Signum Function Method for Generation of Correlated Dichotomic Chains
S.S.Apostolov, F.M.Izrailev, N.M.Makarov, Z.A.Mayzelis, S.S.Melnyk, O.V.Usatenko
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Signum Function Method for Generation of Correlated Dichotomic Chains
S. S. Apostolov, F. M. Izrailev, ∗ N. M. Makarov, † Z. A. Mayzelis, S. S. Melnyk, and O. V. Usatenko ‡ A. Ya. Usikov Institute for Radiophysics and Electronics,Ukrainian Academy of Science, 12 Proskura Street, 61085 Kharkov, Ukraine Instituto de F´ısica, Universidad Aut´onoma de Puebla,Apartado Postal J-48, Puebla, Pue., 72570, M´exico (Dated: November 11, 2018)We analyze the signum-generation method for creating random dichotomic sequences with pre-scribed correlation properties. The method is based on a binary mapping of the convolution ofcontinuous random numbers with some function originated from the Fourier transform of a binarycorrelator. The goal of our study is to reveal conditions under which one can construct binarysequences with a given pair correlator. Our results can be used in the construction of superlatticesand waveguides with selective transport properties.
PACS numbers: 05.40.2a, 02.50.Ga, 87.10.1e
I. INTRODUCTION
The study of properties of disordered complex systemswith spatial and/or temporal correlations is one of thehot topics in modern physics. Recently, much attentionwas paid to the related problem of how to construct dis-ordered materials with specific transport properties thatare due to underlying correlations in a disorder. One ofthe important applications of this problem is a creation ofelectron nano-devices, optic fibers, rough surfaces, acous-tic and electromagnetic waveguides with selective trans-port properties.It is known (see, for instance, [1, 2, 3, 4]) that manyof properties of systems with weak disorder are de-termined by the binary (pair or two-point) correlationfunction of a corresponding random process. Recentlyit was found [5, 6, 7] that specific long-range corre-lations in disordered potentials can lead to anomaloustransport properties. To date, there exist a numberof algorithms for generating long-range correlated se-quences with prescribed correlations. Among such al-gorithms the most widespread one is the convolutionmethod [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In thismethod random elements in the generated chain can beof any value from −∞ to ∞ . However, in many appli-cations it is more convenient to construct the sequenceswith finite number of random values. An important ex-ample of such system, being occurred in nature, is a se-quence of nucleotides in a DNA molecule, consisting offour elements only.The simplest case of a random sequence of finite ele-ments is a stochastic dichotomic (binary) chain of only ∗ Electronic address: [email protected] † Electronic address: [email protected]; On sabbatical leavefrom Instituto de Ciencias, Universidad Aut´onoma de Puebla, Priv.17 Norte No. 3417, Col. San Miguel Hueyotlipan, Puebla, Pue.,72050, M´exico. ‡ Electronic address: [email protected] two elements. In contrast with the case of sequences ofcontinuous elements, the problem of a construction of bi-nary sequences with given correlation properties turnsout to be tricky. As was recently shown in Ref.[16],there is a serious restriction on the type of pair correla-tors in binary sequences, in contrast with the sequenceswith continuous distribution of their elements. Manyof related results for binary sequences can be found inRefs.[17, 18, 19].A direct way to create dichotomic correlated sequencesis to apply the signum function to the sequence of con-tinuous values, obtained with the convolution method.This method is based on the convolution of a white-noise sequence with some function that is determinedby the desired pair correlator. However, as was numeri-cally found in [20], the created binary sequence turns outto have the pair correlator different from the expectedone. To date, it remains unclear how to construct binarysequences having the same pair correlators as in the se-quences with continuous values of their elements. In spiteof its quite simple form, the signum function method isnot rigorously analyzed in the literature.The present paper makes an effort to clarify this prob-lem. Our aim is to understand under what conditions thediscussed method allows to construct binary sequenceswith a desired pair correlator. We perform a detailedtheoretical analysis of the restrictions arising for binarysequences with long-range correlations. Main attentionis paid to the step-wise power spectrum resulting in apower decay of correlations. This type of correlations isextremely important in various applications, such as acreation of devices with selective transport properties.The paper is organized as follows. Section 2 is devotedto general properties of binary sequences. In particu-lar, we derive some of the conditions restricting the formof pair correlators that a binary sequence can have. InSection 3 we describe in details the signum-generationmethod and derive basic relations needed for a furtheranalysis. In next Section 4 we analyze the case of a bal-anced (unbiased) dichotomic sequence, i.e., the sequencewith the zero mean-value. Here we also display the re-strictions related to the discussed method. Our mainfindings are reported in Section 5 where we consider themost interesting case of long-range correlations with thestep-wise spectrum. In Section 6 we ask a question aboutgeneral restrictions appearing in the case of power de-cay of correlations for binary sequences. In last Sectionwe give some additional remarks concerning binary se-quences, and summarize our results. The Appendicescontain some of details of analytical and numerical cal-culations.
II. NECESSARY CONDITIONS
Let us start with generic properties of dichotomic se-quences, not associated with specific choice of a genera-tion method. In view of constructing the sequences witha given pair correlator, it is of great importance to knowwhat are restrictions on the type of correlators allowedfor dichotomic sequences. To shed light on this problem,here we consider a statistically homogeneous random se-quence of symbols s n consisting of the values “ −
