\mathrm{i}}}\right) ^n\,(1+z^2)^{-1-q/2} \,\,\,\text{ and }\,\,\, \phi _n^\mathrm {L}(z) = \frac{1}{\sqrt{\pi }}\,\left( \frac{z+{\mathrm{i}}}{z-{\mathrm{i}}}\right) ^n\,(1+z^2)^{q^*/2} \end{aligned}$$

(B.13)

for eigenvalue $\lambda _n=-2{\mathrm{i}}n$. They satisfy the orthonormalisation condition

$$\begin{aligned} \int _\mathbb {R}{\mathrm{d}}z\,\phi _n^\mathrm {L}(z)^*\phi _m^\mathrm {R}(z)=\delta _{n,m} . \end{aligned}$$

(B.14)

The vectors $\phi _n^\mathrm {R}(z)$, which are labelled by the two numbers (q, n), play the same role as spherical harmonics for the group $\mathrm {SO}(3)$.

To complete the analysis we introduce the ladder operators

$$\begin{aligned} \mathscr {D}_\pm (q) = \mathscr {D}_{\widetilde{K}}(q) \pm \,{\mathrm{i}}\, \mathscr {D}_{A}(q) =\pm {\mathrm{i}}\left( \frac{{\mathrm {d}}}{{\mathrm {d}}z} (z\pm {\mathrm{i}})^2 + q \,(z\pm {\mathrm{i}})\right) . \end{aligned}$$

(B.15)

In summary we have

$$\begin{aligned} \mathscr {D}_K(q) \phi _n^\mathrm {R}(z)&= -2{\mathrm{i}}n\,\phi _n^\mathrm {R}(z) \end{aligned}$$

(B.16)

$$\begin{aligned} \mathscr {D}_\pm (q) \phi _n^\mathrm {R}(z)&= -2\left( n \pm \frac{q+2}{2} \right) \phi _{n\pm 1}^\mathrm {R}(z) . \end{aligned}$$

(B.17)

Starting from the vector $\phi _0^\mathrm {R}(z)$, the ladder operators allow to construct an infinite number of eigenvectors of $\mathscr {D}_K(q)$, hence the irreducible representation has an infinite dimension, unless $(q+2)/2=-N$ is a negative integer, leading to only $2N+1$ vectors, i.e. irreducible representation of finite dimension.

More can be found about group theoretical aspects of the problem in Ref. [36].

C Boundary Conditions for the Spectral Problems (3.33) and (6.17): Frisch–Lloyd Case (Matrices KN or $\widetilde{K}N$)

1.1 C.1 Behaviour of $\widehat{\varPhi }^\mathrm {R}_q(s)$ for $s\rightarrow 0$

Let us discuss the $s\rightarrow 0^+$ behaviour of the solution of Eq. (6.17) (the analysis for Eq. (3.33) is similar). We expect that the solution presents analytic terms in s and also non analytic terms. Assume that the first terms of the $s\rightarrow 0$ expansion are $\widehat{\varPhi }^\mathrm {R}_q(s)\simeq 1+c\,s + \omega \,s^{a+1}$. Injecting this expansion in (6.17) and assuming ${\mathcal {L}}(s)\sim s$ for $s\rightarrow 0$ (i.e. finite first moment of the weight $v_n$), we obtain $-{\mathrm{i}}\,\omega \,a(a+1)\,s^{a}+\mathcal {O}(s)=\varLambda (q)-{\mathrm{i}}\,q\big (c+\omega \,(a+1)s^{a}\big )+\mathcal {O}(s)$, thus $c=-{\mathrm{i}}\varLambda (q)/q$ and $a=q$. Hence we conclude that

$$\begin{aligned} \boxed { \widehat{\varPhi }^\mathrm {R}_q(s) \underset{s\rightarrow 0}{=} 1 - \frac{{\mathrm{i}}\,\varLambda (q)}{q}\, s + \omega _q\, |s|^{q+1} +\mathcal {O}(s^2) } \end{aligned}$$

