SciPost Phys. 15, 211 (2023)

Traveling/non-traveling phase transition and non-ergodic
properties in the random transverse-field
Ising model on the Cayley tree
Ankita Chakrabarti1,2,3⋆ , Cyril Martins4 , Nicolas Laflorencie3 ,Bertrand Georgeot3 , Éric Brunet5 and Gabriel Lemarié2,3,6†
1 Department of Physics, National University of Singapore,
2 Science Drive 3, Singapore 117542, Singapore
2 MajuLab, International Joint Research Unit IRL 3654, CNRS, Université Côte d’Azur,
Sorbonne Université, National University of Singapore,
Nanyang Technological University, Singapore
3 Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France
4 Laboratoire de Chimie et Physique Quantiques, Université Paul Sabatier Toulouse III,
CNRS, 118 Route de Narbonne, 31062 Toulouse, France
5 Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS,
Sorbonne Université, Université Paris Cité, F-75005 Paris, France
6 Centre for Quantum Technologies, National University of Singapore,
117543 Singapore, Singapore⋆ chakrabarti@irsamc.ups-tlse.fr ,† lemarie@irsamc.ups-tlse.fr

Abstract
We study the random transverse field Ising model on a finite Cayley tree. This enables
us to probe key questions arising in other important disordered quantum systems, inparticular the Anderson transition and the problem of dirty bosons on the Cayley tree,or the emergence of non-ergodic properties in such systems. We numerically investigate this problem building on the cavity mean-field method complemented by state-ofthe art finite-size scaling analysis. Our numerics agree very well with analytical results
based on an analogy with the traveling wave problem of a branching random walk inthe presence of an absorbing wall. Critical properties and finite-size corrections for thezero-temperature paramagnetic-ferromagnetic transition are studied both for constant(independent of the system volume) and algebraically vanishing (scaling as an inverse
power law with the system volume) boundary conditions. In the later case, we reveal a
regime which is reminiscent of the non-ergodic delocalized phase observed in other systems, thus shedding some light on critical issues in the context of disordered quantum
systems, such as Anderson transitions, the many-body localization or disordered bosons
in infinite dimensions.
Copyright A. Chakrabarti et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

1

Received 22-02-2023
Accepted 09-10-2023
Check forupdatesPublished 28-11-2023
doi:10.21468/SciPostPhys.15.5.211

SciPost Phys. 15, 211 (2023)

Contents
1 Introduction

22 Main objectives

5

3 Method

54 Traveling/non-traveling wave phase transition
4.1 Traveling wave predictions for the disordered phase
4.2 Numerical evidence for the traveling wave regime
4.3 Stationary distribution for the ordered phase
4.4 Critical properties
4.5 Numerical characterization of the critical behavior
4.6 Finite size scaling of the critical properties

7
8
9
10
11
12
12

5 Non-ergodic phase

14

6 Strong spatial inhomogeneities

17

7 Conclusion

19

A Analytical description of the traveling–non-traveling phase transition
A.1 A Fisher-KPP equation
A.2 Properties of travelling wave solutions
A.3 Properties of the front
A.4 Prediction for the disordered phase
A.5 Slow travelling waves
A.6 Prediction for the ordered phase
A.7 At criticality

21
21
22
23
24
24
25
26

References

27

1

Introduction

Characterization of non-ergodic properties has been of utmost importance for studying disorder induced phases of quantum matter ever since Anderson’s discovery of the localization of
non-interacting electrons in an imperfect crystal [1,2]. It has also emerged as an important issue more recently in the study of isolated quantum many-body systems where both interaction
and disorder are present and lead to a new type of phase of matter, the many-body localization(MBL) [3–12]. The localization of all the many-body eigenstates of a system in presence ofstrong disorder prevents thermalization and has striking consequences on out-of-equilibrium
properties [4, 13–19].
In this context, there has been a renewed interest on the Anderson transition on random graphs [10, 11, 20–29, 29–46]. One of the major motivations is an approximate mapping between the MBL problem formulated in the Fock space [47–50] and Anderson localization on random graphs [4, 6, 10–12, 51–53]. A topic of much debate has been the existenceof a non-ergodic delocalized phase, where states, while not localized as in the strong disordered regimes, only occupy an algebraically small fraction of the system [24–37, 40, 41, 54].
This phase is characterized by multifractal properties. Following its initial discovery, a sub2

SciPost Phys. 15, 211 (2023)

stantial body of work has been dedicated to this phase, spanning from mathematical proofs
and random matrix models to investigations of Floquet-driven or quasi-periodic systems, seee.g. [55–64]. Note that in finite dimensional system, there is no non-ergodic delocalized phase,and multifractal properties only occur at the Anderson transition [65].
While not random by construction, tree-like structures are likewise good testbed for exploring these ideas. Indeed, classical and quantum systems on the Cayley tree (see Fig. 1)
are known to demonstrate strong spatial inhomogeneities in the presence of disorder. Eachsites on this tree has K + 1 neighbors apart from the leaves at the boundary which have 1
neighbor. The paradigmatic directed polymer model [66] studied in terms of a traveling wave
problem [67–69], is known to show a glass transition. In the glassy phase, only a few paths are
explored by the polymer from the exponentially many of the tree. For the quantum problem ofAnderson localization on the Cayley tree, it was shown that there exists a non-ergodic delocalized phase [27–29, 32]. A noteworthy point is that boundary conditions play a crucial role on
the Cayley tree: they must depend on the system size in a certain way for such a non-ergodic
phase to exist [27–29,32]. In many quantum disordered systems an analogy with the directed
polymer problem can be drawn [11, 20, 23, 27, 33, 45, 70–74] facilitating the understandingof universal features of the glassy physics [75]. Consequently, such systems with a tree-likestructure are particularly illustrative for studying non-ergodic properties of glassy physics.

Figure 1: Schematic representation of the Cayley tree. The transverse random field
Ising model Eq. (1) lives on a centered Cayley tree of finite depth n and branchingnumber K (on the figure, n = 4 and K = 2). We label generations by the letter d,with d = 0 being the boundary and d = n being the site at the center of the tree. The
different colors of the nodes for d > 0 correspond to a given random configuration of
the transverse fields εi in the Hamiltonian Eq. (1). The key observable of this workis the cavity mean field Bd at any site within generation d. We study the response
B = Bn−1 closest to the center of the tree as a function of a boundary condition B0 .
In this paper we study the effects of disorder on a quantum system: the random-field Isingmodel on the Cayley tree. This model has been widely used to study quantum phase transitions at zero temperature [76] in the presence of frozen disorder. Important methods havebeen devised to describe its properties, including the strong-disorder renormalization procedure [77, 78] and, in one dimension, the mapping through the Jordan-Wigner transformation
to a free fermion problem [79–81]. The first renormalization approach predicts a transition,
between an ordered and a paramagnetic phase, of infinite disorder type, where quenchedrandomness increases as the system undergoes coarse graining [77, 78, 82–84]. The second
approach allows for exact numerical simulations of large 1D systems and maps them to theAnderson localization problem [80,81]. At first glance, it might seem that the absence of loops
on the Cayley tree could enable such an approach. However, the Jordan-Wigner transformation there introduces non-trivial and non-local phase terms, which hinder the mapping to asimple free fermion problem [85,86].We will use another approach well suited for tree-graphs:
3

SciPost Phys. 15, 211 (2023)

the cavity mean-field (CMF) description of this problem introduced by Feigel’man, Ioffe andMézard [70], see also [71, 87–89].
An initial motivation to consider this problem was to understand the large spatial inhomogeneities emerging at the superconductor-insulator transition in certain strongly disorderedmaterials where the transition is driven by the localization of Cooper pairs [70,71,89,90]. Suchphysics can be modelled by disordered hard-core bosons or equivalently X Y spin-1/2 systems
in the presence of a random magnetic field [91–93]. It was argued in Refs. [70, 71, 89] thatsome features of this transition can be captured by the random transverse-field Ising model on
the Cayley tree, as studied using the CMF approach [70, 71].
In Refs. [70, 71], this problem was mapped to that of directed polymers on the Cayley tree
which brought to the fore interesting glassy and Griffiths properties at strong disorder andeven in the superconducting phase close to the transition. This provides qualitative explanation for the large spatial fluctuations of the local order parameter observed experimentally indisordered superconducting films close to the transition [89, 90].Moreover, hard-core bosons
in a random potential have recently been studied on the Cayley tree [94] where the observed
Bose-glass phase [95, 96] was found to share some common features with a delocalized nonergodic regime.
In this paper, we first aim at investigating in detail the zero-temperature paramagnetic ferromagnetic quantum phase transition as disorder decreases for the above problem on the
Cayley tree. As we will demonstrate, in the cavity mean-field approach, one can draw an analogy between this transition and the traveling/non-traveling phase transition for a branchingrandom walk in the presence of an absorbing wall, as described in [97]. Such an analogy
was also suggested for the Anderson transition on the Cayley tree [23, 28, 33]. In fact, recursive equations for the cavity mean fields closely resemble that of Abou-Chacra, Thoulessand Anderson for the Anderson localization problem [20], but are simpler in the sense thatthey involve only real quantities. Interestingly, the analogy between cavity mean field and thetraveling/non-traveling phase transition [97] allows us to describe analytically and to verify
numerically a range of critical properties for the Ising transition. Some of these properties are
identical to those known for the Anderson transition, while others, including critical behavior,have not yet been explored in the context of the Anderson transition.
Furthermore, we will show that the strong similarity between the Anderson transition andthat for the random-field Ising model extends to non-ergodic properties, particularly in termsof their crucial dependence on boundary conditions. We primarily focus on the case of a finite
Cayley tree, which, in the Anderson model, allows for a clear characterization of a non-ergodic
delocalized phase [27–29, 32]. When considering boundary conditions in the Ising problem
analogous to those in the Anderson localization problem that are suitable for non-ergodic
properties, we identify signs of a “non-ergodic ordered phase”, that we will define precisely inthe following.
The rest of the paper is organized as follows. In Sec. 2 we recall the context of the study
and its main objectives. In Sec. 3 we present the model and the CMF method. In Sec. 4 we
draw an analogy with a traveling wave problem with a wall considered in [69, 97] in order toreach a better understanding of the critical properties and the conditions of existence of theequivalent of a non-ergodic delocalized phase in this context, that we will call non-ergodicordered phase in the following. We first recall the analogy in the disordered regime between
the cavity mean field equation with the directed polymer problem [70]. In turn, this problem
maps onto a traveling wave problem. Close to the transition and in the ordered phase, thenonlinearity of the recursion cannot be neglected and the problem maps instead to a travelingwave problem with a wall, which was considered in [69,97]. Numerical results corresponding
to a uniform boundary field are presented. Behaviors in the traveling and stationary regimes
are compared with the analytical predictions made. Then a finite-size scaling study of these4