1” and“1”, s n = {− , } , n ∈ Z = . . . , − , − , , , , . . . (1)The canonical definition of the correlation function readsas follows C s ( r ) ≡ s n s n + r − s = C s (0) K s ( r ) , (2)where s ≡ s n and C s (0) ≡ s n − s are the mean valueand variance of s n , respectively. Taking into account apeculiar property s n = s n = 1 (3)of our dichotomic sequence s n , one obtains C s (0) = 1 − s . (4)It should be emphasized that the direct relation betweenthe mean value and higher one-point moments is a spe-cific property of any dichotomic chain. For instance, forthe binary chain ε n consisting of “0” and “1” we have therelation C ε (0) = ε (1 − ε ) since ε n = ε n in this case. Onthe contrary, for a sequence of continuous random num-bers of the Gaussian type the variance and mean valueare independent parameters .To proceed with the two-point moments, we associatethe correlator C s ( r ) with the probabilities of two symbolswith the same or opposite signs, occurring at the distance r , 4 P ( ± , . . . |{z} r − , ±
1) = ( s ± + C s ( r ) , (5a)4 P ( ± , . . . |{z} r − , ∓
1) = 1 − s − C s ( r ) . (5b) These relationships also are peculiar properties solely ofdichotomic sequences, they are drastically distinct fromthe analogous relations for other random processes (see,e.g., Eq. (B8) for the two-point probability density ofthe Gaussian process). This fact is strictly confirmed bystraightforward calculations of Eqs. (5).In order to obtain the expression (5a), one should writethe average ( s n ± s n + r ±
1) via the correlator (2),( s n ± s n + r ±
1) = s n s n + r ± s n ± s n + r + 1= C s ( r ) + ( s ± . (6)On the other hand, the same average can be calculatedwith the use of two-symbol probabilities,( s n + 1)( s n + r + 1) = 2 · · P (1 , . . . |{z} r − , · · P ( − , . . . |{z} r − ,
1) + 2 · · P (1 , . . . |{z} r − , − · · P ( − , . . . |{z} r − , −
1) = 4 P (1 , . . . |{z} r − , . (7)( s n − s n + r −
1) = 0 · · P (1 , . . . |{z} r − , − · · P ( − , . . . |{z} r − ,
1) + 0 · ( − · P (1 , . . . |{z} r − , − − · ( − · P ( − , . . . |{z} r − , −
1) = 4 P ( − , . . . |{z} r − , − . (8)Then, the combination of Eqs. (6), (7), and (8) results inthe equality (5a).Similarly, in order to derive the expression (5b), onecan write,( s n ± s n + r ∓
1) = s n s n + r ∓ s n ± s n + r − C s ( r ) + s − . (9)Again, the above average can be calculated employingtwo-symbol probabilities. Specifically,( s n + 1)( s n + r −
1) = 2 · · P (1 , . . . |{z} r − , · · P ( − , . . . |{z} r − ,
1) + 2 · ( − · P (1 , . . . |{z} r − , − · ( − · P ( − , . . . |{z} r − , −
1) = − P (1 , . . . |{z} r − , − . (10)( s n − s n + r + 1) = 0 · · P (1 , . . . |{z} r − , − · · P ( − , . . . |{z} r − ,
1) + 0 · · P (1 , . . . |{z} r − , − − · · P ( − , . . . |{z} r − , −
1) = − P ( − , . . . |{z} r − , . (11)From Eqs. (9), (10), and (11) it follows the equality (5b).Now, with the use of expressions (5a) we express thecorrelation function C s ( r ) via the probabilities to occurthree symbols, C s ( r ) + ( s ± = 4 X a = − , P ( ± , . . . |{z} r ′ − , a, . . . |{z} r − r ′ − , ± . (12)Probability P ( ± , . . . |{z} r ′ − , a, . . . |{z} r − r ′ − , ±
1) is smaller thanboth probabilities P ( ± , . . . |{z} r ′ − , a ) and P ( a, . . . |{z} r − r ′ − , ± C s ( r ) + ( s ± X a = − , min (cid:8) P ( ± , . . . |{z} r ′ − , a ) , P ( a, . . . |{z} r − r ′ − , ± (cid:9) = min (cid:8) − s − C s ( r ′ ) , − s − C s ( r − r ′ ) (cid:9) + min (cid:8) ( s ± + C s ( r ′ ) , ( s ± + C s ( r − r ′ ) (cid:9) = 1 − s + min (cid:8) − C s ( r ′ ) , − C s ( r − r ′ ) (cid:9) +( s ± + min (cid:8) C s ( r ′ ) , C s ( r − r ′ ) (cid:9) . (13)Here we again have used Eqs. (5). Then, according tothe evident relationmin { x, y } + min {− x, − y } = −| x − y | , (14)we arrive at the condition | C s ( r ′ ) − C s ( r − r ′ ) | + C s ( r ) − s . (15)Finally, it is convenient to rewrite Eq. (15) for the nor-malized correlator K s ( r ), | K s ( r ′ ) − K s ( r − r ′ ) | + K s ( r ) . (16)Although we have derived this inequality for 0 < r ′ < r , asimple analysis reveals its validity for arbitrary values of r and r ′ . We would like to stress that the condition (16)is applicable for any mean value s of the dichotomic se-quence s n . However, without a loss of generality in whatfollows we consider binary sequences with the zero mean,since the statistical properties of considered sequences donot depend on mean values.In a similar manner one can obtain second inequalitywith the use of Eq. (5b),1 − s − C s ( r ) = 4 X a = − , P ( ± , . . . |{z} r ′ − , a, . . . |{z} r − r ′ − , ∓ X a = − , min (cid:8) P ( ± . . . |{z} r ′ − , a ) , P ( a, . . . |{z} r − r ′ − , ∓ (cid:9) = min (cid:8) − s − C s ( r ′ ) , ( s ∓ + C s ( r − r ′ ) (cid:9) + min (cid:8) ( s ± + C s ( r ′ ) , − s − C s ( r − r ′ ) (cid:9) . (17) Since C s ( r ) = K s ( r ) in the case of s = 0, Eq. (17) gets asimpler form, | K s ( r ′ ) + K s ( r − r ′ ) | − K s ( r ) s = 0 . (18)Thus, applying the necessary conditions (16) and (17),or (18) if s =0, one can identify the functions that can notbe treated as binary correlators of a dichotomic sequence.Inequalities (16), (17) and (18) have to be met for anyvalues of indices r and r ′ and, therefore, they actuallyrepresent an infinite set of necessary conditions. Evi-dently, in every particular case one should choose thestrongest condition. On the other hand, Eqs. (16), (17)and (18) are automatically satisfied if one of the indicesequals zero, r = 0 or r ′ = 0. The same takes place when r = r ′ . Summarizing all these facts, we can combineEqs. (16) and (18) as followsmax {| K s ( r ′ ) ± K s ( r − r ′ ) | ∓ K s ( r ) } , (19) r = 0 , r ′ = 0 , r = r ′ for s = 0 . The symbol max { . . . } implies the absolute maximum ofa function with respect to the indices r , r ′ . Note thatEq. (19) is automatically fulfilled for two limit cases,namely, for the delta-correlated (white noise) chain, andfor the sequences with infinitely long-range correlations.As one can see, for both these cases K s ( r ) = δ r, or K s ( r ) = 1, respectively. III. BINARY VERSUS GAUSSIAN
