Anderson localization and extreme values in chaotic climate dynamics
AAnderson localization and extreme values in chaotic climatedynamics
John T. Bruun ∗ College of Engineering, Mathematics and Physical Sciences,University of Exeter, Exeter, UK. College of Life and Environmental Sciences,University of Exeter, Penryn Campus, Penryn, UK.
Spiros N. Evangelou † Physics Department, University of Ioannina, Greece. (Dated: November 12, 2019)
Abstract
This work is a generic advance in the study of delocalized (ergodic) to localized (non-ergodic)wave propagation phenomena in the presence of disorder. There is an urgent need to better un-derstand the physics of extreme value process in the context of contemporary climate change.For earth system climate analysis General Circulation Model simulation sizes are rather small, 10to 50 ensemble members due to computational burden while large ensembles are intrinsic to thestudy of Anderson localization. We merge universal transport approaches of Random Matrix The-ory (RMT), described by the characteristic polynomial of random matrices, with the geometricaluniversal extremal types max stable limit law. A generic ensemble based random Hamiltonianapproach allows a physical proof of state transition properties for extreme value processes. In thiswork Anderson localization is examined for the extreme tails of the related probability densities.We show that the Generalized Extreme Value (GEV) shape parameter ξ is a diagnostic tool thataccurately distinguishes localized from delocalized systems and this property should hold for allwave based transport phenomena. a r X i v : . [ n li n . C D ] N ov NTRODUCTION
We introduce a new way to view Anderson localization by focusing at the extremes ofphysical quantities rather than at the mean values and apply it to the climate. Our worldsclimate is changing at an alarming rate [1]. These changes can and appear to be causingmultifaceted impacts which can alter societal resilience [2]. The future state of the climatecan now be simulated based on highly sophisticated Global Circulation Models (GCM’s)that involves a substantial computational effort to study extreme climate phenomena inthe presence of disorder. The chaotic and turbulent phenomena of the Earth require ageneric understanding to enable predictive capacity [3–5]. Universal extreme propertieshave been studied for random energy models and Burgers turbulence in settings whereeigenmode interactions are specified [6–8]. However the physical explanation of the typeof extreme value processes that occurs remains incomplete and we need to better establishhow these alter through system state transitions κ → κ , such as in tipping points of theclimate system [9]. Anderson localization for the absence of wave propagation in solids wasestablished in the late 1950s (Anderson, 1958) [10]. The corresponding Anderson transitionin the presence of disorder is explained via the scaling theory of Abrahams et al. (1979) [11].This ergodicity breaking transition is related to destructive wave interference and for strongenough disorder implies the absence of wave transmission.It also provides us with a transportframework through which state transitions and extreme phenomena driven by the level ofdisorder can be studied [12]. On one side of the transition the delocalized random systemshave correlated energy eigenvalues and ergodic wave flow called quantum chaotic which isunderstood via RMT [13, 14]. In the opposite limit of Anderson localization the systemeigenvalues are uncorrelated described by Poisson statistics. Recent work by Fyodorov et al.(2008, 2012, 2016) [8, 15, 16] has looked at extreme processes from the RMT perspective.The corresponding random matrices are described via their characteristic polynomial D N ( E )which encapsulates all the system N eigenvalues and energies E .The Extreme Value Process (EVP) of any system can be characterised by the shapeparameter ξ that represents the extreme edge of the hysteresis characteristic [17–20] and itis widely used in design engineering such as flood prevention work to assess extrapolationand risk properties. The EVP distribution is justified from a linear renormalisation andextremal types or max stable limit law based on geometric universality [21–24]. A result2f this max stable limit law is that EVP systems, represented by the largest measurements Z (such as the maximum of an ensemble), are described by the GEV distribution function G ( z ; ξ ) where the tail type: Weibull, Gumbel or Frchet is distinguished by ξ . The typeof distribution tells us the sensitivity of the tail process and provides a measure of howextreme variability changes in general for any dynamical system. In the context of systemstate changes κ → κ , (Young, 2011) [3] the extremes can alter. More erratic extremesunder an altered scenario κ such as in climatic change would correspond to a heavier tail ξ ( κ ) > ξ ( κ ). We extend the characteristic polynomial approach [8, 15, 16] to generallyestablish the shape ξ of extremes. We examine state transition properties for Andersonlocalization including superconductivity [25–27] and for a chaotic RMT diffusive system,such as a model of a black hole with added disorder [28] EQUIVALENCE BETWEEN CLIMATIC PHENOMENA AND ELECTRON DIS-ORDER PROPERTIES
It is natural first to ask and discuss why classical wave systems for climate should haveanything to do with properties related to the flow of electron waves in disordered media?The answer is quite simple: the wave properties of quantum electrons and the classicalwaves in climate follow the same physical laws so that techniques developed for wave prop-agation in the quantum world can be also explored in climate. Anderson localization forwave propagation in the presence of disorder also appears for ultrasound waves, microwaves,light, etc.