(C.1)

(these are the first terms when $0\le q<1$). The crucial point is that $\omega _q$ is real as we now discuss. The $s\rightarrow 0$ behaviour of the Fourier transform selects the asymptotic behaviour (3.35): writing

$$\begin{aligned} \widehat{\varPhi }^\mathrm {R}_q(s)&= \int {\mathrm{d}}z\, \varPhi ^\mathrm {R}_q(z) - \int {\mathrm{d}}z\, \varPhi ^\mathrm {R}_q(z) \left( 1-{\mathrm {e}}^{-{\mathrm{i}}sz}\right) \end{aligned}$$

(C.2)

$$\begin{aligned}&\underset{s\rightarrow 0}{\simeq } \int {\mathrm{d}}z\, \varPhi ^\mathrm {R}_q(z) - 2\mathcal {A}_q\,\varGamma (-1-q)\,\sin \left( \frac{\pi q}{2}\right) \,|s|^{q+1} +\mathrm {regular\,terms} . \end{aligned}$$

(C.3)

The coefficient controlling the power law tail (3.35), is proportional to $\omega _q$ controlling the non analytic behaviour in (C.1): $\omega _q=-2\varGamma (-1-q)\,\sin \left( \frac{\pi q}{2}\right) \mathcal {A}_q$. Hence $\omega _q$ is real.

For $q\rightarrow 0$, we get $\omega _0=-\pi \,\mathcal {A}_0=-\pi \,{\mathcal {N}}$, where ${\mathcal {N}}$ is the integrated DoS of the disordered model. Thus (3.35) corresponds with the well-known expansion of the Fourier transform of the invariant density (see [58] and references therein) $\hat{f}(s) = \widehat{\varPhi }^\mathrm {R}_0(s) = 1 - \pi \,{\mathcal {N}}\, |s| - {\mathrm{i}}\,\gamma _1\,s + \mathcal {O}(s^2)$, as it should.

1.2 C.2 Expansion in Powers of q

Let us discuss the consequences for the $s\rightarrow 0$ behaviour of the functions in the expansion (5.2). The term $\mathcal {O}(q^n)$ of (C.1) has the form

$$\begin{aligned} \widehat{R}_n(s) \underset{s\rightarrow 0}{\simeq } |s|\sum _{m=0}^n \beta _{n,m}\,\ln ^m|s| +\mathrm {regular\,terms} . \end{aligned}$$

(C.4)

The coefficients $\beta _{n,m}$ are real since $\omega _q$ is real. In particular, since $\omega _0=-\pi {\mathcal {N}}$, we find $\beta _{n,n}=-\pi {\mathcal {N}}/n!$. Let us apply these considerations to $\widehat{R}_1(s)$ and $\widehat{R}_2(s)$. We find

$$\begin{aligned} \widehat{R}_1'' (s) \underset{s\rightarrow 0}{\simeq } \frac{\beta _{1,1}}{|s|} + 2\beta _{1,0}\,\delta (s) +\mathrm {regular\,terms} \end{aligned}$$

(C.5)

so that ${\mathrm{i}}s\widehat{R}_1'' (s)\simeq {\mathrm{i}}\beta _{1,1}{{\mathrm {sign}}}(s)$ for $s\rightarrow 0$; this shows that ${{\mathrm {Re}}}[{\mathrm{i}}s\widehat{R}_1'' (s)]\rightarrow 0$ in this limit.

We have also

$$\begin{aligned} \widehat{R}_2'' (s) \underset{s\rightarrow 0}{\simeq } 2\beta _{2,2}\frac{\ln |s|+1}{|s|} + \frac{\beta _{2,1}}{|s|} +2\beta _{2,0}\,\delta (s) +\mathrm {regular\,terms} \end{aligned}$$

(C.6)

As a result, ${\mathrm{i}}s\widehat{R}_2'' (s)$ is logarithmically divergent for $s\rightarrow 0$, however ${{\mathrm {Re}}}[{\mathrm{i}}s\widehat{R}_2'' (s)]\rightarrow 0$. These remarks were used in Sects. 5.1 and 6.