Now we analyze the construction of a dichotomic se-quence γ n by means of the signum function genera-tion (SFG) method. It uses an intermediate correlateddisorder β n obtained as a convolution of the uncorre-lated Gaussian noise α n and modulation function G ( n ).Specifically, the γ -sequence is defined by γ n = sign( β n ) , (20a) β n = β + ∞ X n ′ = −∞ G ( n − n ′ ) α n ′ . (20b)The initial Gaussian white-noise chain consists ofstochastic variables α n with the zero mean, unit varianceand corresponding probability density. Respectively, α = 0 , α n α n ′ = δ n,n ′ , (21a) ρ A ( α n = α ) = 1 √ π exp( − α / . (21b)The bar over a random symbol or function implies thestochastic average.Evidently, the constructed dichotomic sequence γ n does not change if one normalizes the intermediate se-quence β n by an arbitrary factor. Hence, without anyloss of generality, we can admit the variance of β n beequal to unity. Then, β is the mean value of β n and itscorrelation function K β ( r ), K β ( r ) = ( β n − β )( β n + r − β ) , (22)is normalized to unity, K β (0) = 1. By direct substitutionof Eq. (20b) into definition (22), the correlator K β ( r ) isreadily associated with the modulation function G ( n ), K β ( r ) = ∞ X n = −∞ G ( r − n ) G ( n ) . (23)Owing to evenness of K β ( r ) = K β ( − r ) and in accordancewith Eq. (23), one can also restrict the function G ( n )to the class of even functions, G ( − n ) = G ( n ). Notethat the condition K β (0) = 1 gives rise to the followingnormalization for G ( n ), K β (0) = ∞ X n = −∞ G ( n ) = 1 . (24)It is convenient to pass to the Fourier transform ofEq. (23) with the use of the standard expressions, K β ( r ) = 12 π Z π − π dk K β ( k ) exp( ikr ) , (25a) K β ( k ) = ∞ X r = −∞ K β ( r ) exp( − ikr ) . (25b)The Fourier transform K β ( k ) of the pair correlator K β ( r )introduced here, is known as the power spectrum of ran-dom β -chain. Since the correlator K β ( r ) is real and evenfunction of r , the power spectrum (25b) is real, even, K β ( − k ) = K β ( k ), and non-negative function of the wavenumber k . Analogously to Eq. (25), for the modulationfunction G ( n ) one can define its Fourier transform G ( k ),which is also real and even function, G ( − k ) = G ( k ).The Fourier representation of Eq. (23) reads K β ( k ) = G ( k ) . (26)Thus, we arrive at the following expression for the mod-ulation function G ( n ), G ( n ) = 1 π Z π dk K / β ( k ) cos( kn ) . (27)Evidently, the solution (27) automatically satisfies thenormalization condition (24).Since the initial chain α n is a delta-correlated Gaus-sian noise, the intermediate variables β n also constitute aGaussian random sequence with single-point distributionfunction ρ B ( β ) (see Appendix A), ρ B ( β n = β ) = 1 √ π exp h − (cid:0) β − β (cid:1) / i . (28)In order to reveal statistical properties of the signum-generated dichotomic sequence γ n , one should associate its mean value γ , variance C γ (0) and pair correlator C γ ( r ) with the corresponding independent characteris-tics, namely, the mean value β and the modulation func-tion G ( n ) [or, the same, with the intermediate correlator K β ( r )]. According to the definition of average, one canexpress the mean value γ in terms of β via the errorfunction [21], γ ≡ γ n = Z ∞−∞ dβ ρ B ( β ) sign( β )= r π Z β dx exp( − x / ≡ erf (cid:0) β/ √ (cid:1) . (29)Similarly to Eq. (4), the variance C γ (0) is written inthe form C γ (0) ≡ γ n − γ = 1 − γ . (30)An important characteristic of the stochastic sequence γ n is the correlation function C γ ( r ), C γ ( r ) ≡ γ n γ n + r − γ = C γ (0) K γ ( r ) . (31)Its calculation is performed in Appendix B. Here wewrite down only the final equation that relates the cor-relator K γ ( r ) to K β ( r ),(1 − γ ) K γ ( r )= 2 π Z K β ( r )0 dx √ − x exp (cid:16) − β x (cid:17) . (32)Note that r.h.s. of the latter equation is not elemen-tary function, therefore its analytical study is not simple.However, the case β = 0 allows one to perform completeanalytical analysis. IV. UNBIASED SEQUENCE
If the mean value of intermediate chain β n vanishes, β = 0, then due to Eq. (29), the mean value of generatedsequence γ n vanishes also, γ = 0. In this case the relation(32) turns out to be remarkably simplified, K γ ( r ) = 2 π Z K β ( r )0 dx √ − x = 2 π arcsin[ K β ( r )] . (33)Another equivalent form is K β ( r ) = sin h π K γ ( r ) i . (34)From the above relations one can conclude that the γ -sequence generated with the discussed signum functionmethod, is random. Indeed, the decay of correlationswith an increase of | r | in the intermediate β -chain, alsoresults in the decay of correlations in the generated di-chotomic γ -sequence.Now it is suitable to rewrite Eq. (34) in the Fourierrepresentation, K β ( k ) = S{ K γ } ( k ) . (35)Here the symbol S{·} ( k ) stands for the operator thattransforms the function K ( r ) by the following rule, S{ K } ( k ) ≡ ∞ X r = −∞ sin h π K ( r ) i exp( − ikr ) (36a)= (cid:16) − π (cid:17) + π K ( k )+2 ∞ X r =1 n sin h π K ( r ) i − π K ( r ) o cos( kr ) . (36b)It is important to note that the series (36a) can convergevery slowly. Therefore, in the analytical and numericalanalysis one has to take into account a lot of terms inthe sum in order to obtain correct result. To avoid thisproblem, we have used the following trick that is basedon the second equality (36b). Namely, since K ( r ) → | r | → ∞ , the latter sum converges quite rapidlyaccording to the asymptotic relationsin h π K ( r ) i − π K ( r ) → π K ( r ) , | r | → ∞ . (37)The substitution of Eq. (35) into Eq. (27) yields thefollowing final relation between the modulation func-tion G ( n ) and the correlator K γ ( r ) of the generating di-chotomic noise γ n , G ( n ) = 1 π Z π dk q S{ K γ } ( k ) cos( kn ) . (38)Since the correlator K γ ( r ) is supposed to be known, therelation (38) should be regarded as the expression deter-mining the modulation function G ( n ).As one can see, the SFG method for constructing thecorrelated dichotomic sequence γ n with the zero mean,unit variance and prescribed two-point correlator K γ ( r )reduces to the following steps. First, starting from a de-sirable profile of K γ ( r ) and employing Eqs. (36) and (38),one has to obtain the modulation function G ( n ). After,the correlated sequence γ n can be generated in accor-dance with Eq. (20). However, it is important to takeinto account the restriction that directly follows fromEq. (35). Specifically, since the power spectrum of anyrandom process, in particular K β ( k ), is a non-negativefunction of the wave number k , the function S{ K γ } ( k )also has to be non-negative, S{ K γ } ( k ) ≥ | k | π. (39)This condition becomes apparent from Eq. (38), in whichthe function S{ K γ } ( k ) enters as a radicand. In otherwords, with this method the function K γ ( r ) can be con-sidered as a correlator of a dichotomic random sequence γ n , if and only if S{ K γ } ( k ) is a non-negative function of k . In view of the revealed restriction (39), let us makea qualitative analysis of Eq. (36b). Note that the firstsummand (1 − π/