The question of classical localization: a theory of white paint?
Anderson (1985)[29] explains the possibility of observing light localization in TiO2 samples. Anderson lo-calization of classical waves turns out to be rather difficult to observe since nature appliessome severe constraints, such as rather small cross-sectional scattering areas and absorption[30, 31]. These properties of disordered systems are thought to occur in many other settings:Anderson localization also occurs in many-body settings other than the real space of oneparticle. In strange metals Patel and Sachdeev (2019) [28] recently showed that significantamounts of disorder are present and electrical insulators governed by similar laws to chaoticmetals exist. The Sachdeev-Ye-Kitaev model which connects Hamiltonian disorder modelsto the physics of the black holes offers a link of many-particle quantum entanglement tomany-body localization [32]. By analogy the physics and wave transport phenomena in3he Earth system and the climate is regulated through disorder introduced through smallscale wave activation processes and surface roughness. This can lead to phenomena suchas convective self-aggregation and spatial clumping transport processes [33]. Recently therehas been a resurgence of interest to assess wave properties of geophysical climate systems(Delplace, et al., 2017; Bruun et al., 2017; Skkala and Bruun, 2018) [20, 34, 35]. In theseanalyses universal and scale invariant properties help to identify the dynamics and the re-sulting wave processes. Equatorial waves in oceans which regulate climate have been shownto be driven by dynamics similar to the so-called topological insulators [34] where unlikenormal insulators a flow of states protected by symmetry occurs only on the surface of asample. The bulk transport medium is insulating having localized states and delocalizedwaves unaffected by disorder travel around the edges of the system due to topology. Forexample in the 2D quantum Hall effect [36] such edge states of electrons are protected by astrong magnetic field which breaks time-reversal symmetry and determine the highly accu-rate Hall conductance. In climate the equivalent role of the magnetic field is played by theCoriolis effect caused by planets rotation, and topological waves that are important for thedominant climatic processes on the Earth flow around the equator [34]. In particular windinduced oceanic Kelvin and Rossby waves travel along the Pacific equator, scatter and reflectat its edges and combine to create the Pacific El Nio Southern Oscillation (ENSO) resonanceproperty [20, 35, 37–40]. Recently Bruun et al. (2017) [20] and Skkala and Bruun (2018)[35] established that low frequency eigenmodes appear to be part of the ENSO process.They occur as a sub-harmonic resonance property and can alter systematically througha state transition parameter κ that represents non-linear ocean-atmospheric coupling [3–5, 20, 41, 42]. A future warmer climate could alter the ENSO resonance through a changeto κ promting the question: Is the current instability of the ENSO modes an example ofthe hysteresis characteristic changing in the industrial period?
