C. Itzykson and J-M. Drouffe, "Statistical Field Theory," Vol. 2, Cambridge University Press, Cambridge, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....change. The critical behaviour of statistical mechanical models of unitary matrices as N 1 and fi=N c has been studied by Periwal and Shevitz [29] and critical phenomena have been studied by Distler and Vafa [7] For details about these topics the reader is referred to Itzykson and Drouffe [19], Mehta [26] and Ginsparg [10] 1.3 The main integral R(x; N; z) The objects of study of this paper are the series R 4 (x; y; z) and R 1 2 (x; y; z) and their relationship to graphs in surfaces. These completely determine the partition function ZN (U ) for Hermitian complex random matrix ....

C. Itzykson and J-M. Drouffe, "Statistical Field Theory," Vol. 2, Cambridge University Press, Cambridge, 1990.

....of random lattices, commonly used in the study of quenched disorder, is Poissonian random lattices. They are constructed by distributing vertices uniformly on a two dimensional manifold and link them together to form a triangulation, usually following a prescription by Dirichlet and Voronoi [28]. We have compared the properties of our ensemble of graphs to that of Voronoi triangulations by looking at the probability distribution of vertex orders p n . In 11 0 0.1 0.2 0.3 0.4 p n 0 5 10 c = 5 Voronoi n c = 0 c = 5 Figure 3: The (normalized) curvature distribution pn for an ....

C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Cambridge University Press, Cambridge, 1989.

.... even if the statement of the conjectures are clear, the understanding of these predictions and of the non rigorous techniques (renormalisation group, conformal field theory, quantum gravity, the link with highest weight representation of some infinite dimensional Lie algebras, see e.g. [20, 11]) used by physicists has been limited. Our aim in the present review paper is to present some results derived in joint work with Greg Lawler and Oded Schramm [33, 34, 29, 30, 31, 32] that proves some of these conjectures, and improves substantially our understanding of others. The systems that we ....

C. Itzykon, J.-M. Drouffe, Statistical Field Theory, Vol. 2, Cambridge University Press, 1989.

....which become critical at the transition point. The prediction of a metallic behavior at zero temperature created a lot of interest both on the experimental and the theoretical side. The universal conductance in a model with no disorder was considered in Ref. 30] 10 by means of 1=N expansion [63] and Monte Carlo simulations and in Ref. 37] by means of an ffl expansion [63] The dirty boson system and the transition to the Bose glass phase (including the case of long range Coulomb interaction) was extensively studied in [35,31] Wen employed a scaling theory of conserved currents at ....

....behavior at zero temperature created a lot of interest both on the experimental and the theoretical side. The universal conductance in a model with no disorder was considered in Ref. 30] 10 by means of 1=N expansion [63] and Monte Carlo simulations and in Ref. 37] by means of an ffl expansion [63]. The dirty boson system and the transition to the Bose glass phase (including the case of long range Coulomb interaction) was extensively studied in [35,31] Wen employed a scaling theory of conserved currents at anisotropic critical points [64] identifying many universal amplitudes. One of these ....

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C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Cambridge University Press (1989). 15

....optimization abound, ranging from random packing of spheres to minimum spanning trees. Such problems also occur naturally in physics and have been considered in applications ranging from stellar dynamics [1] to interactions in liquid systems, to cellular objects such as foams and random lattices [2]. Here we consider the case of N sites placed randomly, with a uniform distribution, on a 2 D surface of fixed area. Let the random variable D k N represent the distance between a given point x and its kth nearest site. The expectation value D k N taken over the ensemble of randomly placed ....

....only for the moments of point to point distances, similar properties hold for higher order simplices such as areas of triangles associated with nearby points. The problem is thus a natural one to consider further in the context of random triangulations, foams, or other physical problems [2] tightly connected to geometry. ACKNOWLEDGMENTS We are grateful to E. Bogomolny, J. Houdayer, and C. Kenyon for sharing with us their valuable insights on this topic, and to O. Bohigas for having introduced us to the problem. O.C.M. acknowledges support from the Institut Universitaire de France. ....

