arXiv:cond-mat/0701699 · Physical Review E

Directed polymer in a random medium of dimension 1+3 : Multifractal properties at the localization-delocalization transition

Cécile Monthus, Thomas Garel

Published May 30, 2007• 13 citations

Abstract

We consider the model of the directed polymer in a random medium of dimension $1+3$, and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights ${w}_{L}(\stackrel{P\vec}{r})$ of the endpoint of the polymer of length $L$ via the moments ${Y}_{q}(L)={\ensuremath{\sum}}_{\stackrel{P\vec}{r}}\phantom{\rule{0.2em}{0ex}}{[{w}_{L}(\stackrel{P\vec}{r})]}^{q}$. We measure the generalized exponents $\ensuremath{\tau}(q)$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ governing the decay of the typical values ${Y}_{q}^{\mathrm{typ}}(L)={e}^{\overline{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{Y}_{q}(L)}}\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\tau}(q)}$ and disorder-averaged values $\overline{{Y}_{q}(L)}\ensuremath{\sim}{L}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)}$, respectively. To understand the difference between these exponents, $\ensuremath{\tau}(q)\ensuremath{\ne}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ above some threshold $q>{q}_{c}\ensuremath{\sim}2$, we compute the probability distributions of $y={Y}_{q}(L)∕{Y}_{q}^{\mathrm{typ}}(L)$ over the samples: We find that these distributions becomes scale invariant with a power-law tail $1∕{y}^{1+{x}_{q}}$. These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent $\ensuremath{\nu}\ensuremath{\sim}2$.

DOI: 10.1103/physreve.75.051122

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Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization in two and three dimensions

Cécile Monthus, Thomas Garel

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Multifractal statistics of the local order parameter at random critical points: Application to wetting transitions with disorder

Cécile Monthus, Thomas Garel

arXiv:0705.0223Physical Review EAug 21, 200712 citations

Random transverse field Ising model in dimensiond> 1: scaling analysis in the disordered phase from the directed polymer model

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Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class

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Universality in Anderson localization on random graphs with varying connectivity

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Jan 1, 1994172 citations

arXiv:cond-mat/0701699 · Physical Review E

Directed polymer in a random medium of dimension 1+3 : Multifractal properties at the localization-delocalization transition

Cécile Monthus, Thomas Garel

Published May 30, 2007• 13 citations

arXiv HTML PDF

Abstract

We consider the model of the directed polymer in a random medium of dimension

1+3

, and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights

{w}_{L}(\stackrel{P\vec}{r})

of the endpoint of the polymer of length

L

via the moments

{Y}_{q}(L)={\ensuremath{\sum}}_{\stackrel{P\vec}{r}}\phantom{\rule{0.2em}{0ex}}{[{w}_{L}(\stackrel{P\vec}{r})]}^{q}

. We measure the generalized exponents

\ensuremath{\tau}(q)

and

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)

governing the decay of the typical values

{Y}_{q}^{\mathrm{typ}}(L)={e}^{\overline{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{Y}_{q}(L)}}\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\tau}(q)}

and disorder-averaged values

\overline{{Y}_{q}(L)}\ensuremath{\sim}{L}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)}

, respectively. To understand the difference between these exponents,

\ensuremath{\tau}(q)\ensuremath{\ne}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)

above some threshold

q>{q}_{c}\ensuremath{\sim}2

, we compute the probability distributions of

y={Y}_{q}(L)∕{Y}_{q}^{\mathrm{typ}}(L)

over the samples: We find that these distributions becomes scale invariant with a power-law tail

1∕{y}^{1+{x}_{q}}

. These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent

\ensuremath{\nu}\ensuremath{\sim}2

DOI: 10.1103/physreve.75.051122

Related papers

Statistics of renormalized on-site energies and renormalized hoppings for Anderson localization in two and three dimensions

Cécile Monthus, Thomas Garel

For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions $d=2,3$ , we study numerically the statistical properties of the renormalized on-site energies $ϵ$ and of the renormalized hoppings $V$ as a function of the linear size $L$ . We find that the renormalized on-site energies $ϵ$ remain finite in the localized phase in $d=2,3$ and at criticality $(d=3)$ , with a finite density at $ϵ=0$ and a power-law decay $1/{ϵ}^{2}$ at large $|ϵ|$ . For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality $(d=3)$ , the statistics of renormalized hoppings $V$ is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions $d=2,3$, we study numerically the statistical properties of the renormalized on-site energies $ϵ$ and of the renormalized hoppings $V$ as a function of the linear size $L$. We find that the renormalized on-site energies $ϵ$ remain finite in the localized phase in $d=2,3$ and at criticality $(d=3)$, with a finite density at $ϵ=0$ and a power-law decay $1/{ϵ}^{2}$ at large $|ϵ|$. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and $u$ a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality $(d=3)$, the statistics of renormalized hoppings $V$ is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

arXiv:0905.4127Physical Review BJul 16, 200928 citations

Multifractal statistics of the local order parameter at random critical points: Application to wetting transitions with disorder

Cécile Monthus, Thomas Garel

Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).