In other words is the ENSOextreme value process shape parameter changing? Here, we set up a novel framework thatsuch questions can be addressed and possibly answered. With current GCMs the ability tostudy large ensemble extremes systematically across changes of system state is not possible:a typical GCM ensemble has of the order of 10 to 50 members [1]. As such, a theoreticalexplanation of the EVP for a generic physical framework is prompted and the physics ofelectron wave transport provides such a framework. Electrons and their absence (holes)have distinct wave dispersion properites, and the way in which they combine define the4ransport encountered in superconductivity. By analogy, in climate science, tropical oceanKelvin and Rossby waves combine to produce the ENSO phenomena. Andreev reflection(Andreev, 1965) [43] in superconductivity is a wave interaction property where an electronentering a medium forms a Cooper pair which consists of an electron plus a hole which isretro-reflected as a hole outside [26, 27, 44]. The Andreev wave scattering and interactionproperties for normal s -type superconductors can extended to topological current p -typedisordered superconductors [45]. The Bogoliubov-de Gennes (BdG) Hamitonian (Bruun etal., 1994 and 1995)[26, 27] represents this process. The full dynamical system structure isgiven by appropriate N × N random matrices which discretize the available space to N points. In the presence of a magnetic field a complex Hermitian GUE of RMT approximatesa high-dimensional disordered systems, in this setting all to all interactions are included anddistance plays no role. This GUE system exhibits highly correlated eigenvalues leading touniversal statistical features [13, 27, 31, 46] and for appropriate distributions for the max-ima of the characteristic polynomial we identify the ergodic and non-ergodic state of themany-body system. HAMILTONIAN EIGENMODE STRUCTURE AND EXTREMES
The Bogoliubov-de Gennes mean-field Hamiltonian equation reads H o ( W ) ∆ − ∆ ∗ − H ∗ o ( W ) ψ = λψ (1) H o ( W ) represents the electron-wave Hamiltonian in the presence of disorder specified by W , − H ∗ o ( W ) is the Cooper paired electron in the form of an Andreev reflected hole, ∆is the order parameter which represents the superconducting energy-gap that opens in thetransport band structure when the temperature T becomes lower than T c and the materialbecomes superconducting. The coupling matrix between electrons and holes for normal s -type and for topological p -type superconductors is∆ = ∆ , ∆ = − ∆ ...∆ ∗ − ∆ ... ∆ ∗ . . . (2)respectively. In the case of ∆ = 0 the system is not superconducting and the electronHamiltonian H o ( W ) can exhibit Anderson localization properties depending on the disorder5 . The studied system is shown in Fig. 1. In the delocalized regime the N × N Hermitian
FIG. 1.
The finite quasi-1D setup.
The transport flows in and out from the left to the right hand side ofthe conductor. The resistance and conductance properties which enable assessment of Anderson localizationproperties are studied by the characteristic polynomial of a similar closed system. matrix H (the Hamiltonian) has a probability density function for Gaussian random matrixensembles of RMT P ( H ) ∝ exp {− ( N β/ T r ( H ) } (3)and takes various forms depending on the universality parameter β = 1 , ,
4. The β = 2case studied here corresponds to the unitary GUE limit where time-reversal symmetry isbroken. The characteristic polynomial of the matrix H at an energy E is obtained from thedeterminant D N ( E ) = det ( EI − H ) = N Y j =1 ( E − E j ) , (4)where I is the N × N unit matrix. D N ( E ) is the basic quantity of interest encoding alleigenvalues { E j ; j = 1 , ..., N } and the roots of H are obtained from matrix diagonalization.6he transport quantity of interest is the resistance of the system [47, 48] R N ( E ) = | D N ( E ) | − . (5)Following the definition by Fyodorov and Simm (2016) [16] D N ( E ) is expressed as f { D N ( E ) } = | D N ( E ) | exp {− < log | D N ( E ) | > } (6)where < ... > is average over the ensemble. The extreme value processes (EVP) is an intrinsicproperty of the characteristic polynomial and the maximum embeds the characteristic of the N eigenvalues for a given ensemble i as M i,N = max E ∈ [ − , { log | D i,N ( E ) | − < log | D i,N ( E ) | > } . (7)We create n ensembles of this process so the set of ensemble maxima are M n = { M ,N , M ,N , ..., M n,N } . (8)A linear renormalisation extremal types theorem (Leadbetter, 1983) [21] converts thesemaxima (8) into a more useful representation by scaling the variable as M ∗ n = ( M n − b n ) /a n .The max stable limit theorem (similar to the central limit theorem) gives P r { ( M n − b n ) /a n ≤ z } → G ( z ) , n → ∞ , (9)where the selection of { a n } and { b n } results in the max stable limiting distribution G ( z ) forlimit of M ∗ n . This distribution is generic and does not depend on any individual generatingdistribution function such as the disorder