C. Itzykson and J. Drouffe, "Statistical Field Theory," Cambridge Univ. Press, Cambridge, UK, 1989.

.... will consider a class of discrete undirected graphical models, Boltzmann machines (Ackley, Hinton, Sejnowski, 1985) These have application in artificial intelligence as stochastic connectionist models (Jordan et al. 1998) in image restoration (Geman Geman, 1984) and in statistical physics (Itzykson Drouffe, 1989). The potential of a Boltzmann machine, with binary random variables s i 2 f0; 1g, is given by H(s; w) X i w i s i 1 2 X i;j w ij s i s j (25) with w ij j w ji and w ii j 0. Unfortunately, Boltzmann machines are in general intractable since calculation of the normalizing constant ....

Itzykson, C., & Drouffe, J.-M. (1989). Statistical Field Theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press.

....hypothesis that the transition at T p is in the universality class of pure percolation. In practice, this is quite difficult. The value of the magnetic exponent ffi in the 3 d Ising model (as determined through renormalization group methods, for instance) yields the prediction FK = 2:207(1) [21]. The value of for pure percolation that one would infer from recent series expansions is = 2:189(5) 9] which is not so different from the FK value. In fact, the power law fits to N(V cl ) are not very precise, due to large finite volume effects and corrections to scaling. The values we ....

C. Itzykson and J-M. Drouffe, Statistical Field Theory, Cambridge University Press, Cambridge (1989).

....variables with a appropriately and then letting a tend to zero. The fermion propagator is defined as the Dirac operator [oe m 0 ] to the power ( Gamma1) and is diagonal in momentum space. With this lattice fermion action the propagator reads in momentum representation [KaSm81] [ItDr89], Ro92] 4(p ) i i 1 a fl sin(p a) m 0 j Gamma1 : 4.11) We denote the points (0; 0) 0; a ) a ; 0) and ( a ; a ) by p . For k = p Gamma p the propagator looks like 4(p ) m 0 Gamma P i a fl sin(p a) m 2 0 P 1 a 2 sin 2 (p ....

....p . The fact that there are four poles and not a unique one is called fermion doubling. This fermion species multiplication reflects a discrete symmetry of the naive fermion action. This discrete symmetry group consists of transformations which transform the points p into each other [KaSm81] [ItDr89]. To obtain the correct propagator in the naive continuum limit the doublers need to be removed. CHAPTER 4. FERMIONS ON THE LATTICE 20 4.1 Wilson fermions One way to remove the extra particles is giving them a mass of the order of the cut off. This can be done by adding the Wilson term [Wi74] ....

C. Itzykson, J. Drouffe, Statistical Field Theory, Cambridge University Press 1989

....However, for some architectures more efficient learning rules exits, as we shall see. 3 Decimation Recently an algorithm has been described [8] which computes the correlations of a BM with a tree structure in linear time (for the free and the clamped phase) The algorithm is based on decimation [3]. This method can also be applied on some networks with cycles. Before specifying exactly the class of network structures on which decimation can be applied we will first show how decimation works. i i i y S 2 S 1 S 3 w 12 w 13 w 10 i i y y S 2 S 3 w 23 w 20 w 30 Figure 1: Two network ....

C. Itzykson and J. Drouffe. Statistical Field Theory. Cambrigde University Press, Cambridge, 1991.

....be interpreted as a maximimum a posteriori approximation (see Appendix B.6) We remark that we will not discuss in the following problems of infinite spaces. This holds especially for the nonlinear models in the next sections for which the question of a continuum limit is highly non trivial [30,75]. Hence, if necessary, integrals can in the following be considered as convenient notation for sums. Analogously, derivative operators can be replaced by their discretized lattice versions. This reflects also the fact that finally numerical calculations have to be done in a finite dimensional ....

....t j (x) j y j . The measured features are h(x) hhjxi. Example 4 (Prior template) A typical smoothness functional in regularization theory, corresponding to the Wiener measure for stochastic processes [16,20,32,43] and to the kinetic energy or a free massless scalar Euclidean field in physics [25,30,75], is Z 1 Gamma1 d d x d X l=1 h(x) x l 2 = Gamma Omega h Gamma t 0 fi fi Delta fi fi h Gamma t 0 ff : 11) Here partial integration has been used under the assumption of vanishing boundary terms. This quadratic concept has the zero function t 0 (x) j 0 as template. It ....

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Itzykson, C. & Drouffe, J--M.: Statistical field theory. Vol 1 & vol 2, Cambridge University Press, 1989.