2) is negative. Therefore, if the spec-trum K ( k ) vanishes within some interval of k and thewhole function S{ K } ( k ) is non-negative there, conse-quently, the third summand is positive, exceeding thevalue ( π/ − ε n = { , } obtained according to theconditional probability P ( ε n = 1 | ε n − ) = ε + ( ε n − − ε ) exp( − k c ) , (40)where the parameter k c is the inverse correlation length.As is known [22], this chain has an exponential pair cor-relator, K exp ( r ) = exp (cid:0) − k c | r | (cid:1) . It is instructive thatthe function S{ K exp } ( k ) takes negative values for small k c below k ∗ c ≈ . . . . , see the data in Fig. 1. Thus,the SFG method can reproduce exponential correlatoronly for k c > k ∗ c , however, not for small k c in the mostinteresting region of long-range correlations. *c k=0k=k= k=3 k c S { e x p (- k c | r | ) } ( k ) k= FIG. 1: (Color online) Dependence S ˘ exp( − k c | r | ) ¯ ( k ) on thecorrelation parameter k c for several values of wave number k . V. LONG-RANGE CORRELATORS WITHSTEP-WISE SPECTRUMA. Maximal Jump
Here we demonstrate that the discussed method can-not be applied for a construction of dichotomic sequenceswith long-range correlators resulting in the step-wisepower spectrum K γ ( r ) = sin( k c r ) k c r , (41a) K γ ( k ) = πk c Θ( k c − | k | ) , < k c π, | k | π. (41b)This kind of correlations is of specific interest in view ofapplications to 1D disordered superlattices with a selec-tive transport, see, e.g. [6, 7]. Here k c is the correla-tion parameter (inverse correlation length) to be speci-fied, and Θ( x ) implies the Heaviside unit-step function,Θ( x <
0) = 0 and Θ( x >
0) = 1. c k c k c k c k S { s i n ( k c r ) / k c r } ( k ) k FIG. 2: (Color online) Dependence S ˘ sin( k c r ) /k c r ¯ ( k ) on k for different values of k c . First, we analyze the values of k c within the interval0 < k c < π . Two specific cases of k c = 0 and k c = π willbe considered afterwards.In the analysis of the SFG method the crucial charac-teristic is the function S{ K γ } ( k ) defined by Eq. (36). Ithas to be non-negative for any value of the argument k within the interval | k | π , see Eq. (39). In the case ofthe long-range correlator (41a) it is suitable to use thefollowing explicit expression for S{ K γ } ( k ), S (cid:26) sin( k c r ) k c r (cid:27) ( k ) = (cid:16) − π (cid:17) + π k c Θ( k c − | k | )+2 ∞ X r =1 n sin h π k c r ) k c r i − π k c r ) k c r o cos( kr ) . (42) It is remarkable that the summand in the last term be-haves as π / k c r when r → ∞ . Hence, at finite k c thesum converges quite rapidly and uniformly . Therefore,the sum is a continuous function of k , in particular, at k = k c , and can be easily calculated numerically.In Fig. 2 the behavior of the radicand (42) in Eq. (38) isshown for several values of the inverse correlation length k c . Since S{·} ( k ) is an even function of the wave number k , the discussion can be restricted by the interval 0 k π . From Eq. (42) and Fig. 2 one can draw the followingconclusions.1. Due to the last term in expression (42) the function S (cid:8) sin( k c r ) /k c r (cid:9) ( k ) increases with an increase of k for all k c within both intervals (0 , k c ) and ( k c , π ).2. The negative jump of S (cid:8) sin( k c r ) /k c r (cid:9) ( k ) occursat k = k c , at the same point where the power spec-trum (41b) has a jump. The maximal and minimalvalues of the function are achieved at k = k c − k = k c + 0, respectively. The jump is exclusivelyformed by the second term in Eq. (42). Therefore,its value reads S (cid:8) sin( k c r ) /k c r (cid:9) ( k = k c − −S (cid:8) sin( k c r ) /k c r (cid:9) ( k = k c + 0) = π / k c . (43)3. The minimal value S (cid:8) sin( k c r ) /k c r (cid:9) ( k = k c + 0)is negative for all finite values of k c within the in-terval 0 < k c < π . The complete dependence ofpositive function −S (cid:8) sin( k c r ) /k c r (cid:9) ( k c + 0) on k c is depicted in Fig. 3. - S { s i n ( k c r ) / k c r } ( k c + ) k c FIG. 3: (Color online) Dependence −S ˘ sin( k c r ) /k c r ¯ ( k c +0)vs k c in log-scale. Item 3 displays clearly that for all finite values k c < π the correlator and corresponding power spectrum (41)cannot be created by making use of the discussed method.Indeed, the modulation function G ( n ) turns out to be ofcomplex value, see Eq. (38).Now we determine the values of k c for which the func-tion (41a) can not be the pair correlator of a dichotomicsequence regardless of the method of generation. Thiscan be done with the use of the results of Sec. II.First, we demonstrate analytically that this functioncannot be the correlation function of a dichotomic se-quence s n with arbitrary mean value s for small but fi-nite values of k c , namely, for 0 < k c ≪
1. To this end,we take Eq. (16) at r ′ = 1 and r = r a = [ a/k c ], where [ x ]is the integer part of x and a is a constant, (cid:12)(cid:12)(cid:12) sin k c k c − sin k c ( r a − k c ( r a − (cid:12)(cid:12)(cid:12) + sin k c r a k c r a . (44)Being expanded in small parameter k c , this conditionreads (cid:16) cos aa − sin aa (cid:17) k c + O ( k c ) . (45)It is evident that the l.h.s. of Eq. (45) can be positiveat some values of a (e.g., for a = 2 π + π/ k c for the case s = 0. Rewriting itat r = 2 and r ′ = 1, we get4 sin k c − sin 2 k c k c . (46)Numerical analysis shows that this condition does nothold true for all finite k c from the interval 0 < k c < k ∗ ,where k ∗ ≈ . . . . . It is met only at k c > k ∗ . However,since Eq. (18) is just necessary condition, one cannotguarantee an existence of the correlator (41a) even at k c > k ∗ .At the critical point k c = π the correlator and spec-trum (41) reduce to K γ ( r ) = δ r, and K γ ( k ) = 1.This gives rise to the relations, S (cid:8) δ r, (cid:9) ( k ) = 1 and G ( n ) = δ n, , hence, β n = α n . Consequently, one canapply the SFG. However, this specific case of k c = π is not interesting because from the Gaussian white-noise α n -sequence the method fabricates the dichotomic chain γ n again of white-noise type.As to the singular point k c = 0, here we have K γ ( r ) =1 for the correlator, and K γ ( k ) = 2 πδ ( k ) for the powerspectrum, therefore, the radicand is S (cid:8) (cid:9) ( k ) = 2 πδ ( k ).One can see that the SFG is formally applicable. Be-sides, the necessary conditions (45) and (46) are satisfiedautomatically. However, this case is a singular one sincefor any arbitrarily small but finite values of k c it is notpossible to create a dichotomic sequence with the corre-lation properties (41) neither by the SFG or by any othermethod. B. Partial Jump