process of strength W . A distribution is calledmax-stable if for every n = 2 , , ... the constants a n > b n exist such that G n ( a n z + b n ) = G ( z ) , n → ∞ , (10)so the max stability property is satisfied by distributions for which the operation of takingsample maxima leads to an identical distribution, apart from a change of scale and location.The apparent difficulty that normalising constants will be unknown in practice is easilyresolved due to the limit in (10) as P r { ( M n − b n ) /a n ≤ z } ≈ G ( z ) (11)7or large enough n , so equivalently P r { M n ≤ z } ≈ G { ( z − b n ) /a n } = G ∗ ( z ) , (12)where G ∗ is another member of the same GEV family. This extremal types theorem enablesapproximation of the distribution of M ∗ n by a member of the GEV family for large n , and sothe distribution of M n itself can be approximated by a different member of the same family.This is a useful property for the estimation stage. As the parameters of the distributions G and G ∗ have to be estimated, it is irrelevant in practice that the location and scale parametersof the distributions will be different. The type of GEV distribution, defined by its shape ξ ,will be the same. Due to this the properties of a max stable process are estimated using theGEV parameterisation G ( z ) = exp {− [1 + ξ ( z − µ ) /σ ] − /ξ } = GEV ( z ; µ, σ, ξ ) (13)using likelihood or rank based inference [17, 19, 22–24, 49]. As the shape is the invariantterm we refer to (13) as G ( z ; ξ ). The location parameter has µ ∈ R , the scale parameter σ > ξ ∈ R and 1 + ξ ( z − µ ) /σ >
0. For ξ = 0, the distributionsimplifies to a Gumbel or Type I distribution: G ( z ) = exp {− exp [ − ( z − µ ) /σ ] } (14)which is unbounded. For ξ >
0, it is known as a Frchet or Type II distribution, with abounded lower tail at z = µ − σ/ξ and infinite upper end point. A Frchet distribution hasa heavy upper tail. For ξ <
0, it is known as the Weibull or Type III distribution, with abounded upper tail at z = µ − σ/ξ and infinite lower end point. The GEV type properties ofthese maxima are evaluated below. To estimate the parameters µ , σ , ξ we use the maximumlikelihood estimation approach. The likelihood L is the joint probability of all the ensemblemembers occurring with the given probability is: L ( µ, σ, ξ ) = n Y i =1 P r { Z = z i ; µ, σ, ξ } . (15)Given the GEV parametrisation of Eq. (13) and the ensemble of maximum data, the EVPis estimated by optimising l = logL the log-likelihood function of (15) as l ( µ, σ, ξ ) = − nlogσ − (1 + 1 /ξ ) n X i =1 log [1 + ξ ( z i − µ ) /σ ] − n X i =1 log [1 + ξ ( z i − µ ) /σ ] − /ξ . (16)8he estimates are obtained numerically as analytical maximisation is not possible. Thestandard errors, confidence intervals and profile log-likelihood of the shape l ( ξ ) (which allowsspecific testing of the shape parameter sign) are obtained from (16) using standard likelihoodtheory [49]. The principle of maximum likelihood estimation for a suitable ensemble is toadopt the set of parameters with the greatest likelihood, since of all the range of parametercombinations, this is the one which assigns highest probability to the observed ensemble.This likelihood estimation approach is asymptotically fully efficient, i.e. no other estimationapproaches can have a smaller estimation variance [49, 50], so we can be confident thatthe estimated values (and the standard errors) of µ , σ , ξ are highly accurate. In practiceensemble sizes n ∼ WAVE SCATTERING AND STATE TRANSITIONSLocalization
The system is described by H o ( W ) consists of N sites arranged in a chain with random sitepotential V j , j = 1 , , ..., N and t j,j ± hoppings between the nearest-neighbour sites j, j ± D Anderson model [10] is defined by a tridiagonal random matrix with diagonal matrixelements V j , j = 1 , , ..., N and off-diagonal matrix elements t j − ,j above and below the maindiagonal. We take all the hoppings t j − ,j = 1 which defines the energy scale and the sitepotentials V j ∈ [ − W/ , W/
2] are independent random variables identically distributed witha uniform probability distribution of width W which represents the strength of the disor-der. The eigenvalues of the random system described by H are E j , j = 1 , , ..., N and thecharacteristic polynomial is evaluated via (4). In Fig. 2 a) for a system of size N = 3000 theeigenvalue level spacing P ( s ) distribution with S = ( E j − E j − ) / < S > is shown. The W = 1setting corresponds to a Poisson configuration with P r { S = s } = exp ( − s ). In b) the eigen-value density ρ ( E ) shows the system is close to the ballistic limit ρ ( E ) = 1 /π p / (4 − E )with singularities at E = ±
2. Fig. 2 c) shows ensemble property < log | D N ( E ) | > for n = 3000 verses energy and two single ensemble members of the characteristic polynomialare shown for d) N = 50 and d) N = 3000. Fig. 3 shows the superconductivity s -type casefor N = 500 in the 1 D limit (with uniform disorder, W = 1 and ∆ = 0 . c) d) c) d) c) d)) c) e)) c) c)) c) FIG. 2.