SciPost Phys. 15, 211 (2023)

results is performed and compared with the critical behavior predicted theoretically. In Sec. 5
we show that this system exhibits a non-ergodic ordered phase for an appropriate choice ofthe boundary conditions. This phase is studied in the light of the analogy with the travelingwave problem. The strongly inhomogeneous spatial distributions of cavity fields observed indifferent phases is discussed in Sec. 6. We summarize our results in Sec. 7.

2Main objectives

In this paper, we study the ordered/disordered phase transition in a spin system, the randomtransverse-field Ising model on the Cayley tree, addressing two important questions whichwere recently debated in the delocalized/localized Anderson transition: (i) What are the critical properties of the transition, in particular how to describe the finite-size scaling properties
in the neighborhood of the transition? (ii) Is there a phase in such a spin system which isanalogous to the non-ergodic delocalized phase of the Anderson problem, and what are theconditions for its existence?
In the first question, we consider the system on a Cayley tree with fixed boundary conditions. In this setting, there is no non-ergodic ordered phase (analogous to the non-ergodic
delocalized phase in Anderson), and we wish to study the critical properties of the direct disordered to ergodic ordered phase transition, akin to the direct localized to ergodic delocalized
AL transition. The first question is important because of a recent debate on the value of critical exponents of the Anderson localization transition [29, 29, 32, 33, 36, 41, 98]. In the CMF
problem we consider, we will draw an analogy with a traveling/non-traveling phase transition
whose critical properties (in particular the critical exponents) are analytically known. This
will allow us to perform a finite-size scaling of this problem which could in turn be useful toanalyse the Anderson transition on the Cayley tree.
The second question of whether a non-ergodic ordered phase exists for the quantum Isingmodel that we consider, similar to the non-ergodic delocalized phase of the Anderson localization problem, is also crucial. Indeed, the recent study of disordered bosons on the Cayley
tree [94] points towards the existence of a Bose-glass phase with properties very similar to
those of a delocalized non-ergodic phase, and also showing some multifractality. In the recursive approach for the Green’s function of the Anderson localization problem on tree graphs, it
was understood that the condition for observing such a non-ergodic delocalized phase is thatthe boundary conditions must tend to zero as the inverse of the number of sites [27,29,32,99].
Indeed, in the Cayley tree, the boundary hosts a finite fraction of the total number of sites, andtherefore plays a preponderant role. In this paper, we will show that by considering boundary
conditions vanishing as a power-law ∼ 1/N φ , where N is the total number of sites, we canalso observe the equivalent of a non-ergodic delocalized phase, and an ergodicity breaking
transition.

3

MethodWe study the random transverse-field Ising model described by the following HamiltonianX
gX x xH =−ε(i) σzi −σi σ j ,
(1)
K
i
〈i j〉

where σz , σ x are the Pauli matrices. The lattice considered is the Cayley tree with a branchingnumber K. The summation in the second term couples the spin at each site i to that of theK + 1 nearest neighbours. The transverse fields ε(i) at each site i are independent random
5

SciPost Phys. 15, 211 (2023)

variables uniform in [−1, 1]; in other words, they are drawn from a box distribution,P(ε) =1
θ (1 − |ε|) .
2(2)

We aim at analyzing the properties of the paramagnetic-ferromagnetic transition at zerotemperature, obtained by decreasing the relative strength of the disorder. This is achieved byvarying the coupling g from small to large values. The phase diagram of this transition for
the infinite, boundaryless, Cayley tree (also called the Bethe lattice), has been investigatedusing a CMF approach [70, 71] where it was found that the disordered phase at sufficiently
low temperature has characteristic glassy properties, in particular replica symmetry breaking
and large distributions of the local order parameter, which persist in the ordered phase close
to the transition.
In the CMF approach, we consider a spin at site j from which the neighbor closer to theroot has been removed (cavity). It is described by a local Hamiltonian:g
H CMF= ε( j) σzj −jKX〈σlx 〉σ xj ,

K l=1

(3)

where 〈σlx 〉 is the magnetization at the site l obtained (recursively) from the CMF approach
itself: to compute 〈σlx 〉, we exclude the site j and only consider the sites further away fromthe center. In a finite Cayley tree of depth n, it is useful to introduce the distance d from theboundary: it increases from 0 to n as we move inwards from boundary to root, see Fig. 1.
Assume site j is at level d + 1; then all the sites l in Eq. (3) are at level d. We introduce thecavity mean field for j as
K
gX x
( j)
Bd =
〈σ 〉 ,(4)
K l=1 lwhere the index d recalls that this quantity is an average of magnetizations of some sites at2( j)( j) 2 1/2
level d. At zero temperature, solving Eq. (3), one finds 〈σ xj 〉 = Bd / ε( j) + Bd
. Then, the′cavity mean field of a site j at level d + 2 is given by
K
Bd(l)
gX
,
Bd+1 =
Ç2
K l=1
ε(l) 2 + Bd(l)(5)

where we have dropped the ( j ′ ) superscript of the left hand side to shorten notation. The
quantities Bd(l) are random variables with a distribution depending only on the index d. For
a given site l at level d + 1, the quantities Bd(l) and ε(l) are independent. For a given d, thequantities Bd(l) and Bd(m) corresponding to two different sites l and m are independent.
In this paper, we will mainly consider the response of the system at the root of the tree
to a uniform and non-random magnetization 〈σlx 〉 = B0 /g for all the sites l at the boundary
d = 0: this corresponds to having cavity field B0 for all the sites at level d = 1. We use Eq. (5)( j)for each of the preceding generation up to the root to obtain the field Bd for all j and d. We
analyze the critical properties by studying the distribution of the cavity field at the root, ford = n − 1. (Remark: at the root, the cavity mean field is still defined by Eq. (4), with the sum
running over an arbitrary choice of K out of K + 1 sites at level n − 1.) For simplicity, we write
henceforth(root).
(6)
B = Bd=n−1
The distribution of the order parameter B depends on the coupling g, the depth of the tree n
and the boundary field B0 . We will consider a branching number K = 2. The notations areillustrated in Fig. 1.
6

SciPost Phys. 15, 211 (2023)

When the depth n of the Cayley tree goes to infinity, we expect the distribution of Bd toconverge, as d → ∞, to some stationary distribution P(B). As the random variables in theright hand side of the recursion relation Eq. (5) are all independent from each other, we seethat Eq. (5) determines a self-consistent equation for the distribution P(B). For numericalstudies, approximate methods like the belief propagation method [23, 32] are used to findsolutions for the distribution P(B). This approach is however accurate only in the bulk of alarge Cayley tree where the boundary can be ignored, as previously considered in [70, 71] forthe random transverse field Ising model.
We consider in this paper the case of a finite Cayley tree of depth n for two main reasons.
First because we want to investigate the finite-size scaling properties of this transition to extract
its critical properties. Finite-size scaling in the belief propagation method approach is notclearly understood (see however [23, 33]). Critical exponents are then determined from theasymptotic behavior (by definition far from the critical regime) of the considered observable,see [36]. Although solving exactly CMF recursive equation Eq. (5) for a tree of finite size
restricts the depth of the tree quite strongly (up to n = 29 in our case), we will see that
finite-size scaling allows to determine precisely the critical exponents of the transition. The
second reason is that the finite Cayley tree case shows strong and interesting boundary effects.
Indeed, the sites at the boundary have a connectivity 1 instead of K + 1, which makes them
qualitatively different. Moreover, the number of sites on the boundary is a significant fraction
1 − K −1 for large trees of the total number of sites N = (K + 1)(K n − 1)/(K − 1) + 1. Similarly
to the Anderson transition on the Cayley tree, we will see that boundary conditions control
the properties of the ordered phase, and that a non-ergodic regime can appear in some cases.
The CMF approach is in fact similar to that used to describe Anderson localization in the
Cayley tree [20,21,23,25,26,29,32,36,38,41,72], where we consider the local cavity Green’s( j)function Gii of the problem where the site j, neighbor of i closer to the root, has been removed.
It can be rigorously shown for the Cayley tree that this observable satisfies a recursion equation
similar to Eq. (5). The crucial quantity is the imaginary part of the local Green’s function ℑG
at the root of the tree in response to a boundary condition η (the imaginary part of the cavity
Green’s function at the boundary sites). If ℑG reaches, in the limit of large system sizes,a finite value (i.e. almost surely non-zero), then the system is in a delocalized regime. It
was understood recently [25–30, 32] that the non-ergodic delocalized phase of the Andersontransition can only be seen by considering a boundary condition η ∼ N −φ , which vanishes asa power law of the system volume with φ a constant exponent. In that latter case, the typicalvalue of ℑG ∼ N D−1 varies as a power-law of the system volume with a disorder-dependent
exponent 0 < D < 1. D can be interpreted as a fractal dimension, and D = 1 in the ergodic
regime while it reaches 0 at the localization transition.
We wish here to revisit the problem of the ferromagnetic-paramagnetic transition in the
random transverse-field Ising model Eq. (1) in the light of this discovery. In our problem, B
corresponds to ℑG and the boundary condition B0 to η. We will therefore see that here too we
can have the equivalent of a non-ergodic delocalized phase if the boundary condition satisfies
B0 ∼ N −φ .