arXiv:0705.0223Physical Review EAug 21, 200712 citations

Random transverse field Ising model in dimensiond> 1: scaling analysis in the disordered phase from the directed polymer model

Cécile Monthus, Thomas Garel

For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.

arXiv:1110.3145Journal of Physics A Mathematical and TheoreticalFeb 16, 201226 citations

Multifractal finite-size scaling at the Anderson transition in the unitary symmetry class

Jakob Lindinger, Alberto Rodríguez

We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to ${L}^{3}={150}^{3}$ , yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.We use multifractal finite-size scaling to perform a high-precision numerical study of the critical properties of the Anderson localization-delocalization transition in the unitary symmetry class, considering the Anderson model including a random magnetic flux. We demonstrate the scale invariance of the distribution of wave-function intensities at the critical point and study its behavior across the transition. Our analysis, involving more than $4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ independently generated wave functions of system sizes up to ${L}^{3}={150}^{3}$, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.

arXiv:1707.05701Physical review. B./Physical review. BOct 5, 201723 citations

Universality in Anderson localization on random graphs with varying connectivity

Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio

We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$ , and that of other works which suggest that there is no such region. We find that, although no separate $W_{E}$ exists from $W_C$ , the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$ , while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$ . The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.We perform a thorough and complete analysis of the Anderson localization transition on several models of random graphs with regular and random connectivity. The unprecedented precision and abundance of our exact diagonalization data (both spectra and eigenstates), together with new finite size scaling and statistical analysis of the graph ensembles, unveils a universal behavior which is described by two simple, integer, scaling exponents. A by-product of such analysis is a reconciliation of the tension between the results of perturbation theory coming from strong disorder and earlier numerical works, which seemed to suggest that there should be a non-ergodic region above a given value of disorder $W_{E}$ which is strictly less than the Anderson localization critical disorder $W_C$, and that of other works which suggest that there is no such region. We find that, although no separate $W_{E}$ exists from $W_C$, the length scale at which fully developed ergodicity is found diverges like $|W-W_C|^{-1}$, while the critical length over which delocalization develops is $\sim |W-W_C|^{-1/2}$. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.

arXiv:2205.14614May 29, 20221 citation

MULTIFRACTAL ANALYSIS OF BROADLY DISTRIBUTED OBSERVABLES AT CRITICALITY

Martin Janßen

The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments. 1.The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent β and the correlation length exponent ν is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent ν are obtained by the typical order parameter exponent α0 and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments. 1.

Jan 1, 1994172 citations

We consider the model of the directed polymer in a random medium of dimension

1+3

{w}_{L}(\stackrel{P\vec}{r})

of the endpoint of the polymer of length

L

via the moments

{Y}_{q}(L)={\ensuremath{\sum}}_{\stackrel{P\vec}{r}}\phantom{\rule{0.2em}{0ex}}{[{w}_{L}(\stackrel{P\vec}{r})]}^{q}

. We measure the generalized exponents

\ensuremath{\tau}(q)

and

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)

governing the decay of the typical values

{Y}_{q}^{\mathrm{typ}}(L)={e}^{\overline{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{Y}_{q}(L)}}\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\tau}(q)}

and disorder-averaged values

\overline{{Y}_{q}(L)}\ensuremath{\sim}{L}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)}

, respectively. To understand the difference between these exponents,

\ensuremath{\tau}(q)\ensuremath{\ne}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)

above some threshold

q>{q}_{c}\ensuremath{\sim}2

, we compute the probability distributions of

y={Y}_{q}(L)∕{Y}_{q}^{\mathrm{typ}}(L)

over the samples: We find that these distributions becomes scale invariant with a power-law tail

1∕{y}^{1+{x}_{q}}

\ensuremath{\nu}\ensuremath{\sim}2

d=2,3

, we study numerically the statistical properties of the renormalized on-site energies

ϵ

and of the renormalized hoppings

V

as a function of the linear size

L

. We find that the renormalized on-site energies

ϵ

remain finite in the localized phase in

d=2,3

and at criticality

(d=3)

, with a finite density at

ϵ=0

and a power-law decay

1/{ϵ}^{2}

at large

|ϵ|

. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and

u

a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality

(d=3)

, the statistics of renormalized hoppings

V

is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.

{L}^{3}={150}^{3}

, yields accurate estimates for the critical exponent of the localization length, $\ensuremath{\nu}=1.446(1.440,1.452)$, the critical value of the disorder strength, and the multifractal exponents.

W_{E}

which is strictly less than the Anderson localization critical disorder

W_C

, and that of other works which suggest that there is no such region. We find that, although no separate

W_{E}

exists from

W_C

, the length scale at which fully developed ergodicity is found diverges like

|W-W_C|^{-1}

, while the critical length over which delocalization develops is

\sim |W-W_C|^{-1/2}

. The separation of these two scales at the critical point allows for a true non-ergodic, delocalized region. In addition, by looking at eigenstates and studying leading and sub-leading terms in system size-dependence of participation entropies, we show that the former contain information about the non-ergodicity volume which becomes non-trivial already deep in the delocalized regime. We also discuss the quantitative similarities between the Anderson transition on random graphs and many-body localization transition.