Now we extend our analysis to a more general corre-lation function that may have various applications. Thisfunction also results in a step-wise power spectrum, how-ever, with an additional parameter h that determines the height of step, K γ,h ( r ) = h δ r, + (1 − h ) sin( k c r ) k c r , (47a) K γ,h ( k ) = h + (1 − h ) πk c Θ( k c − | k | ) > , (47b)0 h , < k c π, | k | π. Eq. (47) coincides with Eq. (41) if the step-parameter h = 0. Otherwise, when h = 1 the generated γ -sequenceturns into a dichotomic white noise independently of k c .Also, γ n becomes delta-correlated at k c = π for arbitrary h . Therefore, at k c = π the conclusions of the previoussubsection are also valid.In what follows, it is convenient to analyze finite valuesof k c < π . The power spectrum (47b) is an even functionof the wave number k and has two symmetric jumps atthe points k = ± k c . For positive 0 < k π the spectrumabruptly falls down at k = k c from the maximal value K γ,h ( k < k c ) = h + (1 − h ) π/k c to the minimal one, K γ,h ( k > k c ) = h . Evidently, this jump can be regardedas a mobility edge of disordered 1D conductors, if themaximal value h + (1 − h ) π/k c is much larger than theminimal one, h , 1 + 1 − hh πk c ≫ . (48)One can see that for finite h well above zero, this is pos-sible only for small k c . What is more tricky, for k c ≪ γ -sequence is close to a dichotomic white noise,i.e., when 1 − h ≪
1. Therefore, one should have,0 < k c ≪ − h ≪ . (49)The reason of existence of the mobility edge under theconditions (49) is a significant contribution of the secondterm in the coorelator (47a). In spite of the fact that ithas a quite small amplitude 1 − h ≪
1, this term providesextremely long-range correlations with the characteristicscale k − c ≫ (1 − h ) − ≫ S (cid:8) K γ,h (cid:9) ( k ) that mustbe non-negative in order to construct the correlated se-quence γ n with the use of the SFG method. In accor-dance with the definition (36), an appropriate analysiscan be done with the following explicit expression, S{ K γ,h } ( k ) = 1 − π − h ) + (1 − h ) π k c Θ( k c − | k | )+2 ∞ X r =1 n sin h π − h ) sin( k c r ) k c r i − π − h ) sin( k c r ) k c r o cos( kr ) . (50)As in the previous case (41), the summand in the lastterm behaves as π / k c r if r → ∞ . Hence, at finite k c the sum converges rapidly and uniformly . It is a contin-uous function of k , in particular, at k = k c .The numerical calculations of S (cid:8) K γ,h ( r ) (cid:9) ( k ) per-formed for finite 0 < k c < π and 0 k π , are shown inFig. 4. Together with Eq. (50) they provide us with thefollowing empirical results.1. Due to the last term in Eq. (50), the function S (cid:8) K γ,h (cid:9) ( k ) increases with k at arbitrary values of k c and h within both intervals (0 , k c ) and ( k c , π ).2. The negative jump of S (cid:8) K γ,h (cid:9) ( k ) occurs at thesame point k = k c as for the jump of the prescribedpower spectrum (47b). The maximal and minimalvalues of the function are reached, respectively, at k = k c − k = k c + 0 for all values of thestep-parameter h within 0 h <
1. The jump isexclusively related to the third term in Eq. (50).Therefore, its value reads S (cid:8) K γ,h (cid:9) ( k c − − S (cid:8) K γ,h (cid:9) ( k c + 0) = (1 − h ) π k c . (51)3. Depending on h , the minimum S (cid:8) K γ,h (cid:9) ( k c + 0)can be either negative or positive. Also, the valueof S (cid:8) K γ,h (cid:9) ( k c +0) monotonically increases with anincrease of h . h =0.6 h =0.2 h min S { K , h (r) } ( k ) k h =0 FIG. 4: (Color online) Function S ˘ K γ,h ( r ) } ( k ) vs k for fewvalues of h at k c = 1 .
89 (this k c is very close to the minimumpoint of the curve in Fig. 3 and to the minimum point ofthe upper curve in Fig. 5). The function is entirely positive if h > h min . When h = h min , the function goes to zero solely atone point k = k c + 0 and is positive otherwise. For h < h min ,the function is negative within the whole interval k c < k < π . The presented numerical analysis can be supplementedwith the following two points. First, from the treatmentof the case (41), we know that S (cid:8) K γ,h (cid:9) ( k c + 0) < h = 0 . (52)Second, from the definition (50) one can easily reveal that S (cid:8) K γ,h (cid:9) ( k ) = 1 for h = 1 . (53) Summarizing our results, one can conclude that thereexists a threshold value of the step-parameter h that werefer to as h min , that separates the region 0 h < h min in which S (cid:8) K γ,h (cid:9) ( k ) has negative values, from the regionwith non-negative values, S (cid:8) K γ,h (cid:9) ( k ) > h min h . (54)It is clear that the threshold h min obeys the equation(see Fig. 4) S (cid:8) K γ,h min (cid:9) ( k c + 0) = 0 , (55)and depends on the correlation parameter k c . By sub-stitution of Eq. (50) into Eq. (55), one can rewrite it inexplicit form, π − h min ) −
12 = ∞ X r =1 n sin h π − h min ) sin( k c r ) k c r i − π − h min ) sin( k c r ) k c r o cos( k c r ) . (56)The numerical solution h min ( k c ) of this equation is dis-played in Fig. 5 by the upper curve.Eq. (56) can be solved analytically at small k c , result-ing in h min ( k c ) = 1 − (cid:18) k c π (cid:19) / + O (cid:0) k / c (cid:1) for k c ≪ . (57)This expression exhibits the limit h min → k c → γ n with long-range correlatorand step-wise power spectrum (47) can be constructedby the SFG method only if its parameters k c and h areplaced onto or above the upper line in Fig. 5. Only in thisarea of the ( k c , h )-plane the condition (54) holds true.Unfortunately, practically within this whole area the pa-rameters k c and h have values of the order of one, and,therefore, one cannot satisfy a quite strong requirement(48) in order to clearly observe a mobility edge. The onlyexception is a narrow vicinity of the point ( k c = 0 , h = 1).Remarkably, in this vicinity due to specific dependence(57) of h min ( k c ), the conditions (49) can be satisfied,0 < k c ≪ − h − h min ( k c ) ≈ (cid:18) k c π (cid:19) / ≪ , (58)and, consequently, the mobility edge can be achieved. So,Eq. (58) gives us the only (perhaps, just purely theoreti-cal) possibility to arrange a mobility edge in the transportthrough the γ -sequence constructed by the SFG method.Now we analyze the consequences of the necessary con-ditions formulated in Sec. II. For the long-range cor-relator (47a) the inequalities (19) lead to the followingrestriction with respect to the step-parameter h ,1 − max − { R ( r, r ′ ) } h (59) correlator may exist h k c correlator does not existsign-generation is not allowed, butsign-generation is allowed FIG. 5: (Color online) Space of control parameters k c and h . Upper curve is the dependence h min ( k c ) while low curvedepicts h ( k c ). Within the lowest area in which 0 h < h , adichotomic sequence γ n with the correlator K γ,h ( r ) does notexist. The area with h min h γ n with K γ,h ( r ) by the discussed method. In the intermedi-ate region h h < h min the SFG method does not work,and an existence of a dichotomic sequence with the step-wisespectrum remains an open problem. at r = 0, r ′ = 0 and r = r ′ . Here we have introduced thefunction R ( r, r ′ ) = (cid:12)(cid:12)(cid:12) sin k c r ′ k c r ′ ± sin k c ( r − r ′ ) k c ( r − r ′ ) (cid:12)(cid:12)(cid:12) ∓ sin k c rk c r . (60)Since by the definition h >