....This problem is particularly important for BM learning. This is because the BM learning rule requires the computation of correlations between neurons. Thus, learning in BMs requires exponential time. A well known approximate method to compute Z, or any other statistics, is by importance sampling (Itzykson and Drouffe, 1989). Glauber dynamics is an example of importance sampling. Importance sampling is more effective than the exact computation because the sampling is biased towards the parts of the configuration space that will give the dominant contribution to Z, but is still very time consuming. This is the ....

....field energy GammaE mf ( s) X i s i fW i i g (7) where we introduce n mean fields W i . The mean fields approximate the lateral interaction between neurons. The values of W i must be chosen such that this approximation is as good as possible. Following the standard mean field approach (Itzykson and Drouffe, 1989) the approximate free energy is given as GammaF = log Z 0 = X i log(2 cosh( i W i ) Gamma X i W i m i 1 2 X i;j w ij m i m j (8) with m i = tanh(W i i ) The mean fields are given by minimizing the free energy which gives the coupled set of mean field equations: m i = ....

Itzykson, C. and Drouffe, J.-M. (1989). Statistical Field Theory. Cambridge monographs on mathematical physics. Cambridge University Press, Cambridge, UK.

....for some architectures more efficient learning rules exits, as we shall see. 3 Decimation Recently an algorithm has been described [8] which computes the partition function of a BM with a tree structure in linear time (for the free and the clamped phase) The algorithm is based on decimation [3], and can be described as follows. Consider the two network fragments in Figure 1. In the left fragment S 2 and S 3 have an arbitrary number of connections with the rest of the network. S 1 , however, is only connected to S 2 and S 3 . By integrating out S 1 we obtain the network on the right. ....

C. Itzykson and J. Drouffe. Statistical Field Theory. Cambrigde University Press, Cambridge, 1991.

....systems [17] 18] 19] Kosterlitz and Thouless (KT) provided several descriptions of the transition, one of which rests on an analysis of the low fugacity dielectric response of a gas of charged dipoles or of vortex pairs. This analysis has since been very widely used and quoted (see e.g. 7] [14], 15] 16] 20] 23] in particular because it is so simple and has far reaching consequences. The renormalization parameter flow that has been derived by this theory is very well known, and constitues a basic paradigm in statistical physics. Extensions to three dimensions [11] 21] 25] 26] ....

....and contrast the results. The framework provided by equation (1.1) is not questioned any further. Among our major conclusions are that different additional assumptions lead to genuinely different theories, and that well known arguments or proofs that these assumptions are equivalent (see e.g. [14], 15] 18] 22] 27] are in error, for reasons we shall exhibit. Kosterlitz and Thouless were already aware that they produced at least two different theories, and felt that the corresponding results were close to each other. In the very useful paper by Young [27] some differences between the ....

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C. Itzykson & J.M. Drouffe, Statistical Field Theory, Cambridge University Press, Cambridge, 1989.

....over the phase space region of size Delta. It was observed [3] that the variety of scaling laws, Eqs. 1 4,6 ) follow from the single scaling law of the free energy density in two variables f( y t t; yh h) d f(t; h) 8) which is refered to as the generalized homogeneity principle [4]. Similar twodimensional scaling was also postulated for the correlation functions. The generalized homogeneity was later justified theoretically using the renormalization group approach [5, 6] Note that there are only two independent exponents y t ; y h in Eq. 8) All phenomenological exponents ....

....generalized homogeneity was later justified theoretically using the renormalization group approach [5, 6] Note that there are only two independent exponents y t ; y h in Eq. 8) All phenomenological exponents (1 6) can be derived from these two and consequently there are relations between them [4], 7] There is an important class of 2 In the particular case of the two dimensional Ising model ff = 0 and c diverges logarithmically. relations between phenomenological exponents of statistical systems which contains the dimensionality of the space d. Such relations are called hyperscaling ....

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C. Itzykson, J.-M. Drouffe, Statistical Field Theory, Cambridge University Press, 1989; and references to the Chapter 4.

....field approximation is to consider larger and larger clusters of spins (see Fig. 3) It can be shown that this modification indeed allows a more and more precise determination of the critical coupling. The first improvement (step 2 in the notation of Fig. 3) is also known as Bethe approximation [17]. In Tab. V we report the results of our analysis. It is easy to see that both with and without the spacelike plaquettes the plain mean field approximation is affected by an error which ranges from 10 to 15 (depending on the role which is played by the spacelike contribution in the Montecarlo ....