4Traveling/non-traveling wave phase transition

In this section, we discuss the analogy with the traveling/non-traveling wave phase transitionof Derrida and Simon [97]. We consider the case where the value of the boundary field B0
is independent of the size N of the system. (We will investigate the case of B0 ∼ N −φ coresponding to the non-ergodic delocalized phase in the Anderson transition on the Cayley tree
in Sec. 5.)

7

SciPost Phys. 15, 211 (2023)

We argue that, when B0 is independent of N , the CMF recursion Eq. (5) describes a transition between a disordered phase for g < g c where the field B = Bn−1 goes to zero as N → ∞,and an ergodic ordered phase for g > g c , where the field converges to some fixed distribution
as N → ∞.
We will first describe analytically the critical properties of this transition. Then, we willverify these predictions by analysing the numerical results obtained for a uniform boundary
field, B0 = g. Note that this choice corresponds to the upper bound of the cavity field B ≤ g.
We have considered, unless stated, Ns = 104 independent realizations of the disorder for eachvalue of n and g.

4.1

Traveling wave predictions for the disordered phase

In the disordered phase g < g c , because B vanishes, we can linearize the recursion relation
Eq. (5), which then gives:(l)K
g X BdBd+1 =.
(7)
K l=1 |ε(l) |This recursion is similar to that of a directed polymer problem on a Cayley tree [66], a paradigmatic model of pinned elastic manifolds with characteristic glassy properties, in particular 1step replica symmetry breaking. More precisely, Eq. (7) is equivalent to the recursion relation
found for the partition function of a directed polymer:
Zd+1 =KX(l)e−β U Zd(l) ,

(8)

l=1

where the partition function Z plays the role of B, the inverse temperature β is arbitrarily set
to 1 and the on-site disorder is U (l) ≡ ln(Kε(l) /g). In turn, the problem of directed polymer on
the Cayley tree can be mapped to a traveling wave problem [66].
This analogy allows us to deduce a number of analytical results, which are well known
in the context of the traveling wave problem but new for the problem considered here. We
explain these results in more details and give some references in Appendix A. When Eq. (7)holds, then3
ln B = −V (g)n −ln n + ln B0 + ln bn ,(9)2γ
where bn is a random variable of order 1 and where γ and V (g) are given below. Furthermore,
the random variable bn converges (in distribution) as n → ∞ to some random variable b
bn −−−→ b ,

(10)

n→∞

and the density of probability of b has a fat tail for large b:
P(b) ≃ (C1 log b + C2 )b−(1+γ) ,for large b ,(11)for some constants C1 and C2 . Finally, the velocity V (g) can be written as
V (g) = ln

gc
,
g

(12)

for some critical value g c . The values of g c , γ and the full distribution of b depend on the
choice of K and on the distribution of the noise ε. When K = 2 and for ε uniform in [−1, 1],
as in Eq. (2), one finds (see Eq. (A.12) and Eq. (A.13) with γ ≡ λc and V (g) ≡ −vc , and [70])γ ≃ 0.626635 .

g c ≃ 0.137353 ,8(13)

SciPost Phys. 15, 211 (2023)

Notice in particular that the exponent γ in the disordered phase is independent of the valueof g.
Fat tails as given by Eq. (11) are a typical feature of a non-ergodic phase [23, 66, 70, 100].
The above value (γ < 1) corresponds to the replica symmetry broken glassy phase of theauxillary directed polymer problem [66, 70], where the average value is dominated by therare events. γ also controls the logarithmic corrections to the linear decay of ln B with n. As
we will show, these corrections are particularly important at small n < 30, as considered inthis work.

4.2

Numerical evidence for the traveling wave regime

P(ln B)

Prob(ln B > x)

0.3

1.00

(a)
0.2

0.75

0.1

0.50
0.25

0.0
− 40

− 30

− 20

− 10

ln B

0.00

(b)
− 40

− 30

− 20

− 10

x

Figure 2: (a) Distributions of ln B deep in the disordered phase, g = 0.03 < g c , fortrees with total number of generations n = 6, 8, 10, · · · , 20, from right to left. P(ln B)
is a traveling wave propagating to the left with the increase of n, preserving its shape(for n ≥ 14) with increasing n. (b) Corresponding cumulative distribution functions
Prob(ln B > x).
In the disordered paramagnetic phase for g < g c , Eq. (9) predicts a traveling wave moving
with a velocity V (g). In Fig. 2 we have plotted the distributions P(ln B) and the cumulative
distribution functions Prob(ln B > x) for trees with total number of generations n = 6 − 20,
deep in the disordered phase, g = 0.03. With the increase in n the distributions move to theleft to progressively smaller values preserving their shape (for sufficiently large trees), which
well reflects the traveling wave solution.

P(z)eγzz
γz

Figure 3: Plot of P(z)e as a function of z, where z := ln B − 〈ln B〉, for g = 0.03
(deep in the disordered phase) and system sizes n = 10, 12, . . . , 24 with n = 10 at
the bottom and n = 24 at the top. The number of disorder configurations considered are 107 for system sizes n ≤ 20 and 106 for n = 22 and 24. From Eq. (9),one has z = ln bn − 〈ln bn 〉. From Eq. (11), one expects for fixed large z to haveP(z) ≃ (C1′ z + C2′ )e−γz as n → ∞. We observe indeed that the curves for P(z)eγz for
z > 0 collapse onto a straight line (indicated by a dashed line) as n increases.
9

SciPost Phys. 15, 211 (2023)

We now focus on the centered shape of the distribution, i.e. on the distribution ofz = ln B − 〈ln B〉. First, notice from Eq. (9) that z can also be written as z = log bn − 〈log bn 〉.
Then, the prediction Eq. (10) means that the distribution of z converges as n becomes large
to some limiting distribution. This is consistent with the observation in Fig. 2 that for large n,
the shape of the distribution of ln B relative to its peak converges. Furthermore, the prediction
Eq. (11) implies that, for a fixed large z, the distribution of z should satisfy, in the n → ∞limit, the following behaviour: P(z) ≃ (C1′ z +C2′ )e−γz . We check that prediction in Fig. 3 where
we plot P(z)eγz as a function of z for system sizes n = 10–24. As n increases the curves forz > 0 collapse to a straight line corresponding to the predicted behaviour C1′ z + C2′ .

<lnB > +23γ lnn

V(g)
1.6

− 10

1.2

− 20

0.8

(a)

− 30
6

10

14

18

n

(b)

0.4
0.03

0.05

0.07

0.09

g

Figure 4: (a) Decay of 〈ln B〉 with system size n in the disordered phase, for g taking
the value g = 0.03, 0.045, 0.06, 0.075, 0.09 and 0.1 from bottom to top (g c ≈ 0.137).
3
We plot 〈ln B〉 + 2γ
ln n with γ ≈ 0.627 (see text and Eq. (9)) and the corresponding
linear fits (shown by solid lines) give the velocity V . (b) Behavior of the velocity V
as a function of g as compared to the theoretical prediction Eq. (12) (red curve) in
the linear approximation. Fitted V values are in good agreement with Eq. (12) for g
far enough from the transition, where the linearization is a good approximation for
the range of n considered.
In Fig. 4 we test numerically another prediction for the disordered behavior, Eq. (9), describing the linear decay of 〈ln B〉 with n and Eq. (12) giving the g-dependence of the velocity
in the linear approximation. Numerical data in the disordered regime g ≤ 0.1, for system sizes
limited to n ≤ 20, clearly show that logarithmic corrections described by Eq. (9) are important.
Taking them into account, the determined velocity V is in good agreement with the prediction
Eq. (12), for g far enough from the transition where the linearization is a good approximation
for the range of sizes considered. Near the transition, the deviations observed are probably
due to finite size effects.

4.3

Stationary distribution for the ordered phase

Notice in Eq. (9) that V (g) > 0 and B → 0 if g < g c , whereas V (g) < 0 and B → ∞ as
g > g c . We claim that this same value of g c also controls the transition between the ordered
and disordered phase in the original model Eq. (5):• if g < g c , then B as obtained from Eq. (5) goes to zero with the system size in the way
described by Eq. (9); the system is in the disordered phase.
• if g > g c , then B as obtained from Eq. (5) cannot go to zero and Eq. (9) no longer
holds because the non-linearity of the recursion Eq. (5) can no longer be neglected. As
it is clear from Eq. (5) that B ≤ g, the only possibility is that B reaches a stationary
distribution in the limit of large sizes; the system is in the ordered phase.
We therefore have a phase transition between a traveling and a non-traveling regimes.
As detailed in the appendix, we believe [23, 33] this transition to be similar to the one for a
10

SciPost Phys. 15, 211 (2023)

branching random walk in presence of an absorbing wall described in [97]. We now simply
list the predictions explained in detail in the appendix.

