F. Tanaka and S.F. Edwards, Analytic theory of the ground state properties of a spin glass: I. Ising spin glass, Journal of Physics F: Metal Physics 10 (1980) 2769--2778.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... has been shown that there are on the average asymptotically 1:05 2 0:2874s many stable states in a binary Hopfield net of size s whose feedbacks and biases are zero (w jj = w j0 = 0 for j 2 V ) and whose other weights are independent, identically distributed zero mean Gaussian random variables [46, 73]. For a particular binary symmetric network, however, the issue of deciding whether there are e.g. at least one (when negative feedback weights are allowed) 13] two [41] or three [13] stable states, is NP complete. Indeed, the problem of determining the exact number of stable states for a given ....

F. Tanaka and S.F. Edwards, Analytic theory of the ground state properties of a spin glass: I. Ising spin glass, Journal of Physics F: Metal Physics 10 (1980) 2769--2778.

.... shown that there are on the average asymptotically 1:05 2 0:2874s many stable states in a binary Hop eld net of size s whose feedbacks and biases are zero (w jj = w j0 = 0 for j 2 V ) and whose other weights are independent identically distributed zero mean Gaussian random variables [84] [131]. For a particular binary symmetric network, however, the issue of deciding whether there are e.g. at least one (when negative feedback weights are allowed) 24] two [66] or three [24] stable states, is NP complete. Indeed, the problem of determining the exact number of stable states for a given ....

F. Tanaka and S.F. Edwards, Analytic theory of the ground state properties of a spin glass: I. Ising spin glass, Journal of Physics F: Metal Physics 10 (1980) 2769-2778.

....whether a randomly generated x # X is a local minimum. Numerical data of this kind are reported e.g. in [181, 180, 63] Methods from statistical mechanics can be used, however, to obtain the scaling of the expected value E[M] with the system size for a variety of disordered systems, see e.g. [191, 189, 19, 73, 36, 156, 34, 46]. A non rigorous result is particular interest in this context. The correlation length conjecture [181] suggests that the number of local optima of a typical landscape can be estimated from its correlation length #, eqn. 4.4) More precisely, one expects on the order of one local optimum on a ....

F. Tanaka and S. F. Edwards, Analytic theory of the ground state properties of a spin glass: I. Ising spin glass, J.Phys.F:Metal Phys., 10 (1980), pp. 2769--2778.

....minima is 0.086 (2) Hence, so far as the statistics of metastable states is concerned, the mean field Hamiltonian H d yields in fact a very close approximation to the pure Hamiltonian H. For the purpose of comparison we note that # = 0.1992 and # = 0. 3552 for the binary 1 2 spin glass [30, 5] and 4 spin glass models [16, 29] respectively, while # = ln 2 # 0.6931 for the random energy model [11] In Fig.1 we show the exponent # as a function of the energy density #. For the sake of clarity we present only the region of positive values of #. The lowest value of # at which the ....

....# as a function of the energy density #. For the sake of clarity we present only the region of positive values of #. The lowest value of # at which the exponent # vanishes, denoted by # 0 , gives a lower bound to the ground state energy density of the spin model defined by the Hamiltonian (2) [30]. We find # 0 = 0.0202845 which, within the numerical precision, is exactly the value predicted by the first step of replica symmetry breaking [4] as well as by Golay s ergodicity hypothesis [15, 3] This coincidence between the replica and the density of metastable states predictions for the ....

F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: I. Ising The low autocorrelated binary string problem 15 spin glass. J. Phys. F, 10:2769--2778, 1980.

....for Kauffman s Nk landscapes [41] A few counter examples are known as well; all of them strongly violate the maximum entropy assumption. 9 Garc ia Pelayo Stadler: The XY Hamiltonian 5. The Discrete XY Hamiltonian The discrete version of the XY model was introduced by Tanaka and Edwards [43]. There are n spins in the plane, each of which can point into one of the ff directions oe i = cos i sin i with i = 2 ff x i ; 0 x i ff : 10) The set V of all possible spin configurations contains jV j = ff n points. The interaction energy between two spins i and j is ....