C. Itzykson and J.-M. Drouffe, Statistical Field Theory, CUP 1989.

....expected to behave like exp b oe (5) for 0 j 1 Gamma fi=fi c 1. The value of the exponent is oe = 1 2 and b is a non universal positive constant. At the critical temperature, the asymptotic behavior for r 1 of the two point correlation function should be (cfr. e.g. Ref. [12]) G(r) crit (ln r) 2 r j 1 O ln ln r ln r # ; 6) with j = 1 4 and = 1 16 . Near criticality, i.e. for 0 1, the behavior of the magnetic susceptibility can be deduced from Eq. 6) Z 0 dr G(r) crit 2 Gammaj (ln ) 2 1 O ln ln ln # ....

C. Itzykson and J. M. Drouffe, "Statistical field theory", Cambridge University Press (Cambridge 1989).

....of extreme orders need techniques and formalization so that the explicit computation in the end can be done by a computer. So the main task is the developement of efficient algorithms and implementations. The techniques of LCE are of course not new. An elementary introduction can be found in [1, 2]. The algorithmic formulation of this expansion for general O(N) symmetric scalar field models has been pioneered by Luscher and Weisz in the famous work [3] in the course of the solution of lattice Phi 4 theory in four dimensions [4, 5] They succeeded to compute two and four point ....

C. Itzykson, J.-M. Drouffe, "Statistical field theory", vol.2, Cambridge University Press, 1989.

.... particle as has been observed by Nelson [4] It turns out that the average path of a classical particle carrying out a Brownian motion (steps Deltax and Deltat, such that the diffusion coefficient d = 1 2 Deltax 2 = Deltat is finite) is also a fractal curve of Hausdorff dimension two [5, 6]. The Euclidean path integral of imaginary time quantum mechanics corresponds to a Wiener measure [5] The case of the Euclidean free scalar field is discussed by Itzykson and Drouffe [6] Quite generally, the measure of Euclidean path integrals gives the dominant contributions coming from field ....

.... coefficient d = 1 2 Deltax 2 = Deltat is finite) is also a fractal curve of Hausdorff dimension two [5, 6] The Euclidean path integral of imaginary time quantum mechanics corresponds to a Wiener measure [5] The case of the Euclidean free scalar field is discussed by Itzykson and Drouffe [6]. Quite generally, the measure of Euclidean path integrals gives the dominant contributions coming from field configurations corresponding to pathes of no where differentiable curves [7] By measuring the length L of a curve in terms of an elementary length Deltax, the geometrical property of ....

C. Itzykson and J.B. Drouffe, Statistical Field Theory, Cambridge Univ. Press., Cambridge (1989), Vol.I.

....between two successive iterations [G ii (z) G(z) This mapping onto a local problem coupled to an effective bath is reminiscent of the standard mean field description of spin systems. As is well known, such a theory emerges systematically from the same limit of large coordination number [32]. One major difference between electronic models and spin models is that in the former the Coulomb interaction introduces a nontrivial local dynamic that is preserved in the mean field approach. The molecular field for the electronic mean field theory is not a constant number, but is instead a ....

C. Itzykson and J.M. Drouffe, Statistical Field Theory, Vol. I & II (Cambridge University Press, Cambridge 1989).

....of field theory to polymer theory. For the case of a continuous polymer with a contact interaction (representing a generic short range interaction) the relation to a zero component OE 4 field theory was first demonstrated by de Gennes ( 2] and has been exploited by several people (see e.g. [3, 6, 4], and references therein) There also exists an analogous relation between the statistics of self avoiding walks on the lattice and a certain zero component spin model (see e.g. 9] Another interesting case is that of a polyelectrolyte, i.e. a polymer with a Coulomb pair interaction. It is ....

.... 4 in four dimensions and higher, a typical random walk self intersects too seldom for the interaction to make any fundamental difference. For the physical case of D = 3, the index has been measured with Monte Carlo methods, yielding 0:588, whereas for D = 2, it is known to be 3=4 (see e.g. [4]) For D 4, 0 . 3.2 Relation to a Spin Model Consider a ferromagnetic O(n) symmetric nonlinear sigma model on the lattice. At every site i, an n component spin S i is defined, restricted to the sphere S 2 i = n, with a sperically symmetric measure D[S] Q i d n S i ffi (S 2 i ....

Itzykson, C., Drouffe, J.-M. (1989): Statistical Field Theory (Cambridge Univ. Press, New York), volume 1, chapter 1.