Prob(ln B > x)P(ln B)
0.5
0.4

1.00

(a)

0.75

0.3

0.50

0.2

0.25

0.1
0.0
−7

−5

−3

ln B

0.00

(b)
−7

−5

−3

x

Figure 5: (a) Distributions of ln B deep in the ordered regime, g = 0.28 for trees
with total number of generations n = 6, 8, 10, · · · , 20, from right to left. The distributions have converged to a stationary one for larger trees. The origin of the apparent
bimodal shape of the distribution has been discussed in [89]. (b) Corresponding cumulative distribution functions Prob(ln B > x).
In the ordered phase g > g c , the distribution of B converges to some limiting distribution
as the system size increases, as shown in Fig. 5 for g = 0.28. Moreover, when g is close to g c ,the average value of ln B is a large negative number described by a critical exponent κ〈ln B〉 ≃ −C(g − g c )−κ ,(14)

for g > g c , g close to g c and n → ∞. Moreover, the distribution of ln B is concentrated around
its expected value, with exponential tails.
As explained in subsection A.6 of the appendix, the analogy with travelling waves [97]suggests that κ = 1/2, and a prediction for the value of C is given in Eq. (A.31), with L being−〈ln B〉. Furthermore, the tail of the distribution of ln B is predicted to be (see Eq. (A.33) with
a = L)
πx −γxProb(ln B > −L + x) ≃ Ĉ L sin
e
,
(15)
L
in the region where both x and L − x are large, where L = −〈ln B〉 is given in Eq. (14) and Ĉis some constant.

4.4

Critical properties

At the transition g = g c , one expects the average value of ln B to vanish with the system sizewith another critical exponent δ:〈ln B〉 ≃ −C ′ nδ ,(16)i.e. B typ ≡ exp(〈ln B〉) vanishes with n as a stretch exponential. As explained in subsection A.7of the appendix, the analogy with [97] suggests that δ = 1/3, and gives a prediction for C ′ ,see Eq. (A.36) with d = n and L d = −〈ln B〉.
A particularly interesting property is the tail of the distribution of ln B at criticality g = g c[69, 97]: it is still given by Eq. (15), but now with L = −〈ln B〉 ≃ C ′ nδ , see Eq. (16).
There is another, trivial, critical exponent in Eq. (12); if g < g c but g close to g c , one has
V (g) ≃

1(g c − g) ,gc

g < gc .

(17)

Finally, we expect 〈ln B〉 to follow a single parameter scaling function in the form:〈ln B〉 = −nδ F [n1/ν (g − g c )] .
11

(18)

SciPost Phys. 15, 211 (2023)

The behavior of the scaling function F is such that one should recover the disordered behavior,
Eq. (9) and Eq. (17) when g < g c ; this leads to

F (X ) ∼ X ν(1−δ) ,for X → −∞ ,(19)with the constraint:
1 = ν(1 − δ) .

(20)

With the value δ = 1/3 predicted for the traveling/non-traveling phase transition in [97], we
obtain ν = 3/2. Moreover, Eq. (19) should allow to recover the behaviour Eq. (14) of theordered phase, in the asymptotic limit n ≫ |g − g c |−ν , which implies

F (X ) ∼ X −νδ ,for X → +∞ ,(21)

and, therefore, the following relation between the critical exponents:κ = νδ .

(22)

With the values of the critical exponents ν = 3/2 and δ = 1/3 predicted for the traveling/nontraveling phase transition in [97], we obtain κ = 1/2.

4.5

Numerical characterization of the critical behavior

In Figs. 6 and 7 we test numerically the theoretical predictions for the critical behavior at
g = 0.137 ≈ g c . Fig. 6 (a) first shows the power law decay of 〈ln B(g c )〉, as in Eq. (16).
Incorporating small system size corrections, we find that the critical data are compatible witha value of the critical exponent δ = 1/3. More precisely, we fit the data with the following
function:〈ln B(g c )〉 = −C ′ (n − c1 )1/3 + c2 ,(23)with the theoretical value C ′ ≈ 5.534 given by Eq. (A.36) and two fitting parameters: c1 ≈ 2.1
and c2 ≈ 3.0.
In Figs. 6 (b) and 7 we verify the form Eq. (15) of the cumulative distribution with L
replaced by a n-dependent quantity of the order of −〈ln B(g c )〉 at criticality. We first plot
Prob(ln B > x)eγx as a function of x for different system sizes in Fig. 6(b). These curves havethe form of an arch. They all vanish at x ≈ −D = −2.65 and their maximal value M (n) isreached at x = −D − L(n)/2, thus defining M (n) and L(n). When plotted as a function of thescaled variable π(x + D)/L(n), the different arches corresponding to Prob(ln B > x)eγx /M (n)all collapse onto a single curve, as expected from Eq. (15). This is shown in Fig.7(a).
The prediction from the appendix is that, for large system sizes, one should haveL(n) ∼ −〈ln B〉. Then, the prediction Eq. (15) gives a maximum for Prob(ln B > x)eγx whichis asymptotically M (n) ∝ L(n)e−γL(n) . In Fig. 7 (b) L(n) is shown as a function of −〈ln B(g c )〉.
The observed linear behavior with a slope close to 1 confirms the prediction of Eq. (15). In
Fig. 7(c), M (n)eγL(n) is plotted as a function L(n). We again find a linear behavior, in agreement with the theoretical prediction.
All of these numerical results give a clear confirmation of the non-trivial predictions fromthe analogy with the traveling–non-traveling transition described in [97]. In particular, they
confirm the compatibility of our results with the critical exponent δ = 1/3.

4.6

Finite size scaling of the critical properties

We have inspected the above transition closely by performing a scaling analysis of the dependence of 〈ln B〉 on the size of the trees, in the vicinity of the transition. A key point of the12

SciPost Phys. 15, 211 (2023)

Prob(lnB > x)eγx<ln B(gc)>

(b)

(a)

x

n
Figure 6: Critical behavior at g c = 0.137. (a) 〈ln B(g c )〉 is well described by
Eq. (16), with the predicted critical exponent δ = 1/3, incorporating small system size corrections described by Eq. (23) with two fitting parameters c1 ≈ 2.1
and c2 ≈ 3.0. The corresponding fit is shown by the red curve. (b) The
cumulative distribution Prob(ln B > x) is studied for different system sizesn = 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27 and 29. As expected, when plotted as
Prob(ln B > x)eγx , we obtain a sinus arch which broadens and whose amplitudedecreases as a function of system size (n = 29 corresponds to the sinus arch at thebottom and n = 6 to the topmost). This follows closely the behavior predicted by
Eq. (15) with L replaced by a n-dependent quantity of the order of −〈ln B〉, as shownin Fig. 7.

considered transition is that we know the exact value of g c : g c ≈ 0.137353, Eq. (13). Generally, this information is not known and constitutes a free parameter in the scaling analysis
which can lead to great uncertainty on the critical behaviors, in particular on the values of
the critical exponents. We have checked the validity of the one-parameter scaling hypothesis
Eq. (18) by fitting our numerical data with the following law:〈ln B〉 − ln B0 = c̃2 − C ′ mδ F (∆g m1/ν ) ,m = n − c̃1 ,

(24)
(25)

P3
with ∆g = (g − g c ) + A2 (g − g c )2 + A3 (g − g c )3 and F (X ) = k=0 Ek X k . C ′ = 5.534, δ = 1/3,
g c = 0.137353 are fixed, while ν, c̃1 , c̃2 , A2 , A3 and Ek , k = 0, · · · , 3, are fitting parameters.
The form of the scaling function incorporates the irrelevant small size corrections that we havefound for the critical behavior, see Eq. (23) and Fig. 6. We have also subtracted to 〈ln B〉 its
boundary value ln B0 = ln g whose dependence on g is not related to the critical properties. We
have chosen the minimal orders of the Taylor expansions of ∆g and F which give a non-zero
goodness of fit.
The result of such a fitting procedure is the following. We are able to find a fit compatible
with the data, within error bars, if we restrict to n ≥ 8 (our data go up to n = 27 and g values
are 0.1 ≤ g ≤ 0.17. This is quantitatively assessed by the goodness of fit Q = 0.73 (see [101]).
We find ν ≈ 1.4 quite close to the theoretical value ν = 3/2. E0 ≈ 1.1 also close to the expectedvalue 1 corresponding to Eq. (23). In Fig. 8, we have plotted the quantityΛ≡〈ln B〉 − 〈ln B0 〉 − c̃2,
C ′ mδ

(26)

as a function of m/ξ with ξ ≡ |∆g|−1/ν . The data are seen to collapse nicely onto a single
scaling function with two branches, the lower one corresponding to the disordered regimeg < g c and the upper one to the ordered regime g > g c . The fitted scaling function F is shownby the dark green and magenta lines. This confirms the validity of the single parameter scaling
hypothesis Eq. (18) and the theoretical value of the critical exponent ν = 3/2.
13

SciPost Phys. 15, 211 (2023)

Prob(lnB > x)eγx/M(n)(a)(x+D)/L(n)M(n)eγL(n)

L(n)

(c)

(b)

L(n)

-<ln B>

Figure 7: Test of the theoretical prediction Eq. (15) for the critical behavior of the
cumulative distribution Prob(ln B > x). (a) The distributions for the system sizesn = 6 − 24 (increased in steps of 2), 27 and 29 shown in Fig. 6(b) all collapse onto
a single shape looking like a sinus arch when Prob(ln B > x)eγx /M (n) is plotted asa function of π(x + D)/L(n). The scaling parameters M (n), L(n) and the constant
D have been determined as follows. D corresponds to the upper vanishing of all thecurves for Prob(ln B > x)eγx at x ≈ −D = −2.65. M (n) corresponds to the maximal
value of these curves. This maximum is attained at x = −D − L(n)/2, which in turndefines L(n). (b) L(n) is shown as a function of −〈ln B(g c )〉. The behavior is clearlylinear L ≈ −1.19〈ln B〉 + 0.07 with a slope close to 1 (shown by the dashed line),
which confirms the theoretical expectation Eq. (15). (c) M (n)eγL(n) is plotted as afunction of L(n). As expected from Eq. (15) we observe a linear behavior.

