M. M'ezard, G. Parisi and M.A. Virasoro, "Spin-glass theory and beyond", World Scientific, Singapore (1988). Home/Search Document Not in Database Summary Related Articles Check |
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Modelling markets dynamics: Minority games and beyond - Challet (2000) (1 citation) (Correct)
....hZ ( i. Each replica b has its own set of dynamical variables s i;b . c (A where the average is over the disorder variables a i;s and Tr s is the trace on the variables s i;c for all i and b. With the random history process, 0 for all . Following standard procedures [53, 52], we introduce a Gaussian variable b so that we can linearize the exponent in Eq. 2.67) p i a;z (2.68) This allows us to carry out the averages over a s explicitly ( i a 2 b;d b [1 s i;b s i;d] 2.69) to leading order in N . Then we introduce new ....
....agents. In practice however, to leading order in N , all realizations of a i;s yield the same limit, which then coincides with the average of min 2 N H f g over i;s . The average of ln Z over the a s, which we denote by h: i a , is reduced to that of moments of Z using the replica trick[52]: hln Zi a = lim lnhZ With integer n the calculation of hZ i a amounts to study n replicas of the same system with the same realization of a i;s . To do this we introduce a set of dynamical variables a f i;s;a g for each replica, which are labeled by the additional index a = 1; ....
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M. Mezard, G. Parisi, M. A. Virasoro, Spin glass theory and beyond (World Scienti c, 1987).
Toy Models of Markets With Heterogeneous Interacting Agents - Marsili (2001) (1 citation) (Correct)
....materials are not free to move. But they are neither placed on a regular grid, as in a crystal. Hence the elegant methods devised for studying homogeneous or periodic systems such as uids or regular crystals are of little use. The approach of theoretical physics to disordered systems [1] is based on a statistical approach: A typical disordered system is a realization of a random system in which the interactions between degrees of freedom are drawn from a given probability distribution. As long as the system is large, i.e. composed of a large number N of interacting units, the ....
.... 0 and s;i = 1. Here hAj i = s=1 s;i s;i : 19) The argument is analogous to what happens in the simpler case discussed in section 3.3. However the characterization of the equilibria is more complex in this case and it requires sophisticated techniques of statistical mechanics [1,38]. Without entering into too many details, we can give a simple explanation, based on linear algebra, of why a phase transition occurs for = 0. This not only gives a avor of the results but it also allows one to understand several things. First note that for = 0, H 0 is a non negative de ....
M. Mezard, G. Parisi, M. A. Virasoro (1987) Spin glass theory and beyond (World Scienti c).
Bivariate Uniqueness in the Logistic Recursive Distributional.. - Bandyopadhyay (2002) (Correct)
....related works see [8, 3] Our interest in this problem is from another perspective. In 2001 Aldous [4] showed that E[A n ] #(2) 4) confirming the earlier work of Mezard and Parisi [6] where they computed the same limit using some non rigorous arguments based on the replica method [7]. In an earlier work Aldous [1] showed that the limit of E [A n ] as exists for any cost distribution, and does not depend on the specifics of it, except only on the value of its density at 0, provided it exists and is strictly positive. So for calculation of the limiting constant one can ....
M. Mezard, G. Parisi, and M. A. Virasoro. Spin Glass Theory and Beyond. World Scientific, Singapore, 1987.
Landscapes and Effective Fitness - Stadler, Stephens (2003) (2 citations) (Correct)
....the landscape measures the full reproductive success of a type. Mathematical constructions analogous to a fitness landscape arise naturally in many other areas of scientific study. For instance, in the physics of disordered systems where spin glasses, for example, can be cast into the same form [56, 57]. Each spin configuration is assigned an energy by virtue of the Hamiltonian that specifies the model; the dynamic properties invoke a collection of transitions between configurations. In biophysics energy landscapes govern the folding of biopolymers, including proteins [58, 59, 60] and nucleic ....
Mezard, M., Parisi, G., and Virasoro, M. Spin Glass Theory and Beyond. World Scientific, Singapore, 1987.
Statistical Mechanics of Neural Networks: Enhancement by.. - Dietrich (Correct)
....order parameters are obtained as saddle point values of the (specific) free energy f = ln Z) #N ) averaged over the distribution of the input examples. Unfortunately, this average cannot be calculated rigorously; instead, the replica trick is used as a standard tool of statistical physics [49]: #f# #N #ln Z# = 2.15) So the partition function (2.13) has to be replicated n times: 2.16) leading to two replica indices a = 1, n and b = 1, # for the decision variables V . In the next step, the average over the complete data set D ....
M. Mezard, M. and G. Parisi, and M.G. Virasoro. Spin Glass Theory and Beyond. World Scientific, Singapore, 1987.
Rigorous Results for Random (2+p)-SAT - Achlioptas, Kirousis, Kranakis.. (Correct)
....[22] methods from statistical mechanics are used to derive predictions for the value of the threshold for (2 p) SAT. Monasson et al. observe that the known bounds for 2 and 3 SAT imply the critical value for (2 p) SAT is bounded above by minf1= 1 Gamma p) r 3 =pg. Using the replica method [18, 22] they predict that for p :413: 1= 1 Gamma p) is also a lower bound, i.e. for values of p in the range 0 to :413: 2 p) SAT behaves in a similar manner as 2 SAT. In this paper we provide a rigorous proof of this assertion for p 2=5 by showing that the threshold for values of p between ....