0, the requirement (59) ismeaningful only if its l.h.s. is positive. Otherwise, itis satisfied automatically. The combination of Eq. (59)with the assumption 0 h h h , (61)where new characteristic quantity h is introduced, h = 1 − max − { , max { R ( r, r ′ ) }} . (62)This function h ( k c ) is shown in Fig. 5 by low curve.Its piecewise shape is caused by the fact that different r and r ′ contribute to h within different intervals of k c . Finally, when k c becomes equal or larger than k ∗ ⋍ . . . . [see text after Eq. (46)], we have h = 0 andthe necessary conditions (61) reduce to the initial ones,0 h k c and h is determined by the relation, 0 1, one canconstruct desired dichotomic sequences with the use ofthe discussed method. C. Predefined Intermediate Spectrum With the SFG method, we first generate intermedi-ate Gaussian sequence β n and after, the dichotomic se-quence γ n . Above, we specified the correlator K γ ( r )of a final dichotomic γ -sequence and analyzed the spec-trum K β ( k ) = S{ K γ } ( k ) of the intermediate Gaussian β -chain, keeping in mind that the latter must be non-negative, see Eqs. (35), (38).Below, we ask question about the type of the spectrumof γ n , that emerges if the intermediate Gaussian sequence β n is assumed to have given pair correlator with step-wisepower spectrum of the following form, K β ( r ) = sin( k c r ) k c r , (63a) K β ( k ) = πk c Θ( k c − | k | ) , < k c π, | k | π. (63b)In accordance with the relation (33) and Fourier trans-forms (25), the corresponding correlator and power spec-trum of the dichotomic γ -sequence read, K γ ( r ) = 2 π arcsin (cid:20) sin( k c r ) k c r (cid:21) , (64a) K γ ( k ) = 2 π ∞ X r = −∞ arcsin (cid:20) sin( k c r ) k c r (cid:21) exp( − ikr )= (cid:16) − π (cid:17) + 2 k c Θ( k c − | k | )+ 4 π ∞ X r =1 n arcsin h sin( k c r ) k c r i − sin( k c r ) k c r o cos( kr ) . (64b)Here, last representation for the spectrum K γ ( k ) issimilar to that we have employed for the function S{·} ( k )in its analysis [compare with Eqs. (36) and (42)]. It isnoteworthy to emphasize that the summand in the lastterm of this representation behaves as 2 / πk c r when r → ∞ . Hence, as above, the sum converges rapidly and uniformly . Therefore, it can be easily calculated numer-ically.Fig. 6 presents the behavior of the power spectrum(64b) for dichotomic γ n in the case of the predefinedstep-wise profile (63b) for the spectrum of intermediatesequence β n . Since K γ ( k ) is an even function of the wavenumber k , the presentation is sufficient within 0 k π .Assuming k c < π , from Eq. (64b) and Fig. 6 one can con-clude:0 c = /20 k c = /5k c =3 /5k c = /5 K ( k ) k k c = /5 FIG. 6: (Color online) Power spectrum K γ ( k ) vs k for differ-ent values of the correlation parameter k c . 1. The function (64b) is positive within the whole in-terval | k | π . Therefore, it truly serves as a powerspectrum and its inverse Fourier transform (64a) isvalid correlator of the generated sequence γ n ;2. Due to the second and last terms in Eq. (64b) thespectrum K γ ( k ) decreases with an increase of k forall k c within whole interval (0 , π );3. The negative jump of K γ ( k ) occurs at k = k c , atthe same point as for the jump of prescribed inter-mediate power spectrum K β ( k ). The maximal andminimal values of K γ ( k ) at the jump are definedat k = k c − k = k c + 0, respectively. Thejump is exclusively formed by the second term inEq. (64b). Therefore, its value reads K γ ( k c − − K γ ( k c + 0) = 2 /k c ; (65)4. The value K γ ( k = k c − 0) to the left from the jumpis always positive. The value K γ ( k = k c + 0) to itsright is also positive for all k c . Their ratio K γ ( k c − / K γ ( k c + 0) being of the order of one for finite k c , seems to slightly increase with a decrease of k c .However, it is saturated if the inverse correlationlength k c vanishes. Indeed, from Eq. (64b) one caneasily get, K γ ( k c − K γ ( k c + 0) = π + I ( k c ) + ( π/ − k c I ( k c ) + ( π/ − k c (66a) → π + I (0) I (0) = 6 . . . . if k c → . (66b) Here we have introduced I ( k c ) = 2 k c ∞ X r =1 n arcsin h sin( k c r ) k c r i − sin( k c r ) k c r o cos( k c r ); (67a) I (0) = 2 ∞ Z dx (cid:26) arcsin h sin xx i − sin xx (cid:27) cos x. (67b)In spite of the divergence of the jump (65), theconvergence of the ratio (66) at k c → K γ ( k ) to the left, K γ ( k c − K γ ( k c + 0), from thejump, increase with decreasing of k c exactly in thesame manner, K γ ( k c − ≈ π + I (0)] /πk c and K γ ( k c + 0) ≈ I (0) /πk c .In addition, it can be analytically shown that in thelimit k c → 0, the correlator K γ ( r ) tends to unity, whilethe spectrum K γ ( k ) turns into the Dirac delta-function,lim k c → K γ ( r ) = 2 π lim k c → arcsin (cid:20) sin( k c r ) k c r (cid:21) = 1 ;(68a)lim k c → K γ ( k ) = 2 πδ ( k ) . (68b)Therefore, as the correlation parameter k c vanishes, thefinal dichotomic γ -sequence becomes to have extremelylong-range correlations.On the contrary, for k c = π the correlator K γ ( r ) re-duces to the Kronecker delta-symbol, whereas the powerspectrum K γ ( k ) degenerates into unity, K γ ( r ) = 2 π arcsin [ δ r, ] = δ r, ; (69a) K γ ( k ) = 1 for k c = π. (69b)Thus, the final dichotomic γ n reduces to the white-noisechain.In summary, in spite of the fact that the jump ratio K γ ( k c − / K γ ( k c + 0) in the power spectrum K γ ( k ) is ofthe order of unity for all k c < π , we hope that the mo-bility edge can be observed at small enough correlationparameter k c ≪ 1, due to specific delta-function behaviorof K γ ( k ) itself [see Fig. 6 and Eq. (68b)]. VI. POWER CORRELATORSA. Power Correlator for Dichotomic Sequence Here we consider an important problem of construct-ing a sequence with the power correlation function and1corresponding spectrum, K γ,p ( r ) = δ r, + ( k c | r | ) − p (1 − δ r, ) , (70a) K γ,p ( k ) = 1+ k − pc n Li p (cid:2) exp( ik ) (cid:3) + Li p (cid:2) exp( − ik ) (cid:3)o , (70b) p > , k c > , | k | π. Here p and k c are positive real numbers characterizinghow fast the correlator decreases. Note that the param-eter k c cannot be less than one since K γ,p ( r ) 1. TheFourier transform K γ,p ( k ) of this correlator is expressedvia the