....use of the Wilson loop as an order parameter of the theory. The language used here, as well as the introduction of a lattice, makes it more natural to use Euclidean rather than Minkowski space. Thus until (and including) section 3.3 we work on Euclidean space. As discussed in great detail in ref. [68] there is a natural interpretation of the theory on the lattice in terms of a string theory, which describes the flux tubes, and the string tension is defined by k = lim C 1 Gamma 1 A(C) ln W [C] 3:9) The non confining case thus corresponds to k = 0, which is the case if the Wilson loop ....

C. Itzykson and J.M. Drouffe, Statistical field theory, vol. 1 (Cambridge Univ. Press, 1989)

....L it coincides with Eq. 34) from l = 2 onwards. Of course, this should be expected on the grounds of symmetry on a torus under interchange of its sides L and fi, that is, modular symmetry. Incidentally, the exact free energy on the torus can be computed with some more sophisticated mathematics [19, 17]. Its two cylinder limits yield Eq. 34) or Eq. 40) Nevertheless, to show the modular invariance of the exact expression on the torus is not easy: It can be done performing its expansion in powers of the dimensionless modular invariant parameter m 2 A, with A the area of the torus, but it is ....

....to the critical point. The perturbation parameter is fi y . For the thermal perturbation y = 2 Gamma d Phi equals in general the inverse of the critical exponent . The expansions of Eq. 40) and (45) are instances of it and can be obtained from the respective c = 1 or c = 1=2 CFT [19]. The logarithmic term in them, which contributes to the relative entropy, is due to UV divergencies. The analysis of logarithmic divergences is done by examining the behaviour of the integrals of correlators. In fact, a logarithmic term appears at order 2 only for the Gaussian or the Ising ....

C. Itzykson and J. Drouffe, Statistical field theory, vol. II, Cambridge University Press, (1989).

....an example of such a Boltzmann tree is shown in Figure 1. Modifications to this basic architecture and the generalization to many output units will be discussed later. The key technique to compute partition functions and expectation values in these trees is known as decimation (Eggarter, 1974; Itzykson Drouffe, 1991). The idea behind decimation is the following. Consider three units connected in series, as shown in Figure 2a. Though not directly connected, the end units S 1 and S 2 have an effective interation that is mediated by the middle one S. Define the temperature rescaled weights J ij j w ij =T . We ....

....can incorporate certain intralayer connections into the tree at the expense of introducing a slightly more complicated decimation rule, valid when the unit to be decimated is biased by a connection to an additional clamped unit. There are also decimation rules for q state (Potts) units, with q 2 (Itzykson Drouffe, 1991). The algorithm for Boltzmann trees raises a number of interesting questions. Some of these involve familiar issues in neural network design for instance, how to choose the number of hidden layers and units. We would also like to characterize the types of learning problems best suited to ....

Itzykson, C. and Drouffe, J. (1991), Statistical Field Theory, Cambridge: Cambridge University Press.

....Especially in larger networks it is impossible to compute the expectation values directly from the Boltzmann Gibbs distribution. Instead, they can be estimated through Gibbs sampling. However, this method turns out to be very computation intensive. In the next section the technique of decimation [2, 4, 10] will be presented that allows the exact and efficient calculation of expectation values in certain Boltzmann machines. 2.2 Decimation in Boltzmann Trees We will describe the decimation rules proposed by [10] in a way that they are most easily extensible. The basic idea behind decimation is to ....

C. Itzykson and J. Drouffe. Statistical Field Theory. Cambridge University Press, 1991.

....with the original form (20) of the effective action, to the extent that improvements of the mean field approximation, obtained by considering larger and larger clusters of spins (see Fig. 1) become viable. In this way more precise estimates of the critical coupling can be obtained (see [8] for further details) The results of our mean field analysis for the SU(2) and SU(3) models are reported in Tab. I together with the N = 1 result already obtained in [1] Notice that the complexity of the analysis increases with N ; so we could reach a step 3 cluster in the SU(2) case, while ....

C. Itzykson and J.-M.. Drouffe, Statistical Field Theory, CUP 1989.

....learning times become linear in the number of neurons for tree like architectures. Kappen, 1995) show how strong inhibition between hidden neurons reduces the computation time to polynomial in the number of neurons. A well known approximate method to compute correlations is the Monte Carlo method (Itzykson and Drouffe, 1989), which is a stochastic sampling of the state space. Glauber dynamics is an example of such a method. The terms in the sum over states are proportional to a Boltzmann factor exp( GammaE) Monte Carlo methods can be more effective than the summation of all terms because the sampling is biased ....