....1) neighbors in Q n ff . ii) Two configurations are neighbors of each other if a single spin orientation differs by Sigma(2 =ff) while all the other spins are the same. The corresponding graphs C n ff are the n fold direct products of the cycles C ff . This notion of neighborhood was used in [43]. Each spin configuration has D = 2n neighbors in C n ff if ff 3, and only D = n neighbors if ff = 2. It is trivial to check that the complete graphs Q ff and the cycle graphs C ff are Cayley graphs of the commutative group ZZ ff = f ; j; j 2 ; j ff Gamma1 g with the sets of ....

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F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: II. XY spin glass. J.Phys.F, 10:2779--2792, 1980.

....the spins are arranged on a two or three dimensional lattice with non zero interactions only between lattice neighbors. At least three groups have computed the number of local minima of the SK model by means of what are now considered standard methods in Statistical Mechanics. Tanaka and Edwards [32] computed the expected number of local optima E [N ] while Bray and Moore [33] and De Dominicis et al. 34] used a replica approach to evaluate E[ln N ] For the case of short range spin glasses, in which only a small number z of coupling constants J ij are non zero for any given spin i, a ....

.... [N ] while Bray and Moore [33] and De Dominicis et al. 34] used a replica approach to evaluate E[ln N ] For the case of short range spin glasses, in which only a small number z of coupling constants J ij are non zero for any given spin i, a slightly larger number of local optima has been found [35, 32]. The only known case in which the logarithmic average deviates from the direct average is the linear spin chain [36] Since all Ising models have the same correlation length n = n=4 [37, 17] but somewhat different values of N , we cannot hope for an exact formula relating and E [ for ....

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F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: I. Ising spin glass. J.Phys.F, 10:2769--2778, 1980. -- 22 -- Garc' ia-Pelayo & Stadler: The XY-Hamiltonian

....The only known case in which E[ln N ] 6= ln E [N ] is the linear spin chain [27] Peter F. Stadler 23 For the case of short range spin glasses, in which only a small number z of coupling constants J ij are non zero for any given spin i, a slightly larger number of local optima has been found [15, 147] than for the long range Sherrington Kirkpatrick model [129] Since all Ising models have the same correlation length = n=4 [156, 133] but somewhat different values of N , we cannot hope for a general, exact formula relating E[ln N ] and E [ From the maximum entropy interpretation of ....

F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: I. Ising spin glass. J. Phys. F, 10:2769--2778, 1980.

....model received considerable attention around 1980; at least three groups have computed the number of local minima of the SK model by means of what are now considered standard methods in Statistical Mechanics. Tanaka 15 Garc ia Pelayo Stadler: The XY Hamiltonian and Edwards [38] computed the expected number of local optima E [N ] while Bray and Moore [39] and De Dominicis et al. 40] used a replica approach to evaluate E[ln N ] These papers provide also a detailed analysis of the distribution of local minima as a function of their energies. The common result of the ....

.... short range spin glasses, in which only a small number z of coupling constants J ij are non zero for any given spin i, a slightly larger number of local optima has been found lim n 1 1 n log E [N n ] lim n 1 1 n E[log N n ] fl(2) fl 0 z O(z Gamma2 ) 28) where fl 0 0:0656 [41, 38]. The only known case in which the logarithmic average deviates from the direct average is the linear spin chain. Derrida and Gardner [42] found log E [N n ] n ln(4= 0:2416 and E[log N n ] n (ln 2) 3 0:2310 for this example. Since all Ising models have the same correlation length n = ....

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F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: I. Ising spin glass. J.Phys.F, 10:2769--2778, 1980.