5

Non-ergodic phase

Investigation of the existence of a non-ergodic delocalized phase has been a topic of active
research in the context of both Anderson localization and MBL [24–29,32,54,94]. In the nonergodic delocalized phase of the Anderson transition, the states are delocalized within a large
number of sites ∝ N D where 0 < D < 1; however these sites represent only an algebraicallysmall fraction ∝ N D−1 of the volume of the system. D is called the fractal dimension of
the support of the states. For the Anderson transition in finite dimension, such behavior isobserved only at the transition threshold, where the states are multifractal [65]. However, it
has recently been realized that a number of models, in particular the Anderson problem on
the Cayley tree [27–29, 32], display an extended phase having these properties: 0 < D < 1 ina finite range of disorder strengths W1 < W < W2 . Similar features were found recently inthe case of dirty bosons on the Cayley tree [94], a problem closely related to the system weconsider here. Taking inspiration from these studies, in this section we explore the possibilityof observing such non-ergodic properties in the ordered phase g > g c .

14

SciPost Phys. 15, 211 (2023)

Λ<ln B> - ln B0mnFigure 8: Finite-size scaling analysis of the traveling/non-traveling transition.
(a) Raw data: 〈ln B〉 − ln B0 as a function of n in the vicinity ofthe transition, g = 0.1, 0.11, 0.12, 0.125, 0.127, 0.13, 0.132, 0.134, 0.137, 0.14,
0.142, 0.144, 0.146, 0.15, 0.155, 0.16, 0.17 from below to top. The data are shownby dots with their error bar, while the lines represent the global fit by the scaling
function Eq. (24). The pale green dots (data) and dashed curve (fit) correspond tothe critical behavior g = 0.137 ≈ g c . (b) Scaling function: The data in (a) are fitted by a single parameter scaling function given by Eq. (24) with the critical valueg c = 0.137353 and the critical exponent δ = 1/3 fixed. The best fit gives a valueof ν ≈ 1.4 close to the theoretical expectation ν = 3/2 for the critical exponent of
the analogous traveling–non-traveling transition [97]. When Λ given by Eq. (26) isplotted as a function of m/ξ with m = n − c1 and ξ ∼ |g − g c |−ν (see text), the datacollapse onto a single scaling function. The fitted scaling function F is shown by thedark green and magenta lines. This confirms the predictions for the single paramaterscaling law Eq. (18).
We observed in Sec. 3 that a major fraction of the total number of sites lie on the boundaryof the Cayley tree. Hence, to study the non-ergodic delocalized phase of the Anderson transition on the Cayley tree, it was found that one needs to consider a boundary condition for theimaginary part η of the Green’s function which vanishes as N −φ [27–29,32] with φ a constant
exponent. η corresponds to the boundary field B0 here, so that we need to appropriately scale
the boundary field and choose B0 ∝ N −φ .
This choice can be understood in a simple way in the language of the traveling wave problem of Sec. 4. Note that in the ordered phase (g > g c ), starting from a low value (far fromthe cut off) B0 ≪ g, the recursion can at first be linearized. However, corresponding velocity
will be negative, V (g ≳ g c ) < 0 as per Eq. (12). Hence, B first increases exponentially as thedepthn increases until it reaches a stationary value due to the effect of the nonlinear cutoffp2
ε + B 2 , see Eq. (5). Having B0 = α0 N −φ means that B does not necessarily reach the nonlinear cutoff, even in the limit of large n ≈ log2 N . Indeed, the linearized recursion gives, see
Eq. (9):3ln n + 〈ln bn 〉2γ
3
≈ − (V + φ ln 2) n −ln n + Cste ,2γ〈ln B〉 = −V n + ln B0 −(27)

with Cste = ln α0 + 〈ln b〉. This means that for V + φ ln 2 > 0, 〈ln B〉 (with the log correction)
decreases with n and thus never reaches the upper cutoff value g and a stationary regime. For
our numerical observations, we chose α0 = 100 and φ = 1. Thus for g c < g < g e , whereg e = 2g c ≈ 0.27 is given by |V (g e )| = ln 2, we can expect to find clear deviation from ergodic
behavior.
15

SciPost Phys. 15, 211 (2023)<ln B>−6− 10

g = 0.18
g = 0.15

− 14
6

10

14

18

n

Figure 9: We show the variation of 〈ln B〉 as a function of n, corresponding to different
choices of the boundary field B0 in the early ordered phase. The choice B0 = 100
N(shown in solid lines) corresponds to a power law behavior indicating non-ergodicity.
While, for the choice B0 = g (shown in dashed lines) we find an ergodic orderedphase, where 〈ln B〉 tends to a stationary value at large n. Notice that for n = 8, we
have N = 766 and 100
N = 0.13 ≃ g, and so the dashed and plain curves of either color
cross very close to that point.
In Fig. 9, we look at the contrasting behavior of 〈ln B〉 as a function of n for the two different
choices of boundary field, B0 = g and B0 = 100/N . We have looked at two different values ofg in the early ordered regime: g = 0.15 and g = 0.18. In the B0 = 100/N case, we observe apower law behavior of the typical field with the system size
exp〈ln B〉 ∼ N D1 −1 .

(28)

By analogy with the algebraic behavior found for the typical value of ℑG ∼ N D−1 , with
0 < D < 1, in the non-ergodic delocalized phase of the Anderson localization transition on the
Cayley tree [29,32,99], we interpret the behavior Eq. (28) as indicating a non-ergodic orderedphase. The exponent 0 < D1 < 1, analogous to the fractal dimension D for the Anderson localization problem, is introduced to characterize this property. For the system sizes n ≤ 20,
appropriately incorporating the log correction we find that the numerical data are well fittedby:3〈ln B〉 +ln n ≈ −[1 − D1 ]n ln 2 .
(29)
2γRecall that ln N ∼ n ln 2. In the regime where linearization of the recursion is possible, g < g e ,<lnB > +23γ lnn

D1

−2
−6
− 10

ge

(a)

− 14
10

14

18

g

n

3
Figure 10: (a) 〈ln B〉 + 2γ
ln n (incorporating logarithmic corrections) as a functionof n is shown for selected values of g = 0.3, 0.28, 0.24, 0.20, 0.18, 0.15 and 0.14 from
top to below. The linear fits indicated by the dashed lines give the value of the corresponding exponent D1 as per Eq. (29). (b) Exponent D1 (as obtained from part (a)of the figure) as a function of g. We find a non-ergodic regime for g < g e ≈ 0.27,
characterized by: 0 < D1 < 1. In the regime g c < g ≲ g e we find very good agreement with the theoretical prediction for D1 , Eq. (30) (red solid line).

16

SciPost Phys. 15, 211 (2023)

we find from Eq. (27) with φ = 1:
D1 ≈ −

V
.
ln 2

(30)

3
In Fig. 10 we look at 〈ln B〉 + 2γ
ln n as a function of n for different values of g c < g ≤ 0.3. We
find a non-ergodic ordered regime characterised by: 0 < D1 < 1, where D1 agrees very wellwith the theoretical value for g c < g < g e and tends to unity as g → g e , similarly to what wasobserved in the non-ergodic delocalized phase of the Anderson transition in the Cayley tree.
A note is in order here, regarding the significance of finite-size effects in the present problem compared to the related Anderson problem. In the Anderson transition on random and tree
graphs, drastic and non-trivial finite-size effects have been observed [32–34, 36, 37, 98, 102].
These effects were central to the debate concerning the existence of a non-ergodic delocalized
phase [24–37, 40, 41, 54], which has now been resolved. In our problem, the mapping to thetraveling wave problem allows us to analytically deduce the presence of a universal logarithmic finite-size correction that depends solely on the tail exponent γ, as indicated in Eq. (29).
By incorporating this logarithmic correction, we can accurately determine the exponent D1
which characterizes the non-ergodic ordered phase.
For the related problem of disordered hard-core bosons [94], a non-ergodic phase has also
been identified, but for strong enough disorder, in the so-called Bose-glass regime. There, thecoherent fraction (analogous to the ferromagnetic order parameter) is zero, while the one-bodydensity matrix [103, 104] display some non-ergodic delocalized properties, with for instancean anomalously slow decay of the condensed fraction, suggesting a new form of off-diagonalquasi-long-range order induced by the disorder [94]. Here, since the U(1) symmetry is notpreserved by our quantum Ising Hamiltonian, we cannot further follow the analogy between
the two models, namely there is no equivalent of the condensed fraction.