1D Anderson localization W = 1. a) P ( s ) the level spacing between consecutive eigenvalues(red line is the Poisson distribution exp {− S } for localized states). b) Eigenvalue distribution for 1 D ballisticstates, N = 3000. c) The ensemble < log | D N ( E ) | > property and characteristic polynomial f { D N ( E ) } fortwo 1 D chains: d) N = 50 for ballistic and e) N = 3000 for localized systems. distribution a) shows the Poisson limit. The superconducting energy gap is clearly evidentin Fig. 3 b) the density of eigenvalues ρ ( E ), c) the ensemble property < log | D N ( E ) | > and10 FIG. 3. : Superconductivity with ∆ = 0 .
1. a), b) the level spacing and eigenvalue Poisson distributionfor the 1D localized limit with N = 1500. c) The ensemble < log | D N ( E ) | > property and d) characteristicpolynomial f { D N ( E ) } for N = 500. The superconducting energy gap near E = 0 is visible. d) the characteristic polynomial. It is interesting to note that the density of states ρ ( E )shows both the near ballistic limit Poisson characteristic and the superconducting energygap at E = 0. Also note how the additional symmetry imposed by superconductivity witha real order parameter has reduced the complexity of the characteristic polynomial to besymmetric around E = 0. Delocalization
For a general GUE system N × N random Hermitian matrix H = H † ( † is the conjugatetranspose) the entries are sampled from a Gaussian process, N ( µ, σ ), µ the mean and σ H i,i ∼ N (0 , , i = 1 , ..., N (17)and the off-diagonal matrix elements for i > j have H i,j = x i,j + iy i,j ; x i,j , y i,j ∼ N (0 , / . (18)This sampling approach ensures Hermitian symmetry. To find the eigenvalues E j , j =1 , ..., N within ± √ N . In Fig.4 a) we show theGUE level distribution universal characteristic for N = 3000. Note how P r { S = 0 } = 0, i.e. FIG. 4. : Random matrix theory ergodic limit.
The a) GUE level spacing distribution P ( S ) = S π exp ( − S /π ) and b) the Wigner semicircle eigenvalue density with N = 3000. c) the ensemble
500 3000 W = 1 na 0.077 Frchet(0.046, 0.107)1 D weak disorder
500 1000 W = 0 .
01 na -0.211 Weibull(-0.233, -0.189)
1D superconducting
300 1000 W = 1 ∆ = 0 . W = 1 ∆ = 0 . θ ∼ U ( − π, π ) (0.008, 0.184) delocalized RMT limit
500 3000 GUE na -0.070 Weibull(-0.095, -0.045) the 1 D setting for the electron Hamiltonian with W = 1 (Fig. 5 a) the shape parameter ξ > z = µ − σ/ξ . For thesame system but with s -type superconductivity (Fig. 5 b: real order parameter) the shapeparameter ξ > ξ > a) g(z) z z z z z ξ a) b) c) d) e) f) g(z) g(z) g(z) g(z) FIG. 5.
GEV distributions of ensemble maxima. a) to e) histogram and estimated GEV density(black line). 1 D systems: a) W = 1 electron system, N = 3000 and n = 3000. b) superconductivity with∆ = ∆ real, W = 1, N = 300 and n = 1000. c) superconductivity with ∆ = ∆ e iθ random phases 1 D limit, W = 0 .