polylogarithm function Li q ( z ) that is defined byLi q ( z ) = ∞ X r =1 z r r q . (71)For K γ,p ( r ) to be the correlator of a stochastic process,it is necessary to have K γ,p ( k ) > k . This condi-tion is satisfied if and only if the following inequality isfulfilled, k c > (cid:2) − p ( − (cid:3) /p . (72)This result is due to the shape of spectrum K γ,p ( k ) thatmonotonously decreases with an increase of k within theinterval (0 , π ), and reaches its minimal value at k = π .The r.h.s. of the condition (72) can be calculated in limitcases, (cid:2) − p ( − (cid:3) /p = ( π/ − cp , p ≪ , p − ln 2 , p ≫ , (73) c = 1 . . . . . In Fig. 7 the area where power spectrum (70b) is non-negative, is located above the dotted lowest curve.If K γ,p ( r ) is the correlator of a dichotomic ran-dom sequence generated by the SFG method, then S{ K γ,p } ( k ) > k . Since this function, aswell as K γ,p ( k ), monotonously decreases, the only condi-tion is required, S{ K γ,p } ( π ) > . (74)Expanding in Eq. (36a) the sin-function into series, weget useful expression S{ K γ,p } ( k ) = 1 + ∞ X l =0 π l +1 l +1 (2 l + 1)! k − p (2 l +1) c × n Li p (2 l +1) (cid:2) exp( ik ) (cid:3) + Li p (2 l +1) (cid:2) exp( − ik ) (cid:3)o . (75)For p → ∞ it can be approximately calculated as S{ K γ,p } ( k ) ⋍ (cid:0) πk − pc / (cid:1) cos k. (76) k c p FIG. 7: (Color online) Various borders of the parameters p and k c above which the following relations are fulfilled:(a) Eq. (72) (dotted curve), (b) Eq. (74)) (solid curve), (c)Eq. (80)) (dashed curve), (d) Eq. (84) (dash-dotted curve). Therefore, taking into account that k c > 1, one obtainsthe following asymptotic of the condition (74) k c > p − ln 3 for p ≫ . (77)The area where the discussed SFG method is applicablelocated in Fig. 7 above solid curve.Now we analyze the necessary condition for existenceof the power correlator (70a) for dichotomic sequence re-gardless of the generation method. For the sake of sim-plicity we consider the case γ = 0. From the inequali-ties (19) one can obtain the following relation,max (cid:8)(cid:12)(cid:12) | r ′ | − p ± | r − r ′ | − p (cid:12)(cid:12) ∓ | r | − p (cid:9) k pc , (78) r = 0 , r ′ = 0 , r = r ′ . It is easy to see that the maximum with respect to r ′ ,occurs at r ′ = ± r ′ = r ± 1. Therefore, Eq. (78) canbe rewritten asmax r> (cid:8) ± ( r + 1) − p ∓ r − p (cid:9) k pc . (79)The last condition is equivalent to k c > (cid:0) − − p (cid:1) /p = ( − p ln , p ≪ , p − ln 2 , p ≫ . (80)In Fig. 7 the border corresponding to this condition isdesignated by dashed curve. It should be noted thatEq. (80) is the necessary condition, thus, if it is met, itis still not clear whether the correlator with such valuesof parameters p and k c exists. B. Predefined Intermediate Power Correlator We have found that the dichotomic sequence γ n withthe power correlator (70a) can be constructed by the SFG2method for some values of parameters p and k c that meetthe condition (74). Now let us take the intermediateGaussian sequence β n prescribed to have the power cor-relator and corresponding spectrum K β,p ( r ) = δ r, + ( k c | r | ) − p (1 − δ r, ) , (81a) K β,p ( k ) = 1+ k − pc n Li p (cid:2) exp( ik ) (cid:3) + Li p (cid:2) exp( − ik ) (cid:3)o , (81b) p > , k c > , | k | π. Evidently, the condition (72) is implied to be met. Inaccordance with the relation (33) and Fourier transforms(25), the correlator and power spectrum of the generateddichotomic γ -sequence are described as K γ,p ( r ) = δ r, + 2 π arcsin (cid:2) ( k c | r | ) − p (cid:3) (1 − δ r, ) , (82a) K γ,p ( k ) = 1 + 4 π ∞ X r =1 arcsin h ( k c r ) − p i cos( kr ) . (82b)We can assert that since Eq. (72) is satisfied, i.e. thespectrum (81b) of the intermediate β -sequence is non-negative, then the spectrum (82b) of the generated di-chotomic γ n is also non-negative.When | r | → ∞ , the correlator (82a) tends to zero inaccordance with the following asymptotic K γ,p ( r ) ⋍ ( k ′ c | r | ) − p , k ′ c = k c ( π/ /p . (83)As was shown, the allowed values of k c are expressedby Eq. (72). Therefore, the scaling parameter k ′ c shouldsatisfy the condition k ′ c > (cid:2) − π Li p ( − (cid:3) /p . (84)This condition for possible values of k ′ c and p is met inthe area above the dash-dotted curve in Fig. 7.Thus, the mapping of the gaussian sequence with thepower correlation function (81a) into the binary sequenceresult in the same power for the decrease of the final cor-relator (82a) expressed by Eq. (83). However, such a be-havior of the final correlator occurs only asymptotically,for sufficiently large values of | r | . VII. CONCLUSION First, we would like to emphasize the following pointthat was briefly mentioned in the beginning. Our studyof the correlation properties of a random dichotomic se-quence γ n is based on the example (1) in which two el-ements are “ − 1” and “1”. On the other hand, thereis a simple correspondence between this chain and a di-chotomic sequence ε ( n ) consisting of two arbitrary sym-bols ε and ε , ε ( n ) = { ε , ε } , n ∈ Z = . . . , − , − , , , , . . . (85) The correspondence is expressed by the linear relation-ship, ε ( n ) = ε + ε ∓ ε − ε γ n . (86)The choice of the sign is not important. It dererminesonly into what symbol, ε or ε , the initial values “ − γ -sequence, the connection betweenthe mean values and variances is as follows, ε ( n ) = ε + ε ∓ ε − ε γ n ; (87a) ε = ε + ε ∓ ε − ε γ ; (87b) C ε (0) ≡ ε ( n ) − ε = ( ε − ε ) C γ (0) . (87c)Analogously, the two-point correlation function C ε ( r ) ofthe ε -chain is associated with the binary correlation func-tion C γ ( r ) of the sequence γ n as follows C ε ( r ) ≡ ε ( n ) ε ( n + r ) − ε = ( ε − ε ) C γ ( r ) . (88)The comparison of Eqs. (87c) and (88) makes obviousthe equality between the normalized correlators K ε ( r )and K γ ( r ), K ε ( r ) ≡ C ε ( r ) /C ε (0) = C γ ( r ) /C γ (0) ≡ K γ ( r ) . (89)Thus, our analysis is valid for any dichotomic sequence.Our results can be summarized as follows. We haveshown that the statistical properties of random di-chotomic sequences are principally different from thoseknown for sequences with a continuous distribution oftheir elements. We were able to find analytically theconditions (19) that can be used to know whether a bi-nary sequence can have the desired pair correlator. Notethat these two conditions are necessary only.Another important restriction is due to the inequality(39) derived under quite general assumptions. We haveshown that even in the well known case of an exponentialdecay of correlations, there are no binary sequences