6

Strong spatial inhomogeneities

The different phases described above can show strongly inhomogeneous spatial distributions ofcavity fields. Such spatial inhomogeneities are well known in the problem of directed polymers
on the Cayley tree [66], which has a glassy phase at low temperature where the polymer
explores only a finite number of branches instead of the exponentially many at disposal (thesymmetry of replicas is then broken). In this last section, we characterize this property in the
random field Ising model considered.
In Fig. 11, we contrast the spatial distribution of the cavity field within a given level d, inthe disordered and ordered phases. We consider in each case a single realization of disorderand represent the corresponding spatial distribution of the cavity fields as follows. The position inside the finite Cayley tree with n = 18 is specified by the depth 0 < d < n − 1 (excluding
the root and the boundary) increasing from the boundary to the root (where d = n − 1).
Strong spatial inhomogeneity is clearly seen in the disordered case shown in Fig. 11(a) where
the cavity fields amplitudes are large only on a few branches, sufficiently far from the homogeneous boundary. This is reminiscent to the behavior of the directed polymer problem. On
the contrary, in the ordered case shown in Fig. 11(b), the spatial distribution is homogeneous,indicating an ergodic behavior. Finally, in the non-ergodic ordered case shown in Fig. 11(c),corresponding to a boundary condition B0 = 100/N , the spatial distribution is again strongly
inhomogeneous, similarly to the disordered phase.

17

SciPost Phys. 15, 211 (2023)

(a)

(b)

(c)

Figure 11: Spatial distribution of the cavity fields on the vertices of a Cayley tree
far away from the uniform boundary, for a single random disorder configuration.
The total number of generations is n = 18. Only generations 9 ≤ d ≤ 18 are shown(from the root to half of the tree). The radius of the circles at vertex i of generation d
are given by: [Bd(i) − mini (Bd(i) )]/[maxi (Bd(i) ) − mini (Bd(i) )], where the min and max are
taken over all the sites within a given level d. Panel (a) corresponds to g = 0.03, deepin the disordered phase, while panel (b) corresponds to g = 0.3, deep in the orderedphase. Both have boundary condition B0 = g. In panel (c) g = 0.18 and B0 = 100/N .
It describes the non-ergodic delocalized phase. Note that the same randomness wasused in these three simulations. In (a) and (c), cavity fields have large amplitudes
only on few branches, while (b) shows an ergodic delocalized behavior.
ln<I2>ln Nd

Figure 12: The second moment 〈I2 (d = n/2)〉 of the cavity fields B̃di at depthd = n/2, as a function of the total number of site Nn/2 at depth d = n/2,
see Eq. (31), shown in log scale for B0 = 100/N . The values of g considered
are: 0.14, 0.15, 0.18, 0.2, 0.24, 0.3 and 0.45, from the top to bottom. In the nonergodic regime (g c < g < g e ), the second moment tends to a constant, showingthat cavity fields explore only a few branches of the tree. The ergodic behavior〈I2 (n/2)〉 ∼ 1/Nn/2 is restored for g > g e ≈ 0.27. This is indicated by the dashedline shown for the lower curve corresponding to g = 0.45.
To characterize these non-ergodic features more quantitatively, we study the normalized
second moment of the measure given by the cavity fields within a level d:Pl

I2 (d) = P

Bd(l)
(l)

l Bd

2
2 ,

(31)

where the sum is over all the sites at a given level d. I2 (d) is analogous to the inverse participation ratio and gives the inverse of the number of sites occupied by the cavity fields. If one
18

SciPost Phys. 15, 211 (2023)

calls Nd the number of sites at a level d, then one can see that I2 (d) = 1/Nd if all the cavity
fields are equal, and I2 (d) = 1 if one cavity field is much bigger than all the others. Then,the study of 〈I2 (d)〉 as a function of Nd , gives a characterization of non-ergodic features in themodel we study. We choose to study 〈I2 (n/2)〉 as a function of Nn/2 , at the level d = n/2, deep
inside the tree, far enough from the boundary, when the boundary condition is B0 = 100/N
and for g > g c .
Fig. 12 shows that the second moment tends to a constant in the non-ergodic regime,corresponding to small g c < g < g e values. As the second moment corresponds to the inverseof the number of sites with large cavity fields, this behavior indicates that the cavity fields
explore only a few branches of the tree in the non-ergodic regime, confirming our previous
observations, see Fig. 11(c), and the analogy between this phase and the directed polymer
problem [66], see Eq. (27). At large g ≳ g e ≈ 0.27 values, on the other hand, the second
moment has a Nd−1 dependence characteristic of an ergodic regime.

7

Conclusion

In this paper we have investigated the zero temperature quantum paramagnetic-ferromagnetic
phase transition for a random transverse field Ising model in its cavity mean field description
by Feigel’man, Ioffe and Mézard [70] on the Cayley tree. This problem was initially considered to describe the mechanism responsible for the strong spatial inhomogeneities observedin strongly disordered superconductors [70, 89, 90]. In the Cayley tree, the disordered phaseof this system is analogous to the problem of directed polymers and therefore has a glassy
regime characterized by strong inhomogeneities. This is not unlike the localized phase of theAnderson transition, which has similar properties [23,25,33,45]. The emergence of such nonergodic properties due to disorder has been the subject of great interest recently, in particularwith regard to the existence of a non-ergodic delocalized phase [24–37, 40, 41, 54] and inconnection with the problem of many-body localization [4, 6, 10–12, 46, 48–53].
We addressed two open questions in this context: We first described the critical propertiesof the paramagnetic-ferromagnetic transition for the random transverse-field Ising model, andexamined the possibility of a non-ergodic ordered phase. One of the motivations for the firstquestion is the similarity between this problem and the Anderson transition on the Cayley
tree whose critical properties are not yet well understood [29, 33, 36, 38, 41]. The cavity
mean-field equation can be seen as a simpler form of the self-consistent equation for Anderson
localization [20], thus allowing a better understanding of these critical properties. Similarly,
we use the analogy with the Anderson transition, which has a non-ergodic delocalized phaseon the finite Cayley tree [24–29, 32], to understand the conditions for the existence of a nonergodic phase. Such a phase has recently been observed in disordered hard core bosons on
the Cayley tree [94]. The system we have considered can be seen as an approximation of thisproblem [70, 89, 91].
To answer these questions, we have followed an analogy with a traveling wave problem
corresponding to a branching random walk in presence of an absorbing wall [69, 97] anddirect numerical simulations of the cavity mean field equation on a finite Cayley tree. As in
the case of the Anderson transition on the finite Cayley tree, we find that the nature of theordered phase (corresponding to the delocalized phase) depends crucially on the boundary
conditions.
We have first considered constant boundary conditions, which corresponds to a transition
between a disordered and an ergodic ordered phase. We have tested with great precision
the results deriving from the expected analogy with the problem of traveling wave with awall [69,97]. We have found quantitative agreement with the predictions in the paramagnetic

19

SciPost Phys. 15, 211 (2023)

phase which can be identified with the traveling wave regime by appropriately incorporating
the logarithmic corrections for the small system sizes considered. We therefore show that suchcorrections can be very important for numerical studies with finite size systems. The criticalbehavior and in particular the cumulative distribution function agrees very well with the mathematical predictions considered in [69,97], in particular for the critical exponent δ = 1/3. The
critical properties have been assessed using a detailed finite-size scaling approach incorporating irrelevant corrections. We have confirmed the critical exponent predicted in [97], ν ≈ 3/2.
Furthermore, we have explored the appropriate choice of boundary condition for observing
the much studied non-ergodic delocalized phase [24–37, 40, 41, 54] from a traveling wave
perspective. To be able to observe a non-ergodic ordered phase, it is necessary to take boundary
conditions which vanish algebraically with the volume of the system, B0 ∼ N −φ where φ isa positive exponent. In this case, the ordered phase is characterized by an order parameter
which also vanishes algebraically, characterized by a fractal dimension which reaches one in
the ergodic phase. We have characterized the strong spatial inhomogeneities of this phase.
Our work has revealed an intriguing analogy with the Anderson transition, establishingconnections through the traveling/non-traveling phase transition [97]. However, it is important to acknowledge the limitations of this analogy. Specifically, our focus is on ground state
properties, whereas the Anderson problem primarily addresses highly excited states. An extension of the CMF approach we have used is required to describe properties related to excitations
above the ground state, including the presence of a mobility edge or other dynamical properties [70].
An interesting outcome of the analogy between CMF and the Anderson problem lies in theanalytical predictions it provides for critical properties. Notably, our results put forth analyticalpredictions for critical behavior and finite-size scaling, which, to the best of our knowledge,
have not been explored within the framework of the Anderson problem. These predictions
could make a substantial contribution to resolving debates surrounding critical exponents of
the Anderson transition on random graphs [29, 29, 32, 33, 36, 41, 98].
Furthermore, this analogy allows us to explore the extension of non-ergodic properties tosystems with interactions, shedding new light on the non-ergodic behaviors observed in closely
related models like hard-core disordered bosons on the Cayley tree [94]. An open question
remains regarding whether, similar to the Anderson transition, the presence of a non-ergodic
delocalized phase crucially depends on the type of graphs considered [28, 32, 33, 105], suchas Random Regular Graphs or small-world networks instead of the Cayley tree. Our work
reveals a distinction in nature between the ordered phase of the infinite Bethe lattice/Cayley
tree, which is ergodic, and a finite Cayley tree where the delocalized phase is non-ergodic. It
essentially boils down to a matter of the order of limits. Due to the finite boundary fraction
compared to the total volume on a finite Cayley tree, the order of limits B0 → 0 and N → ∞becomes crucial. In the usual order, where N → ∞ precedes B0 → 0 (corresponding to theinfinite Bethe lattice), the ordered phase exhibits ergodic behavior. Conversely, in the invertedorder, where B0 → 0 precedes N → ∞ (corresponding to B0 ∼ N −φ ), the non-ergodic orderedphase emerges.
Since the approach we have used is based on a non-trivial mean-field approximation, it isimportant to compare its outcomes with other approaches. Strong-disorder renormalization
procedures have been used in such graphs of effective infinite dimensionality [82–84, 106].
References [82,84,106] point towards a transition of infinite randomness criticality, but where
the randomness grows only logarithmically with N . It remains unclear to us whether the CMF
approach aligns with this logarithmic form of infinite randomness. On the one hand, the CMF
approach predicts that, at the critical point, both the typical and averaged values of B exhibit stretch exponential decay. While the typical decay mirrors the behavior observed in theconventional 1D infinite randomness criticality, the averaged decay contradicts the expected