1, ∆ = 0 . N = 300 and n = 500. d) 1 D Ballistic electron limit W = 0 . N = 50 and n = 1000. e) GUE system with N = 500 and n = 3000. f) For case e) the profile log-likelihood l ( ξ ) (blackline) of ξ , the 95% confidence interval range is given by the drop of the lower blue line (from the maximum)indicating ξ < has occurred. This heavy tail result is consistent with that previously established for the 1 D localized regime [25, 51, 52]. When the disorder level is reduced to the ballistic limit in the1 D Anderson transport setting (Fig. 5 d) then ξ much less than zero indicating a Weibulltype. Then the upper tail becomes lighter than for a Gumbel situation and the characteristicpolynomial has an upper bound at z = µ − σ/ξ . This appears consistent with random energymodel discussion of Bouchaud and Mezard (1997) [7] for systems with correlated eigenvalues.14or the GUE delocalized electron case (Fig. 5 e) ξ < D N ( E )property is an interesting example of a log-correlated Gaussian random field (Fyodorov etal., 2008) [8] and a partially proved conjecture for max { < log | D N ( E ) | > } , in GUE canbe found in Fyodorov et al. (2016) [16]. The delocalization property and its correlatedeigenvalues appear to support the Weibull type for the GUE setting. We test this propertywith the profile log-likelihood l ( ξ ) obtained from Eq. (16) in Fig. 5 f) (the black line). The95% confidence interval is given by the width of the log-likelihood surface where it intersectsthe lower blue line. This confirms the shape parameter is negative (GEV diagnostics insupplementary: Fig. A1). Our results show that the extreme shape varies with the systemstate property κ . In particular for the transition from localized to de-localized, the maxstable limit property (9,13) implies that the mobility edge corresponds to the Gumbel typewith ξ ∼
0. This work establishes the ξ ( κ ) structure using the full GEV representation andthe EVP for a range of universal symmetry breaking changes κ → κ , known to exist inquantum transport problems. It follows generically from our analysis and the proof abovethat in extreme value process studies it is essential to include an assessment of ξ the shapeparameter. In climate systems we propose that changes in system state in the extremes offuture climate will be measurable in the magnitude and sign of the shape parameter. Forexample a transition to a heavy tailed process could indicate a form of localization (Ludlamet al., 2005) [31] in the associated climatic phenomena. CONCLUSIONS
The interesting possibility arises that some of the phenomena observed due to wavepropagation in climate could be a consequence of disorder. We link Anderson localization,a phenomenon at the heart of quantum physics, to the analysis of the climate. Andersonlocalization wave properties in disordered media, although difficult to observe for classicalwaves, arises from multiple wave scattering through a disordered medium. We define a newway to view the Anderson transition by focusing on the extremes of physical quantitiesrather than at the mean values and their fluctuations. Our approach is developed in lowdimensional localized and infinite dimensional delocalized systems to assess a wide rangeof extreme physical processes. We have shown that ξ > ξ < ξ ∼
0. Our results can beextended to include non-ergodic multifractal states known at criticality between localizedand delocalized regimes. More generally we have established that ξ ( κ ) can change whenthe dynamical system fundamentally changes its physical structure κ → κ and that thisis a universal result. As a consequence we can assess the extreme shape parameter ξ ofother systems, such as in the earths climatic system to help better characterise extreme riskscenarios. In conclusion, we have shown via the max stable law how to study Andersonlocalization encountered for wave phenomena in many random settings including climate. ACKNOWLEDGEMENTS
The authors gratefully acknowledge the UK Research Councils funded Models2Decisionsgrant (M2DPP035: EP/P01677411), ReCICLE (NE/M00412011) and Newton FundedChina Services Partnership (CSSP grant: DN321519) which helped fund this research. Theauthors also acknowledge useful discussion with Katy L. Sheen on the Earth system trans-port context. The data used in this manuscript is simulated from the analytical formulastated in the manuscript using R and the extreme value likelihood inference with the Rlibrary ismev . JTB and SNE designed the analytical and theoretical basis for this study.JTB set the wider geophysical context of the work performed the analyses and wrote the ini-tial manuscript. Both the authors contributed to the analysis discussion and the completedmanuscript. The authors declare no competing interests. ∗ Corresponding author: [email protected]; http://emps.exeter.ac.uk/mathematics/staff/jb1033 † Second author: [email protected][1] M. Collins, R. E. Chandler, P. M. Cox, J. M. Huthnance, J. Rougier, and D. B. Stephenson,Nature Climate Change , 403 (2012).[2] W. Cai, G. Wang, A. Santoso, M. J. Mcphaden, L. Wu, F. F. Jin, A. Timmermann, M. Collins,G. Vecchi, M. Lengaigne, M. H. England, D. Dommenget, K. Takahashi, and E. Guilyardi,Nature Climate Change , 132 (2015).[3] P. C. Young, Recursive estimation and time series analysis , 2nd ed. (Springer-Verlag BerlinHeidelberg., 2011).