D The Distribution $\mathrm {Pf}(1/|x|)$

Consider a regular function $\psi (x)$, decaying at infinity. We define the distribution $\mathrm {Pf}(1/|x|)$ as

$$\begin{aligned} \int _{-\infty }^{+\infty } {\mathrm{d}}x\, \psi (x) \, \mathrm {Pf}\frac{1}{|x|} {\mathop {=}\limits ^{{\mathrm{def}}}}\lim _{\epsilon \rightarrow 0^+} \left[ \left( \int _{-\infty }^{-\epsilon } + \int _{+\epsilon }^{+\infty }\right) {\mathrm{d}}x\,\frac{\psi (x)}{|x|} +2\,\psi (0)\,\ln \epsilon \right] \,, \end{aligned}$$

(D.1)

in the same spirit as Hadamard’s regularization of the integral $\int {\mathrm{d}}x\,\psi (x)/x^2$. Equivalently, using $\Big (\int _{-1}^{-\epsilon }+\int _\epsilon ^1\Big ){\mathrm{d}}x/|x|=-2\ln \epsilon$, we can avoid the regulator and define the distribution as

$$\begin{aligned} \int _{-\infty }^{+\infty } {\mathrm{d}}x\, \psi (x) \, \mathrm {Pf}\frac{1}{|x|} {\mathop {=}\limits ^{{\mathrm{def}}}}\left( \int _{-\infty }^{-1} + \int _{+1}^{+\infty }\right) {\mathrm{d}}x\,\frac{\psi (x)}{|x|} + \int _{-1}^{+1}{\mathrm{d}}x\,\frac{\psi (x)-\psi (0)}{|x|} . \end{aligned}$$

(D.2)

We can check two useful properties:

$$\begin{aligned} \big [ \mathrm {sign}(x)\,\ln |x| \big ] ' = \mathrm {Pf}\frac{1}{|x|} \qquad \text{ and }\qquad \big [ |x|\,(\ln |x|-1) \big ] '' = \mathrm {Pf}\frac{1}{|x|} \end{aligned}$$

(D.3)

where derivation is understood here in the distributional sense.

Application: The distribution can be used in order to write the solution of the equation

$$\begin{aligned} \left( -\frac{{\mathrm {d}}^2}{{\mathrm {d}}x^2} + k^2 \right) \varphi (x) = \frac{1}{|x|} \,, \end{aligned}$$

(D.4)

which is a simplified version of Eq. (5.16). The solution is expected to be continuous at $x=0$, but non differentiable as $\varphi (x) \simeq \varphi (0) -|x|\,\big (\ln |x|-1\big )$ for $x\rightarrow 0$. The finite part allows to write the solution under the integral form

$$\begin{aligned} \varphi (x) = \int _{-\infty }^{+\infty } {\mathrm{d}}y\, G(x-y) \, \mathrm {Pf}\frac{1}{|y|} \qquad \text{ for } x\ne 0 \,, \end{aligned}$$

(D.5)

where $G(x)=(2k)^{-1}{\mathrm {e}}^{-k|x|}$ is the Green’s function. The integral is well defined thanks to the finite part.