....( s) tanh(W i i ) j m i ; 15) hs i s j i mf j X s s i s j pmf ( s) m i m j i 6= j; 16) where we have introduced the parameters m i , which are still to be fixed because of their dependence on W i . The real partition function Z, Eq. 4, can be computed in the mean field approximation (Itzykson and Drouffe, 1989): Z = X s exp( GammaE) X s exp( GammaE mf Emf Gamma E) Zmf hexp(Emf Gamma E)i mf Zmf exp(hEmf Gamma Ei mf ) Z 0 : 17) The mean field approximation is in the last step and is related to the convexity of the exponential function hexp fi exp hfi (Itzykson and Drouffe, ....

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Itzykson, C. and Drouffe, J.-M. (1989). Statistical Field Theory. Cambridge monographs on mathematical physics.

....; hm 2 1 i L Gamma(0:17 Sigma0:01) P max 1 L (0:09 Sigma0:01) 16) These results are in remarkable agreement among them and in good agreement with the spin wave exact value j(fi = 1:3) 0:12. Corrections due to vortices are equivalent to a higher effective temperature [24], that undergoes here a 30 shift. We have also established that a power fit to a non zero infinite volume order parameter of the form m max 1 (L) m max 1 (1) A L B ; 17) with m max 1 (1) different from zero and A and B constant is excluded by the data. Figures (4) and (5) are ....

C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge University Press 1989).

....the bulk critical behaviour be treated appropriately. A convenient way to achieve this is to choose it in such a manner that it reduces for all bulk quantities to a well established standard procedure. In our case this will be the conventional one based on normalization conditions (see, e.g. [35,36,19,20,3]) Alternatively, one could choose a massive approach based on minimal subtraction of poles, as described for the bulk case by Schloms and Dohm [12] 3.1. Bulk normalization conditions Starting from the bare bulk vertex functions Gamma (N;I) bulk ( m 2 0 ; u 0 ) we perform a mass shift ....

....correlation functions at the line c = 0: G (N;M) ren;sp ( m;u) Z Gamma(N M) 2 OE (Z sp 1 ) GammaM=2 G (N;M) m 0 ; u 0 ; c sp 0 ) 31) 4.2. Callan Symanzik type equations By varying m at fixed u 0 and c sp 0 the analogs of the Callan Symanzik equation (see references [3,19,20]) for the correlation functions G (N;M) ren;sp can be derived in a straightforward fashion: m m fi(u) u N M 2 j OE (u) M 2 j sp 1 (u) G (N;M) ren;sp ( m;u) DeltaG (32) with DeltaG j Gamma [2 Gamma j OE (u) m 2 Z V d d X G (N;M;1;0) ren;sp ( m;u) ....

[Article contains additional citation context not shown here]

Itzykson C., Drouffe J.-M. Statistical field theory. Cambridge, Cambridge University Press, 1989.

.... models, Boltzmann machines (Ackley, Hinton, Sejnowski, 1985) These have application in the areas of artificial intelligence as stochastic connectionist models (Russell Norvig, 1995; Jordan, Ghahramani, Jaakkola, Saul, 1998) image restoration (Geman Geman, 1984) and statistical physics (Itzykson Drouffe, 1989). The potential of a Boltzmann machine, with binary random variables s i 2 f0; 1g, is given by H 1 (s; w) X i w i s i 1 2 X i;j w ij s i s j (22) with w ij j w ji and w ii j 0. Unfortunately, Boltzmann machines are in general intractable since calculation of the normalizing constant ....

Itzykson, C., & Drouffe, J.-M. (1989). Statistical Field Theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press.

....lattice gauge theory at the deconfinement transition and the energy operator of the corresponding d dimensional statistical model. Consider for example the 2D Ising model: the shape and size dependence at criticality of the expectation value of the internal energy on a torus is given by [4, 5, 6] hffli = p =m jj( j 2 p AZ 1=2 ( 1) where A and are respectively the area and the modular parameter of the torus, and Z 1=2 is the Ising partition function at the critical point: Z 1=2 = 1 2 4 X =2 fi fi fi fi (0; j( fi fi fi fi (2) where are the Jacobi theta ....