....we shall concentrate on the relation between the number of local optima and pair correlation functions using a particular class of spin glass Hamiltonians, the XY model, as an example. 2. The Discrete XY Hamiltonian The discrete version of the XY model was introduced by Tanaka and Edwards in [13]. There are n spins in the plane, each of which can point into one of the ff directions oe i = cos i sin i with i = 2 ff x i ; 0 x i ff : 1) 2 Garc ia Pelayo Stadler: The XY Hamiltonian The interaction energy between two spins i and j is given by J ij h oe i ; ....

....we obtain all neighbors of a configuration when l runs over all spins and m = Sigma1 (for ff 3; there is only one value of m for ff = 2) The corresponding graphs are the n fold direct products of the cycle graphs C ff , which we shall denote by C n ff . This notion of neighborhood was used in [13]. 1 The (Cartesian or direct) product Gamma 1 Theta Gamma 2 of two graphs Gamma 1 = V1 ; E1 ) and Gamma 2 = V2 ; E2 ) has the vertex set V = V1 Theta V2 . Two vertices (x1 ; x2 ) and (y1 ; y2 ) of the product are connected by an edge if either (i) x1 = y1 and x2 ; y2 are adjacent in ....

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F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: II. XY spin glass. J.Phys.F, 10:2779--2792, 1980.

....are #P complete. It is known that if in a simple Hopfield network with threshold vector h = 0 the elements of the connection matrix are independent identically distributed zero mean Gaussian random variables, the asymptotic estimate for the number of stable vectors is about 1:05 Theta 2 0:2874n [62, 81]. 5.2 Finding Stable Vectors We now turn to the problem of finding a stable vector to which a given initial vector may converge under sequential operation. It is easy to see that a sequential computation by the network itself solves the problem in polynomial time, if the weights are small. Alon ....

F. Tanaka and S. F. Edwards. Analytic theory of the ground state properties of a spin glass: I. Ising spin glass. Journal of Physics F: Metal Phys., 10:2769--2778, 1980.

....larger n. p = 2 corresponds to the Sherrington Kirkpatrick model [14] It received considerable attention around 1980; at least three groups have computed the number of local minima of the SK model by means of what are now considered standard methods in Statistical Mechanics. Tanaka and Edwards [21] computed the expected number of local optima hg 0 i, while Bray and Moore [1] and De Dominicis et al. 3] used a replica approach to evaluate hln g 0 i. These papers provide also a detailed analysis of the distribution of local minima as a function of their energies. The common result of the ....

.... of short range spin glasses, in which only a small number z of coupling constants J ij are non zero for any given spin i, a slightly larger number of local optima has been found lim n 1 1 n lnhg 0 i = lim n 1 1 n hln g 0 i = ff(2) ff 0 z O(z Gamma2 ) 5) where ff 0 0:0656 [2, 21]. The only known case in which the logarithmic average deviates from the direct average is the linear spin chain. Derrida and Gardner [7] found lnhg 0 i=n ln(4= 0:2416 and hln g 0 i=n (ln 2) 3 0:2310 for this example. 3 Local Minima of p Spin Models 3. Numerical Simulations In ....

[Article contains additional citation context not shown here]

F. Tanaka and S. F. Edwards. Analytic theory of ground state properties of a spin glass: I. Ising spin glass. J.Phys.F, 10:2769--2778, 1980.

....are on average E [N ] 1 D 1 N local optima. An extension of these results to landscape with a small correlations can be found in [127] Estimates for the number of local optima are also known for the Nk model [142, 141, 70] SherringtonKirkpatrick model and related short range 2 spin models [19, 20, 28, 33, 134] for the Travelling Salesman Problem with Transpostion metric [127] and for RNA free energy landscapes [47] However, there is no routine method for deriving the number of local optima and for relating this number to the corrlation structure of the landscape. In most cases, the number of local ....

F. Tanaka and S. Edwards. Analytic theory of the ground state properties of a spin glass: I. Ising spin glass. J.Phys.F:Metal Phys., 10:2769--2778, 1980.

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