4] H. Tong,
Non-linear time series, a dynamical system approach (Oxford University Press Inc.,New York., 1990).[5] M. J. Feigenbaum, Annals of the New York Academy of Sciences , 330 (1980).[6] B. Derrida, Physical Review B , 2613 (1981).[7] J. P. Bouchaud and M. M´ezard, Journal of Physics A: Mathematical and General , 7997(1997).[8] Y. V. Fyodorov and J. P. Bouchaud, Journal of Physics A: Mathematical and Theoretical ,372001 (2008).[9] T. M. Lenton, H. Held, E. Kriegler, J. W. Hall, W. Lucht, S. Rahmstorf, and H. J. Schellnhu-ber, Proceedings of the National Academy of Sciences of the United States of America ,1786 (2008).[10] P. W. Anderson, Physical Review , 1492 (1958).[11] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Physical ReviewLetters , 673 (1979).[12] F. Evers and A. D. Mirlin, Reviews of Modern Physics , 1355 (2008).[13] E. P. Wigner, The Annals of Mathematics , 548 (1955).[14] S. N. Evangelou, Physical Review Letters , 2550 (1995).[15] Y. V. Fyodorov, G. A. Hiary, and J. P. Keating, Physical Review Letters , 170601 (2012).[16] Y. V. Fyodorov and N. J. Simm, Nonlinearity , 2837 (2016).[17] J. T. Bruun and J. A. Tawn, Journal of the Royal Statistical Society. Series C: AppliedStatistics , 405 (1998).[18] B. Gouldby, D. Wyncoll, M. Panzeri, M. Franklin, T. Hunt, N. Tozer, U. Dornbusch, D. Hames,T. Pullen, and P. Hawkes, in E3S Web of Conferences (2016) p. 01007.[19] L. de Haan and J. de Ronde, Extremes , 7 (1998).[20] J. T. Bruun, J. Allen, and T. Smyth, Journal of Geophysical Research: Oceans , 6746(2017).[21] M. R. Leadbetter, Zeitschrift f¨ur Wahrscheinlichkeitstheorie und Verwandte Gebiete , 291(1983).[22] L. de Haan, in Statistical Extremes and Applications, NATO ASI Series (Series C: Mathe-matical and Physical Sciences) (Springer, Dordrecht, 1984) pp. 31–48.[23] L. de Haan, Communications in Statistics. Stochastic Models , 765 (1994).
24] J. A. Tawn, Biometrika , 397 (1988).[25] C. J. Lambert and S. J. Robinson, Physical Review B , 10391 (1993).[26] J. T. Bruun, V. C. Hui, and C. J. Lambert, Physical Review B , 4010 (1994).[27] J. T. Bruun, S. N. Evangelou, and C. J. Lambert, Journal of Physics: Condensed Matter ,4033 (1995).[28] A. A. Patel and S. Sachdev, Physical Review Letters , 66601 (2019).[29] P. W. Anderson, Philosophical Magazine B: Physics of Condensed Matter; Statistical Mechan-ics, Electronic, Optical and Magnetic Properties B , 505 (1985).[30] A. Lagendijk, B. Tiggelen, and D. Wiersma, Physics Today , 24 (2009).[31] J. J. Ludlam, S. N. Taraskin, S. R. Elliott, and D. A. Drabold, Journal of Physics CondensedMatter , 321 (2005).[32] E. Altman, Nature Physics , 979 (2018).[33] A. A. Wing and K. A. Emanuel, Journal of Advances in Modeling Earth Systems , 59 (2014).[34] P. Delplace, J. B. Marston, and A. Venaille, Science , 1075 (2017).[35] J. Sk´akala and J. T. Bruun, Journal of Geophysical Research: Oceans , 116602 (2018).[36] S. Sarma and A. Pinczuk, Perspectives in quantum Hall effects: novel quantum liquids inlow-dimensional semiconductor structures (Wiley, New York, 1997).[37] M. A. Cane and S. E. Zebiak, Science , 1085 (1985).[38] M. J. Suarez and P. S. Schopf, Journal of the Atmospheric Sciences , 3283 (1988).[39] E. Tziperman, L. Stone, M. A. Cane, and H. Jarosh, Science , 72 (1994).[40] M. Munnich, M. A. Cane, and S. E. Zebiak, Journal of the Atmospheric Sciences , 1238(1991).[41] R. Hilborn, Chaos and nonlinear dynamics , 2nd ed. (Oxford University Press, Oxford, U.K.,2000).[42] P. Young, P. Mckenna, and J. T. Bruun, International Journal of Control , 1837 (2001).[43] A. Andreev, Soviet Physics Jetp , 6 (1965).[44] C. W. Beenakker, Reviews of Modern Physics , 731 (1997).[45] X. L. Qi, T. L. Hughes, S. Raghu, and S. C. Zhang, Physical Review Letters , 187001(2009).[46] S. N. Evangelou and J. L. Pichard, Physical Review Letters , 1643 (2000).