20

SciPost Phys. 15, 211 (2023)

power-law behavior in such cases. On the other hand, Griffiths effects have been observedin the CMF approach on both sides of the transition [70]. The strong disorder renormalization approach of Ref. [83] establishes a connection with the directed polymer problem in theparamagnetic phase, thus confirming the analogy with the traveling phase we have described.
Lastly, CMF indicates the existence of an infinite disorder fixed point in dimensions d = 2
and 3 [88]. These contrasting findings underscore the need for further investigations into thenature of infinite randomness criticality in systems of effective infinite dimensionality.
Other interesting perspectives include expanding our analysis to describe other quantum
many-body systems on the Cayley tree, for example the metal-insulator transition for disordered fermions in the statistical dynamical mean-field approach [107,108]. This type of system
can now be studied experimentally, for example by means of quantum computers [109]. The
study of their non-ergodic properties should make it possible to understand the emergence ofsimilar properties in finite dimensionality.

Acknowledgments
AC and GL wish to thank J. Gong and C. Miniatura for fruitful discussions. We also thank José
Hoyos for interesting discussions.
Funding information This study has been supported through the EUR grant NanoX n° ANR17-EURE-0009 in the framework of the “Programme des Investissements d’Avenir”, researchfunding Grants No. ANR-17-CE30-0024, ANR-18-CE30-0017 and ANR-19-CE30-0013, and by
the Singapore Ministry of Education Academic Research Fund Tier I (WBS No. R-144-000437-114). We thank Calcul en Midi-Pyrénées (CALMIP) for computational resources and assistance.

A

Analytical description of the traveling–non-traveling phase transition

In order for the paper to be sufficiently self-contained, we recall in this appendix the analyticalderivations of the main results on which we relied in Sec. 4.
This appendix is about travelling waves in the universality class of the Fisher-KPP equation[67,68], invading an unstable phase on the right from a stable phase on the left. As explained
below, there are several possible travelling waves, with different velocities v and spatial decay
rates λ, but the most relevant one for our problem is the critical travelling wave going at velocity
vc and having a spatial decay rate λc .
These critical values vc and λc appear directly in some important results of the paper, but
under another name:
V (g) ≡ −vc ,γ ≡ λc .
(A.1)
See in particular Eq. (9), Eq. (11), etc.

A.1

A Fisher-KPP equation

We discuss first how the linearized cavity mean field recursion Eq. (7) describing the disorderedphase can be mapped onto the traveling wave problem described in [66]. In this analogy, thedistance d from the boundary plays the role of a time.

21

SciPost Phys. 15, 211 (2023)

Assume that Bd obeys Eq. (7) with B0 being some non-random constant. Notice first that
Bd /B0 is independent of B0 and introduce¬¶G(x, d) := exp − e−x Bd /B0 ,(A.2)
where 〈·〉 represents averaging over the random variable Bd . Using Eq. (7), we obtain theevolution equation
¬ 
¶K
g
G(x, d + 1) = G x − ln + ln |ε|, d
,
(A.3)
K
where 〈·〉 represents now averaging over the random variable ε.
There are two constant solutions to Eq. (A.3): G(x, d) = 0 and G(x, d) = 1. We say thatG = 0 is the stable phase and G = 1 is the unstable phase; indeed, if G(x, 0) is a constant in(0, 1), then G(x, d) is at all time d a constant, converging to 0 as d → ∞.
Eq. (A.3) describes the propagation of a front between the stable phase G(−∞, d) = 0 andthe unstable phase G(∞, d) = 1. This equation is in the universality class of the Fisher-KPP
equation [67, 68], first introduced in 1937:
∂F
∂ 2F
=
− F (1 − F ) .
∂t
∂ x2

(A.4)

(Nota: the Fisher-KPP equation is often written for f = 1 − F ; the quantity f follows the same
equation as F , except that the minus sign in front of F (1 − F ) is changed into a plus sign.)
As for Eq. (A.3), Eq. (A.4) has two uniform solutions, F = 0 which is stable and F = 1
which is unstable. For an initial condition that interpolates between 0 and 1, a front builds
up where the stable phase 0 invades the unstable phase 1. While not obvious at first sight,
Eq. (A.3) and Eq. (A.4) share enough features to behave in similar ways: the expectation over
ε in Eq. (A.3) plays the role of the diffusion in Eq. (A.4). The reaction −F (1 − F ) term in
Eq. (A.4) leaves F = 0 and F = 1 unchanged, and pushes any intermediate values towards 0;
similarly, taking the K-th power in Eq. (A.3) leaves G = 0 and G = 1 unchanged, but pushes
intermediate values towards 0.
There is an exhaustive literature on the Fisher-KPP and related equations; in particular,much work has been devoted on giving precise asymptotics on the position of the front [110–115]. More references can be found in a review from 2003 [116] and in a dissertation from2016 [117].
In the following sections, we review quickly and apply to Eq. (A.3) some well-known resultsof Fisher-KPP related equations and obtain in that way some informations on Bd .

A.2

Properties of travelling wave solutions

We first consider travelling wave solutions with velocity v to Eq. (A.3). These solutions satisfy,
for some function ω v ,GTW (x, d) = ω v (x − vd) .
(A.5)
Since, from Eq. (A.2), the quantity G(x, d) for fixed d is an increasing function of x goingfrom G(−∞, d) = 0 to G(+∞, d) = 1, we only consider travelling wave solutions satisfying
ω v ∈ (0, 1), ω v (−∞) = 0 and ω v (∞) = 1.
Such a travelling solution exists for all v ≥ vc , where vc is called the critical velocity. Noticethat the problem is translation invariant; this implies that if ω v (x) represents a travelling wave
solution, then ω v (x + a) for any constant a is also a travelling wave solution.
Plugging Eq. (A.5) into Eq. (A.3), one obtains that the travelling waves ω v satisfy¬¶Kgω v (x − v) = ω v (x − ln + ln |ε|) .
K
22

(A.6)

SciPost Phys. 15, 211 (2023)

We linearize for large x, where ω v is close to the unstable solution 1. Looking for exponential
solutions1 − ω v (x) ∝ e−λx , as x → ∞ ,(A.7)
one obtains a relation between the velocity v and the spatial decay rate λ:v = v(λ) := ln

¶
g
1  ¬
+ ln K e−λ ln |ε| .
K λ

(A.8)

The critical velocity vc is the minimal value reached by v(λ):
vc = v(λc ) = min v(λ) .
λ

(A.9)

The travelling waves ω v ( y) for v > vc behave as in Eq. (A.7), where λ satisfies Eq. (A.8) for
the given value of v and λ < λc . However, the critical travelling wave ωc (for v = vc ) has apolynomial prefactor:
1 − ωc (x) ≃ C(x + a)e−λc (x+a) ,as x → ∞ ,(A.10)

for some constant C which is hard to compute (it depends on the whole non-linear equation
Eq. (A.6)). The value of a is arbitrary; it depends on the choice of the travelling wave, which
is only defined up to translation.
In this paper, we mostly consider the case K = 2 and ε uniform in [−1, 1]. Then Eq. (A.8)
becomes, for λ ∈ (0, 1):g 1
2v(λ) = ln + ln.
(A.11)
2 λ 1−λClearly, λc does not depend on g. Numerically, one finds thatλc ≃ 0.626635 .

(A.12)

The critical velocity depends on g. Numerically,vc ≃ ln

A.3

gg+ 2.67835 = ln ,
2
gc

with

g c ≃ 0.137353 .

(A.13)

Properties of the front

We only consider the case K = 2 and ε uniform in [−1, 1]. Going back to Eq. (A.2), one getsfor d = 0G(x, 0) = exp − e−x ≃ 1 − e−x , as x → ∞.
Thus, the initial condition converges to 1 as x → ∞ much faster than does the critical travelling wave, see Eq. (A.10) and Eq. (A.12). In this case, it is known that the front properly
centered converges uniformly to the critical travelling wave: there exists a choice of ωc such
that


3
G vc d −ln d + x, d → ωc (x) , as d → ∞ .
(A.14)
2λc(Recall that any translation of a travelling wave is another travelling wave. The equation above
selects one of these travelling waves.)
Note that for other choices of K and ε, we could find λc > 1. Then, the initial condition would converge to 1 more slowly than the critical travelling wave. In that case, thebehaviour of the front would be rather different. In the following, we assume to always be in
the case λc < 1.