thatcan be created with the SFG method, unless the decayis sufficiently strong. This fact is very important in viewof many applications.Our specific interest was in a possibility to create, withthe considered method, the binary sequences with long-range correlations described by Eqs. (41a) and (41b). Wehave analytically found that the function (41a) can notbe a pair correlator of any binary sequence. We havealso examined a more general correlation function [seeEq. (47)] that corresponds to the generalization of thestep-wise power spectrum. Our extensive examination ofthe signum-function method, applied to this correlationfunction, has revealed the regions of parameters k c and h VIII. ACKNOWLEDGMENTS This work was partly supported by the CONACYT(M´exico) grant No 43730. APPENDIX A: PROBABILITY DENSITY OF β n The standard way to derive the probability density ρ B ( β ) of the random process β n is due to its charac-teristic function ϕ B ( t ) defined by ϕ B ( t ) ≡ exp[ itβ n ] = Z ∞−∞ dβ ρ B ( β ) exp( itβ ) . (A1)From the last equality in this definition it immediatelyfollows that the probability density ρ B ( β ) is the Fouriertransform of ϕ B ( t ), ρ B ( β ) = 12 π Z ∞−∞ dt ϕ B ( t ) exp( − itβ ) . (A2)To start with, we substitute the explicit expression(20b) for β n into the definition (A1) for characteristicfunction ϕ B ( t ). Then, we rewrite the result as an infiniteproduct of exponential functions and take into accountthe statistical independence of uncorrelated random vari-ables α n . This procedure yieldsexp[ itβ n ] = exp( itβ ) ∞ Y n ′ = −∞ exp (cid:2) itG ( n − n ′ ) α n ′ (cid:3) . (A3)In accordance with the Gaussian distribution (21b) of α n , its characteristic function is ϕ A ( τ ) ≡ exp( iτ α n ) ≡ Z ∞−∞ dα ρ A ( α ) exp( iτ α )= exp( − τ / . (A4)The use of Eqs. (A3), (A4) with τ = tG ( n − n ′ ), andthe normalization condition (24) results in ϕ B ( t ) = exp( iβt − t / . (A5)As is known, this characteristic function corresponds tothe Gaussian probability density (28). One can confirmthis fact by a direct evaluation of the integral in Eq. (A2). APPENDIX B: PAIR CORRELATOR OF γ n Let us derive the pair correlator γ n γ n + r . Employingthe standard integral presentation for the signum func-tion, sign( z ) = 1 π Z ∞−∞ dx sin( zx ) x , (B1)and Eq. (A4), we arrive, in a manner similar to the calcu-lation of the characteristic function ϕ B ( t ) in Appendix A,at the expression, γ n γ n + r = J ( K β ( r ) , β )= 2 π Z ∞ dx x Z ∞ dx x exp (cid:16) − x + x (cid:17) × X s = − , s exp (cid:2) sK β ( r ) x x (cid:3) cos (cid:2) β ( x − sx ) (cid:3) . (B2)Thus, we have reduced the problem to the derivation of J ( K, β ). To solve it, we obtain the derivative of J ( K, β )with respect to K . After some simplifications one gets, ∂∂K J ( K, β ) = 1 π Z ∞ dx Z ∞−∞ dx exp (cid:16) − x + x (cid:17) × exp (cid:2) Kx x (cid:3) X t = − , exp (cid:2) itβ ( x − x ) (cid:3) . To proceed, we write down the following relation that isvalid for arbitrary real quantities a and b , Z ∞−∞ dx exp( − x / 2) exp( ax ) exp( ibx )= √ π exp (cid:2) ( a − b ) / (cid:3) exp( iab ) . (B3)Using Eq. (B3) with a = Kx and b = − tβ we inte-grate over x and make further simplifications, ∂∂K J ( K, β ) = √ ππ Z ∞−∞ dx exp (cid:2) − x (1 − K ) / (cid:3) × exp( − β / 2) exp (cid:2) iβ (1 − K ) x (cid:3) . Now we change the integration variable x , x ′ = x p − K . (B4)Then, applying Eq. (B3) with a = 0 , b = β r − K K , (B5)we perform the integration over x that gives rise to theexpression ∂∂K J ( K, β ) = 2 π √ − K exp (cid:16) − β K (cid:17) . (B6)4The general solution of Eq. (B6) is J ( K, β ) = J (0 , β )+ 2 π Z K dx √ − x exp (cid:16) − β x (cid:17) . (B7)It should be noted that Eq. (B7) can be also obtainedby means of the two-point probability density that forthe correlated Gaussian sequence β n with the correlator K β ( r ) is defined by ρ B ( β n = β, β n + r = β ′ ) (B8)= 12 π q − K β ( r ) exp ( − β + β ′ − K β ( r ) ββ ′ − K β ( r )] ) . The last step we should take, is to calculate J (0 , β ). It can be directly obtained from Eq. (B2), J (0 , β ) = (cid:20) π Z ∞ dx sin( βx ) x exp( − x / (cid:21) = γ . (B9)This result can be easily explained. Indeed, the condition K β ( r ) = 0 implies that the correlations between β n and β n + r disappear, hence, the correlations between γ n and γ n + r are absent as well.As a result of these calculations, we finally get J ( K β ( r ) , β ) = γ + 2 π Z K β ( r )0 dx √ − x exp (cid:16) − β x (cid:17) This expression provides Eq. (32). [1] I. M. Lifshits, S. A. Gredeskul and L. A. Pastur, Intro-duction to the Theory of Disordered Systems ( New York:Wiley, 1988)[2] N. M. Makarov, I. V. Yurkevich, Zh. Eksp. Teor. Fiz. Sov. Phys. JETP 628 (1989)]; V. D. Frei-likher, N. M. Makarov, I. V. Yurkevich, Phys. Rev. B ,8033 (1990).[3] M. Griniasty, S. Fishman, Phys. Rev. Lett. , 1334(1988).[4] N. M. Makarov, Lectures on Spectral and Trans-port Properties of One-Dimensional Disordered Conduc-tors , 3735(1998).[6] F. M. Izrailev, A. A. Krokhin, Phys. Rev. Lett. ,4062 (1999); A. A. Krokhin, F. M. Izrailev, Ann. Phys.(Leipzig) SI- , 153 (1999).[7] F. M. Izrailev, N. M. Makarov, J. Phys. A: Math. Gen. , 10613 (2005).[8] S. O. Rice, Bell Syst. Tech. J. , 282 (1944); S. O. Rice,in Selected Papers on Noise and Stochastic Processes , ed.by N. Wax (Dover, New York, 1954) p. 180.[9] D. Saupe, in The Science of Fractal Images , ed. by H.-O. Peitgen and D. Saupe (Springer, New York, 1988);J. Feder, Fractals (Plenum Press, New York, 1988).[10] C.-K. Peng et al., Phys. Rev. A , 2239 (1991). [11] S. Prakash et al., Phys. Rev. A , R1724 (1992).[12] C. S. West, K. A. O’Donnell, J. Opt. Soc. Am. A , 390(1995).[13] A. Czirok, R. N. Mantegna, S. Havlin, H. E. Stanley,Phys. Rev. E , 446 (1995).[14] F. M. Izrailev, N. M. Makarov, Opt. Lett. , 1604(2001); F. M. Izrailev, N. M. Makarov, Phys. Rev. B , 113402 (2003).[15] F. M. Izrailev, N. M. Makarov, Appl. Phys. Lett. ,5150 (2004).[16] F. M. Izrailev, A. A. Krokhin, N. M. Makarov, andO.V.Usatenko, Phys. Rev. E. , 026140 (2005).[18] S. S. Melnyk, O. V. Usatenko, and V. A. Yampol´skii,Physica A, , 405 (2006).[19] V. A. Yampol´skii, S. Savel´ev, O. V. Usatenko,S. S. Mel´nik, F. V. Kusmartsev, A. A. Krokhin, andF. Nori, Phys. Rev. B, , 014527 (2007).[20] P. Carpena, P. Bernaola-Gal´van, P. Ch. Ivanov, H. E.Stanley, Nature , 955 (2002); Nature , 764 (2003).[21] A. Erdelyi, Higher Transcendental Functions, V. 2,McGraw-Hill, 1953.[22] F. M. Izrailev, A. A. Krokhin, N. M. Makarov, S. S. Mel-nyk, O. V. Usatenko, and V. A. Yampol’skii, Physica A372