E Continuum Limit of the Frisch–Lloyd Model: The Halperin Model

The disordered model (1.7) for a Gaussian white noise potential $\left\langle V(x)V(x') \right\rangle =\sigma \,\delta (x-x')$ corresponds to the so-called Halperin model [62, 69]. It can be recovered from the impurity models in two different manners. As discussed in Sect. 7.2, the continuum limit for fixed angles (regular lattice of impurities) leads to the Halperin model. The high impurity density limit of the Frisch–Lloyd model, $\rho \rightarrow \infty$, with $v_n\rightarrow 0$, also leads to the Halperin model (this limit was already studied by Frisch and Lloyd in [48]). The Halperin model has been extensively studied in the literature. In particular the fluctuations have been analysed in [90, 92] and the generalized Lyapunov exponent in [52]. Hence it is a good case to benchmark the method of the present article. In this appendix, we establish the correspondence with the equations studied in these papers. Considering the limit $\rho \rightarrow \infty$ and $v_n\rightarrow 0$ in Eqs. (3.33,6.15), keeping $\rho \left\langle v_n \right\rangle =0$ and $\rho \left\langle v_n^2 \right\rangle =\sigma$ fixed, leads to

$$\begin{aligned} \left[ \mathscr {G}^\dagger + q\, z \right] \varPhi ^\mathrm {R}_q(z) = \varLambda (q)\, \varPhi ^\mathrm {R}_q(z) \qquad \text{ where } \mathscr {G}^\dagger = \frac{\sigma }{2}\frac{{\mathrm {d}}^2}{{\mathrm {d}}z^2} + \frac{{\mathrm {d}}}{{\mathrm {d}}z}( E + z^2 ) \end{aligned}$$

(E.1)

is the (forward) generator of the diffusion for the Riccati variable $z(x)=\psi '(x)/\psi (x)$ involved in the localisation problem [35, 48, 62, 70]. Eq. (E.1) is precisely the equation given in Refs. [52, 90].

It is interesting to discuss the origin of (6.21) without using Fourier transform. A convenient starting point is the equation in real space for the $q^n$ order contribution to $\varPhi ^\mathrm {R}_q(z)$:

$$\begin{aligned} \mathscr {G}^\dagger R_n(z)=(\gamma _1-z)R_{n-1}(z) + \sum _{m=2}^n\frac{\gamma _m}{m!}R_{n-m}(z) . \end{aligned}$$

(E.2)

In order to get $\gamma _1$, one can integrate the perturbation equation at order $q^1$:

$$\begin{aligned} \lim _{X\rightarrow +\infty }\int _{-X}^{+X}{\mathrm{d}}z\, \left[ \mathscr {G}^\dagger R_1(z)+(z-\gamma _1)R_0(z) \right] = 0 \end{aligned}$$

(E.3)

thus, using the normalisation $\int R_0=\int f=1$,

(E.4)

where the principal part is important as $\mathscr {G}^\dagger R_1(z)\simeq z \, R_0(z) \simeq {\mathcal {N}}/z$ at infinity. This corresponds to (6.19). Similarly, starting from the equation at order $q^2$, $\mathscr {G}^\dagger R_2(z)=(\gamma _1-z)R_1(z)+(1/2)\gamma _2f(z)$, we integrate and get (note that $R_2(z) \sim \ln ^2|z|/z^2$ for $z\rightarrow \infty$, cf. Appendix C, hence the importance of the principal part). This leads to

$$\begin{aligned} \gamma _2 = \lim _{s\rightarrow 0}{\mathrm{i}}\, \left[ \widehat{R}_1'(s)+\widehat{R}_1'(-s)\right] -2\, \gamma _1\,\widehat{R}_1(0) \end{aligned}$$

(E.5)

or more conveniently (6.21). The final result is given in the text, Eq. (6.23) where ${\hat{f}}(s)=\varphi (s)/\varphi (0)$ is given by (5.7).