....the area and the modular parameter of the torus, and Z 1=2 is the Ising partition function at the critical point: Z 1=2 = 1 2 4 X =2 fi fi fi fi (0; j( fi fi fi fi (2) where are the Jacobi theta functions and j is the Dedekind function (for notations and conventions see Ref. [6]) Comparing Eq. 1) with the finite size behavior of the plaquette operator in a 3D lattice gauge theory such that the center of the gauge group is Z 2 provides a stringent test of our identification. The simplest choice is the 3D Z 2 gauge model, for which it is possible to achieve very high ....

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C.Itzykson and J.Drouffe, "Statistical Field Theory", Cambridge 1989, Chap. 9.

....expansions for the polyelectrolyte can be simply derived from the loop expansion for the related field theoretical two point function, for which established computational methods exist. Introduction Relations between Euclidean field theories and random walks or polymers are well known [1, 2, 3, 4, 5, 6]. For a non interacting continuous polymer, equivalent to a random walk, the relation to the propagator of a free field theory is trivial. For the case of a continuous polymer with a delta function pair interaction, corresponding to a self avoiding random walk, de Gennes has shown the relation to ....

....to a random walk, the relation to the propagator of a free field theory is trivial. For the case of a continuous polymer with a delta function pair interaction, corresponding to a self avoiding random walk, de Gennes has shown the relation to a OE 4 field theory (Landau Ginsburg model) [2, 3, 4, 5]. A similar relation has been exploited for an unscreened polyelectrolyte [7] The focus of the present paper is to exploit the generic result for the special case of a screened polyelectrolyte, leading to a specific, very simple field theory. The generic interacting polymer A quite generic ....

C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge Univ. Press, New York 1989), vol. 1, chpt. 1.

....for the nonrigorous conformal field theory predictions for i is the assumption that a conformal field exists whose two point function is given by the Green s function of two Brownian motions starting at one point, ending at another point, conditioned so that the paths have no intersection. See [8, 15] for discussions of the methods of conformal field theory and [11] for the particular example of nonintersecting Brownian motions. While we are far from constructing a field with this two point function, we have made a start by constructing a measure on nonintersecting Brownian motions which ....

Ityzkon, C. and Drouffe, J.-M. (1989). Statistical Field Theory, Vol. 2, Cambridge University Press.

....P (V ) or work directly with P (HjV ) we will resort to an approximation from statistical physics known as mean field theory. 3. Mean Field Theory The mean field approximation appears under a multitude of guises in the physics literature; indeed, it is almost as old as statistical mechanics (Itzykson Drouffe, 1991). Let us briefly explain how it acquired its name and why it is so ubiquitous. In the physical models described by Markov networks, the variables S i represent localized magnetic moments (e.g. at the sites of a crystal lattice) and the sums P j J ij S j h i represent local magnetic fields. ....

Itzykson, C., & Drouffe, J.M. (1991). Statistical Field Theory. Cambridge: Cambridge University Press.

....limit (in the sector w = 0) of the OE 3;1 (integrable) perturbation of M 3;5 was found to be the theory H 3;1 3;5 = H 1;1 Omega H 1;1 ) Phi (H 3;1 Omega H 3;1 ) Phi (H 2;1 Omega H 3;1 ) Phi (H 4;1 Omega H 1;1 ) 5.17) which is genuinely braided. Our notation follows, for example, [16]. In this context our analysis seems to suggest that the conformal limit of the different massive perturbations of a conformal field theory correspond to different quantisations and global structures of the underlying conformal theory. It would be interesting to check this conjecture by analysing ....

C. Itzykson and J.-M. Drouffe, Statistical field theory, Volume 2 (Cambridge University Press, Cambridge, 1989)

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C. Itzykson and J.-M. Drouffe, Statistical Field Theory, (Cambridge University Press, Cambridge 1989).

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C. Itzykson and J.M. Drouffe, Statistical field theory , Vol. 1, Cambridge Univ. Press, 1989.

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Itzykson, C. & Drouffe, J--M. 1989. Statistical field theory. vol 1 & vol 2, Cambridge University Press, Cambridge, MA.

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Itzkyson, C. & Drouffe, J.--M., (1989) Statistical Field Theory. (Vols. 1 and 2) Cambridge: Cambridge University Press.

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C. Itzykon, J.-M. Drouffe (1989), Statistical Field Theory, Vol. 2, Cambridge University Press.

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C. Itzykson and J.-M. Drouffe, Statistical Field Theory, vol. 2 (Cambridge University Press, Cambridge, 1989).

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