M. M'ezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987.
Social Distance, Heterogeneity and Social Interactions - Cont, Löwe (2003) (1 citation) (Correct)
.... the existence and uniqueness of a dynamic equilibrium and provides a description of the distribution of actions at equilibrium: This was rst remarked in the context of neural networks by Hop eld [Hop eld] in 1982 and was the starting point of the statistical mechanics of neural networks, see [Mezard, Parisi Virasoro] Proposition 3.1. The Markov process of agents choices (t) i (t) i=1; N evolving under the rules described in section 3.2 has a unique stationary distribution given by 8 2 f1; 1g HN; HN; 21) where HN; f1; 1g R is de ned by HN; 1 2pN Then for ....
....0.6 0.7 0.8 0.9 1 0.2 0.15 0.1 0.05 0 0.05 F(z (beta) for b=1.7 Figure 3. The value of the minimum of ( as a function of 1. Case where p = 1 (completely connected graph) The main idea in the proof of Theorem 4. 1 consists in using the HubbardStratonovich transformation (see [Mezard, Parisi Virasoro] De ne QN; de ned as the convolution of QN; with a Gaussian measure in R with expectation 0 and covariance matrix ( N) Id (where Id denotes the identity matrix) QN; QN; N (0; N) Id) 33) Since N (0; N) Id) converges weakly to zero it is easily seen ....
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M. Mezard, G. Parisi, and M Virasoro (1988) Spin Glass Theory and beyond, Singapore: World Scienti c.
A Random Walk in Statistical Physics - Svenson (Correct)
....is impossible to have an infinite number of lattice points, so instead Z L is used and one looks at how various quantities scale with the linear size L of the system, hoping to be able extract info on the L # # limit. A model which has both frustration and disorder is called a spin glass (e.g. [6, 7, 8, 19, 20, 21, 22, 23, 24]; see also the web site http: online.itp.ucsb.edu online lnotes balents bignotes.html) Several of the systems considered in this thesis are spin glasses, and concepts related to it provide the underlying motivation for much of the work that does not refer to it. There are different kinds of ....
....After equilibrating both systems, the Hamming distance (the number of different spins) between the spin configurations S and S h#S ;S ## # # (4.11) where is the Kronecker delta function) is measured. The Hamming distance can also be expressed in terms of the Parisi overlap [20] q # i S ### #h: 4.12) In analogy with the ordering temperature T c , the spreading temperature T d is defined as the temperature below which the difference between the systems disappears after again equilibrating the systems [131, 132] Coarsening as well as persistence and damage ....
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M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond. World Scientific, Singapore, 1987.
Damage spreading in small world Ising models - Svenson, Johnston (2001) (Correct)
....After equilibrating both systems, the Hamming distance (the number of different spins) between the spin configurations S and S h#S ;S ## # # (1) where is the Kronecker delta function) is measured. The Hamming distance can also be ex pressed in terms of the Parisi overlap [8] q # S # S ### #h: 2) Most of the work on both spin models and damage spreading place the spins either on a finitedimensional lattice or on a random graph. Here we instead use small world graphs [9, 10] to study the ferromagnetic Ising model on graphs interpolating between 2 and ....
M. Mezard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond, World Scientific 1987.
A Variational Approach to Learning Curves - Malzahn, Opper (2001) (Correct)
....for suitable data averages of posterior expectations. The partition function Zm is Zm = E exp : 2) To perform the average [ln Zm ] D we use the replica trick [ln Zm ] D = lim n 0 ln[Z m ] D n , where [Z m ] D is computed for integer n and the continuation is performed at the end [5]. We obtain Zn (m) Z m ] D = En A ; 3) where En denotes the expectation over the replicated prior measure. Eq. 3) can be transformed into a simpler form by introducing the grand canonical partition function n ( n ( m=0 m m Zn (m) En exp( Hn ) 4) ....
....statistics of the training data. We can use the distribution induced by the prior and e in order to compute approximate combined data and posterior averages. As an example, we first consider the expected local = hf (x)i hf(x)i ] D . Following the algebra of the replica method (see [5]) this is approximated within the variational replica approach as (x) lim hf a (x)i 0 hf a (x)f b (x)i 0 = K 0 (x) K(x) 12) Second, we consider the noisy local mean square prediction error of the posterior mean predictor f(x) hf(x)i which is given by (x; y) f(x) y) ....
M. Mezard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, (1987).
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M. M'ezard, G. Parisi and M.A. Virasoro, "Spin-glass theory and beyond", World Scientific, Singapore (1988).
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M. M'ezard, G. Parisi, and M.A. Virasoro, "Spin-glass theory and beyond", World scientific, Singapore (1988).
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Mezard, M., Parisi, G., & Virasoro, M. (1987). Spin glass theory and beyond. Singapore: World Scientific.
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M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scienti c, Singapore, 1987.
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M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond, World Scientific, 1987.
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M. Mezard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987.
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M.Mezard, G.Parisi, M.A.Virasoro, Spin Glass theory and beyond, World Scien. (1987)
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M. Mezard, G. Parisi, and M.A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
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