The adjoint spectral problem.— In the paper, we have considered the spectral problem (3.24) for $\varPhi ^\mathrm {R}_q(z)$. We can equivalently work with the adjoint problem, which here takes the form

$$\begin{aligned} \left[ \mathscr {G} + q\, z \right] \varPhi ^\mathrm {L}_q(z) = \varLambda (q)\, \varPhi ^\mathrm {L}_q(z) \qquad \text{ where } \mathscr {G} = \frac{\sigma }{2}\frac{{\mathrm {d}}^2}{{\mathrm {d}}z^2} -(E+z^2) \frac{{\mathrm {d}}}{{\mathrm {d}}z} . \end{aligned}$$

(E.6)

We expand the vector as $\varPhi ^\mathrm {L}_q(z)=\sum _{n=0}^\infty q^n\,L_n(z)$ with $L_0(z)=1$. In particular, at order $q^1$ we get

$$\begin{aligned} \mathscr {G} L_1(z)=\gamma _1-z . \end{aligned}$$

(E.7)

Multiplication by f(z) and integration gives (using ). Similarly, imposing the condition $\int {\mathrm{d}}z\,L_1(z)\,f(z)=0$, we get . This is analogous to the general discussion for functionals of stochastic processes provided in Sect. 1.

F Phase Formalism for the Frisch–Lloyd Model

The Frisch–Lloyd model for impurities at random positions can be conveniently studied with the phase formalism of Refs. [7, 69]. This allows to derive some simple asymptotic behaviours. Furthermore, we can obtain approximate formulae beyond the perturbative regime.

1.1 F.1 Phase Formalism

The starting point is to parametrize the wave function as $\psi (x) = {\mathrm {e}}^{\xi (x)} \, \sin \theta (x)$ and $\psi '(x) = k{\mathrm {e}}^{\xi (x)} \, \cos \theta (x)$ for $E=+k^2$. The two variables obey the differential equations $\theta '(x)=k-\big [V(x)/k\big ]\,\sin ^2\theta$ and $\xi '(x)=\big [V(x)/(2k)\big ]\,\sin 2\theta$. The Lyapunov exponent and the variance can be obtained by studying the drift and the diffusion constant of the process $\xi (x)$, corresponding roughly to $\ln |\psi (x)|$ (cf. remark in [32, p. 442]). This process is constant between two impurities, and make a jump of [17, 99]

$$\begin{aligned} \varDelta \xi _n =\xi (x_n^+) - \xi (x_n^-) = \frac{1}{2}\ln \left( 1 + \frac{v_n}{k}\sin 2\theta _n^- + \frac{v_n^2}{k^2}\sin ^2\theta _n^- \right) \end{aligned}$$

(F.1)

through the impurity n. In the $k\gg \rho$ limit, we can assume that the phase $\theta _n^-=\theta (x_n^-)=k\ell _n+\theta (x_n^+)$ modulo $\pi$ is uniformly distributed over $[0,\pi ]$. As a consequence, the increments (F.1) are independent and one can easily compute their moments. Using the independence assumption, we conclude that $\xi (x)$ is a compound Poisson process

$$\begin{aligned} \xi (x) = \sum _{n=1}^{\mathscr {N}(x)} \varDelta \xi _n \,, \end{aligned}$$

(F.2)

where $\mathscr {N}(x)$ is a Poisson process of intensity $\rho$. The drift and the diffusion constant are

$$\begin{aligned} \gamma _1 = \lim _{x\rightarrow \infty }\frac{\left\langle \xi (x) \right\rangle }{x} \simeq \rho \, \left\langle \varDelta \xi _n \right\rangle \qquad \text{ and }\qquad \gamma _2= \lim _{x\rightarrow \infty }\frac{\mathrm {Var}(\xi (x))}{x} \simeq \rho \, \left\langle \varDelta \xi _n^2 \right\rangle \end{aligned}$$

(F.3)

(note that the diffusion constant involves the second moment of the increment due to the fact that the number of impurities fluctuates in a fixed interval [0, x]; if the number of impurities would be constant in the interval, $\gamma _2$ would involve the variance of $\varDelta \xi _n$).

1.2 F.2 Universal (Perturbative) Regime

We fist consider the high energy regime $\rho ,\,v_n\ll k=\sqrt{E}$. In this case we can expand the logarithm in (F.1), and eventually average over the phase. We see that the two moments are equal:

$$\begin{aligned} \gamma _1 \simeq \gamma _2 \simeq \frac{\rho \,\left\langle v_n^2 \right\rangle }{8E} \qquad \text{ for } E\rightarrow +\infty \,, \end{aligned}$$

(F.4)

which is the single parameter scaling property (as noticed in [99], this is only true for the leading order term; see also [94]).