47] V. M. Gasparian, B. L. Altshuler, A. G. Aronov, and Z. A. Kasamanian, Physics Letters A , 201 (1988).[48] B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Phys. Rev. Lett. , 2803 (1997).[49] S. Coles, An introduction to statistical modelling of extreme values (Springer, London, UK,2001).[50] Y. Pawitan,
In all likelihood: statistical modelling and inference using likelihood (Oxford Uni-versity Press, Oxford, U.K., 2013).[51] A. M. Somoza, M. Ortu˜no, and J. Prior, Physical Review Letters , 116602 (2007).[52] J. T. Bruun, Statistical properties of mesoscopic superconductors , Phd thesis, Lancaster (1994). upplemental material:Anderson localization and extreme values in chaotic climatedynamics John T. Bruun
College of Engineering, Mathematics and Physical Sciences,University of Exeter, Exeter, UK. College of Life and Environmental Sciences,University of Exeter, Penryn Campus, Penryn, UK.
Spiros N. Evangelou
Physics Department, University of Ioannina, Greece. (Dated: November 12, 2019) a r X i v : . [ n li n . C D ] N ov D SYSTEM CHARACTERISTIC POLYNOMIAL VIA RECURSION
For the 1 D system assessed in this work the characteristic polynomial can also be derivedvia the recursion relation D j ( E ) = ( E − V j ) D j ( E ) − t j − ,j D j − ( E ) , j = 1 , , ..., N.D − ( E ) = 0 , D ( E ) = 1 . (A.1)The corresponding determinants are D ( E ) = E − V = ( E − V ) D ( E ) D ( E ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E − V − t − t E − V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( E − V ) D ( E ) − t , D ( E ) D ( E ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E − V − t , − t , E − V − t , − t , E − V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( E − V ) D ( E ) − t , D ( E ) , ... (A.2)This gives the same results as for (4). 2 a) b) c) d) Figure A 1.
Diagnostics for GUE ensemble shape parameter (for Fig 5 e and f ).
The a)probability-probability and b) quantile-quantile plots here indicate that the GEV parametric model representthe raw ensemble of maxima well. c) The return level and the 95% confidence interval range show the smallnegative shape parameter as a curvature in that graph. d) The ensemble histogram and fitted GEV (solidline). a) b) c) d) Figure A 2.
Diagnostics for D ballistic case. W = 0 . , n = 1000 , N = 500 . The a) probability-probability and b) quantile-quantile plots here indicate that the GEV parametric model represent the rawensemble of maxima well. c) The negative shape parameter is evident in the curvature of the return level.d) The ensemble histogram and fitted GEV (solid line). a) b) Figure A 3.
Diagnostics for D ballistic case. W = 0 . , n = 1000 , N = 500 . a) The histogramand fitted GEV (solid line). b) This shows the profile log-likelihood of the shape parameter and that thisensemble is a Weibull type.a) The histogramand fitted GEV (solid line). b) This shows the profile log-likelihood of the shape parameter and that thisensemble is a Weibull type.