23

SciPost Phys. 15, 211 (2023)

A.4

Prediction for the disordered phase

From the definition Eq. (A.2) of G, one has

where bd is defined by


 ¬

¶
3
G vc d −ln d + x, d = exp − e−x bd ,2λc

(A.15)

3
Bd = B0 exp vc d −ln d bd .
2λc(A.16)

Then, Eq. (A.14) means that the distribution of bd satisfies¬¶
exp − e−x bd → ωc (x) , as d → ∞ .

(A.17)

This means bd converges (in distribution) to a random variable b = b∞ with generating
function given by
exp − s b = ωc (− ln s) .
(A.18)
When s > 0 is small, Eq. (A.10) gives that
exp − s b ≃ 1 − C e−λc a (− ln s + a)sλc .

(A.19)(The value of a is no longer arbitrary as Eq. (A.17) only holds for one specific choice of ωc ,but determining its value is hard.) By usual tail analysis, this implies that for large b one has
Prob(b > x) ≃ (C ′ ln x + C ′′ )x −λc ,(A.20)

where C ′ and C ′′ are other constants.
To summarize, Eq. (A.16) gives the typical value of Bd for large d in the linearized cavity
mean field recursion Eq. (7), while Eq. (A.20) gives the tail distribution of large values of B.
Note that, to obtain Eq. (7), we first assumed that Bd → 0 as d → ∞. This is consistent
with Eq. (A.16) only if vc < 0 which is the case if g < g c , see Eq. (A.13).
With the substitutions V (g) ≡ −vc and γ ≡ λc , then Eq. (A.12) and Eq. (A.13) are equivalent to Eq. (12) and Eq. (13). Furthermore, Eq. (9) is the same as Eq. (A.16) in the large
d = n − 1 limit where bd ≃ b, and Eq. (11) is obtained by differentiating Eq. (A.20).

A.5

Slow travelling waves

We mentioned that travelling waves satisfying ω v ∈ (0, 1), ω v (−∞) = 0 and ω v (∞) = 1
only exist for v ≥ vc . It turns out that there exists travelling waves for v < vc if one allows
ω v to take values larger than 1. As those travelling waves are relevant to the study of thedisordered/ordered transition, we now quickly review their properties.
From now on, we use the following notation:x ≪ y or y ≫ x ,to mean thaty−x

is large.

(A.21)

From Eq. (A.9), one has that v(λc ) = vc , v ′ (λc ) = 0 and v ′′ (λc ) ≥ 0. Actually, in the caseK = 2 and ε uniform in [−1, 1], one obtains from Eq. (A.11)
v ′′ (λc ) ≃ 11.4477 .

(A.22)For λ close to λc , one has thereforev(λ) ≃ vc +v ′′ (λc )(λ − λc )2 .
224

(A.23)

SciPost Phys. 15, 211 (2023)

One can obtain v(λ) < vc by giving a small imaginary part to λ. Let L be some large parameter,
and takeiπ
λ = λc ±.
(A.24)
L
Then, in (A.23),
v ′′ (λc )π2.
(A.25)
v(λ) ≃ vc −2L 2
It turns out that travelling waves ω v with v < vc given by Eq. (A.25) exist, and that their large
x behavior is obtained by using Eq. (A.24) in Eq. (A.7). One obtainsCLπ(x + a) −λc (x+a)
sine, if x + a ≫ 0 ,(A.26)
πL
where C has some definite value, but a is arbitrary. (Recall that a shifted travelling wave isanother travelling wave.) Notice that this front oscillates around the unstable phase 1. Noticealso that by sending L to infinity, one recovers Eq. (A.10); in particular the C in Eq. (A.26) is
the same as the C in Eq. (A.10).
1 − ω v (x) ≃A.6

Prediction for the ordered phase

Assume now that g > g c , with g very close to g c . Eq. (A.13) gives vc > 0 and Eq. (A.16) would
predict that B → ∞, but of course this is not true as Eq. (5) implies that Bd < g. We see that
Bd must reach some stationary distribution B as d → ∞.
We introduce the function G as in Eq. (A.2) for the long time distribution B, but without
the B0 term which no longer makes sense and, of course, without the dependence in d:¬¶G(x) := exp − e−x B .
(A.27)
When x is a large positive number, given that B is bounded, one can expand the exponential
to obtain
G(x) ≃ 1 − 〈B〉e−x ,for x ≫ 0 .
(A.28)
When x is a large negative number, the value of G is dominated by the small values of B (oforder e x ). But for these small values, the linearized recursion Eq. (7) still holds, and this
implies that the recursion Eq. (A.3) (without the d) still holds:
¬ 
¶K
g
G(x) ≃ G x − ln + ln |ε|, for x ≪ 0 ,(A.29)K
but Eq. (A.29) is actually the same as Eq. (A.6), the equation defining the travelling wave ω v ,for v = 0. We conclude that
G(x) ≃ ω0 (x) ,for x ≪ 0 ,(A.30)

for some choice of the zero-velocity travelling wave ω0 . (Recall that travelling waves aredefined up to some translation.)
Recall from Eq. (A.13) that vc = ln(g/g c ). When g ≃ g c , this gives vc ≃ (g − g c )/g c . Then,from Eq. (A.25), the choice of L which makes v = 0 isv
t v ′′ (λ )π2 gccL≃(g − g c )−1/2 .
(A.31)
2This L gives the length of the sine arches in the travelling wave ω0 , see Eq. (A.26). To summarize, we have at this point the following prediction for G(x):−x
for x ≫ 0 ,1 − 〈B〉e ,π(x+a) −λc (x+a)
CL(A.32)
G(x) ≃ 1 − π sin L e, for −a ≪ x ≪ 0 ,0 ,for x ≪ −a .
25

SciPost Phys. 15, 211 (2023)(More precisely, G(x) interpolates from 0 to 1 in a region of size of order 1 around x = −a.)Notice that, at this point, we still have to determine the correct value of a. (Recall that a,which comes from Eq. (A.26), represents the translation symmetry.) This value of a will be
determined in a self-consistent way later on.
We claim that Eq. (A.32) implies that the distribution of ln B is concentrated around −a,with a tail given by
Prob(ln B > u − a) ≃ Ĉ L sin

πu −λc ue,L

for 0 ≪ u ≪ a ,(A.33)

for some constant Ĉ. To do this, we check that this form gives the correct function G(x) in
Eq. (A.32). Write
Z∞
 −x
d 
G(x) =
Prob(B > y) e−e yd y (−)
dy0Z∞−x(A.34)=1−d y Prob(B > y)e−x−e y=1−Z0

du eu Prob(B > eu )e−x−eu−x.

But Prob(B > eu ) = Prob(ln B > u). Changing u into u − a and x into x − a, we getZ
G(x − a) = 1 −du Prob(ln B > u − a)eu−x−e

u−x.

(A.35)

The function u 7→ Prob(ln B > u − a) is 1 for u ≪ 0 and roughly e−λc u for 0 ≪ u ≪ a, and
0 for u > a. One can then check that the integral in Eq. (A.35) is dominated by u close to x
if x ≪ a. If x ≪ 0, then the term Prob(ln B > u − a) can be replaced by 1 in the integral.
If 0 ≪ x ≪ a, it can be replaced by its expression Eq. (A.33) with sin(πu/L) replaced bysin(πx/L), as the difference between x and relevant values of u is small compared to L. Then,
one checks that these substitutions lead respectively to the third and second line of Eq. (A.32).
We already argued that the first line must hold (independently of the distribution of B, as longas it is bounded) in Eq. (A.28). This validates the claim that Prob(ln B > u − a) is given by
Eq. (A.33).
We still have to determine a. Recall that B is bounded from above by g; this means that
Prob(ln B > u − a) should cancel very quickly when u becomes close to a. But the form
Eq. (A.33) cancels when u = L; this suggests that a should be close to L.
To summarize, in the regime g > g c with g close to g c , we expect the stationary distributionof ln B to be concentrated around −L, with L ≫ 1 given by (A.31). This distribution of ln B
has a power-law tail with an exponent −λc modulated by a sine arch, as given in Eq. (A.33)
with a = L. Fig. 13 illustrates this prediction:

A.7

At criticality

We now consider g = g c . The results for g < g c and g > g c , as exposed in this appendix, arereminiscent of the problem studied [97] of a Fisher front F (u, t) with a drift and a boundary
condition F (0, t) = 0 (with 0 being now the unstable phase). The results of [97] for the front
they study are exactly the same as the properties of Prob(ln Bd > u) (with d playing the roleof time) that we have just presented: depending on the drift (which plays the same role as g),
the front in [97] escapes to infinity (as in our disordered phase), or reaches a stationary statewith a sine arch (as in our ordered phase).

26

SciPost Phys. 15, 211 (2023)

Prob(ln B > u)

1
u

−L

eλc u Prob(ln B > u)u−LFigure 13: Sketch of the distribution of ln B in the stationary regime of the orderedphase.
This remark suggests that Prob(ln Bd > u) has in fact all the properties of a Fisher front
with a boundary condition. Then, if the analogy still holds at criticality, [97] gives a prediction
for the problem we consider, which we now present.
For g = g c , the front escapes at infinity, meaning that Bd goes to zero as d → ∞. Moreprecisely, for large d, The distribution of ln Bd is concentrated around −L d , with
Ld ≃

3
2

π2 v ′′ (λc )d

1/3

,

(A.36)

πu −λc ue,
Ld(A.37)and that, similarly to (A.33),
Prob(ln Bd > u − L d ) ≃ Ĉ L d sin
for 0 ≪ u ≪ L d and d large.

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