1.3 F.3 Intermediate Regime for Weak Density

For weak density, in the intermediate (non perturbative) regime $\rho \ll k\ll v_n$, the phase increment between impurities is $k\ell _n\gg 1$, hence we can use that the phase $\theta _n^-$ is uniformly distributed (phase randomization). We can also simplify the logarithm as $\varDelta \xi _n \simeq \ln \left| \frac{v_n}{k}\sin \theta _n^-\right|$. Using that $\int _0^\pi \frac{{\mathrm{d}}\theta }{\pi }\,\ln \sin \theta =-\ln 2$ and $\int _0^\pi \frac{{\mathrm{d}}\theta }{\pi }\,\ln ^2\sin \theta =\frac{\pi ^2}{12}+\ln ^22$ we deduce

$$\begin{aligned} \gamma _1 \simeq \rho \, \left\langle \ln \frac{v_n}{2k} \right\rangle \qquad \text{ and } \qquad \gamma _2 \simeq \rho \, \frac{\pi ^2}{12} + \rho \, \left\langle \ln ^2\frac{v_n}{2k} \right\rangle \,, \end{aligned}$$

(F.5)

thus we expect $\gamma _2\gg \gamma _1$. For an exponential distribution of weights $\mathrm {Proba}\{v_n>x\}={\mathrm {e}}^{-x/v}$ we use $\int _0^\infty {\mathrm{d}}x\,\ln x\,{\mathrm {e}}^{-x}=-\mathrm {C}$ and $\int _0^\infty {\mathrm{d}}x\,\ln ^2 x\,{\mathrm {e}}^{-x}=\frac{\pi ^2}{6}+\mathrm {C}^2$, where $\mathrm {C}\simeq 0.577$ is the Euler-Mascheroni constant. Eventually we deduce (6.27).

1.4 F.4 Large Negative Energy

For $E=-k^2$, the process $\xi (x)$ is not constant between the impurities, but grows as

$$\begin{aligned} \xi (x_{n+1}^-) - \xi (x_n^+) = k\ell _n + \frac{1}{2}\ln \left( \frac{\tan ^2(\theta _n^++\pi /4)+{\mathrm {e}}^{-4k\ell _n}}{\tan ^2(\theta _n^++\pi /4)+1}\right) \end{aligned}$$

(F.6)

where $\theta _n^+=\theta (x_n^+)$. One must add this contribution to (F.1). Using that the phase is locked $\theta _n^-\rightarrow \pi /4$ for $E\rightarrow -\infty$, we can expand the total increment $\varDelta \xi _n =\xi (x_{n+1}^-) - \xi (x_n^-)$ as

$$\begin{aligned} \varDelta \xi _n = k\ell _n + \frac{v_n}{2k} + \mathcal {O}\left( \frac{v_n^2}{k^2} \right) + \mathcal {O}\left( {\mathrm {e}}^{-4k\ell _n} \right) \end{aligned}$$

(F.7)

As a result

$$\begin{aligned} \gamma _1 \simeq k + \frac{\rho \left\langle v_n \right\rangle }{2k} \simeq \sqrt{-E+\rho \left\langle v_n \right\rangle } \qquad \text{ and } \qquad \gamma _2 \simeq \frac{\rho \left\langle v_n^2 \right\rangle }{4k^2} \end{aligned}$$

(F.8)

Interestingly, one has recovered the same property as for the Halperin model [90]

$$\begin{aligned} \gamma _2 \big |_{E=-k^2} \simeq 2\,\gamma _2 \big |_{E=+k^2} \qquad \text{ for } k\rightarrow \infty . \end{aligned}$$

(F.9)