Lattice recursive

The LST, on the other hand, explicitly takes into account all correlations (up to an arbitrary order) that arise between different cells on a given lattice, by considering the probabilities of local blocks of N sites. For one dimensional lattices, for example, it is simply formulated as a set of recursive equations expressing the time evolution of the probabilities of blocks of length N (to be defined below). As the order of the LST increases, so does the accuracy with which the LST is able to predict the statistical behavior of a given rule. 

Here the technique was first developed in a statistical mechanical framework [141], with in fact applications of the technique to other lattice combinatorial problems going back [145] to the 1940s. In this area the most focus has been on the infinite-length infinite-width limit as the solution for an extended 2-dimensional surface. In the resonance-theoretic context the treatment of some polymer chains of arbitrary width has also been made [142], A flow chart for subroutines for the recursion of the preceding section and its use in developing transfer matrices for finite-width chains is described in [143], Ref. [145] gives a... 

We have generated a chain of variables with diagonal matrix elements a and nearest-neighbor interaction b . The explicit calculation of and b can be performed with standard programs for any operator expressed in a local basis. In Fig. 2 we give a schematic picture of the chain-model variables generated by the recursion method in a simple two-dimensional square lattice. 

Figure 2. Two-dimensional square lattice (a) in the recursion scheme representation (fc). After defining o) = /o> 4o)i I f s)...

In recent years there has been an explosion of interest in the electron properties of disordered lattices. The more common line of approach to this kind of problem is to study the mean resolvent of the random medium, and the memory function methods can be of remarkable help for this purpose. Otherwise one can investigate by the memory function methods (basically the recursion method) a number of judiciously selected configurations this line of approach is particularly promising because it allows one to overcome some of the limitations inherent in the mean field theories. In this section we de-... 

An efficient terminator technique is certainly desirable in the application of recursion methods to the study of disordered systems. It has been shown recently that a self-consistently determined terminator can be fruitfully applied to calculate the electronic states in the Anderson model and to evaluate the vibrational spectrum of lattices with isotopic disorder. The basic idea is to extend the procedures discussed in Section IV to ensemble averages. In this case a useful generalization of Eq. (4.5), satisfied by the terminator t(E), is... 

Observe that needs to be computed only for the nodes at the left-most area of the lattice, because the electric field at the remaining points can be recursively calculated by the two left neighboring cells. This remark confirms the three-band nature of (6.26) and (6.28) and illustrates... 

If one has helical boundary conditions and spins that interact only with their nearest neighbors, one can repeatedly add just a single site and the bonds connecting it to its neighbors above and to the right. Analogously, the transfer matrix G can be used to compute recursively the conditional partition function of a lattice with one additional site... 

The decomposition and reconstruction lattice equations can then be rewritten as a simple recursive matrix multiplication. 

For reasons of convenience, and as will be shown in Chapter 6. the decomposition and reconstruction lattice equations can also be written as a pair of equations of recursive matrix products where we separate out the low-pass and high-pass filter coefficients in the form of infinite matrices Cj and Dy. That is. 

Figure 8.04. Comparison of densities of states for amorphous (solid line) and crystalline rhombohedral (dashed line) As. Experimental data are from XPS of Ley et al. (1973), and theoretical curves are from Kelly (1980) (recursion method) and from Pollard and Joannopoulos (1978) (cluster-Bethe-lattice method, CBLM) (Combined Figure from Elliott, 1984).

After examining hundreds of recursive lattice patterns, such as the full-page examples at the end of this chapter, I find that viewers often prefer high values of the fractal dimension of around 1.8, which is close to the average D for all the patterns in this chapter. The p r) curves for the preferred structures usually do not exhibit global features (bumps and valleys). To date, I have not been able to determine a correlation between perceived beauty and the parameter. Readers may wish to compute the lattice patterns and also look for correlations between perceived beauty and the fractal dimension. 

Not all individuals to whom I showed the recursive patterns found them of esthetic interest. After seeing the 7-by-7 recursive lattice designs. Dr. Michael Frame of Yale University commented ... 

FIGURE 27 J. Recursive lattice designs on a triangular lattice (see text). 

Recursion Recursive Lattices Beauty and the Bits Higher Symmetry Digressions... 

Useful information can be obtained from models amenable to exact analysis even if they look artificial. Real space renormalization group approach can be handled in an exact fashion for a class of tailor-made lattices called hierarchical lattice. Such lattices are constructed in a recursive fashion as shown in Fig. 1. The problem of a directed polymer in a random medium on hierarchical lattices has been considered in Ref. [44,45]. Here we consider the RANI problem on hierarchical lattices. As already noted, the effective... 

In a similar approach on the same lattice DM derive recursive relations involving distributions instead of the constant fugacity A on each bond visited by a SAW which corresponds to a bimodal distribution. This relation for the distribution is... 

The first family we shall discuss is the n-simplex family, defined for all positive integers n > 2. The n = 3 case was first defined by Nelson and Fisher [11], who called it truncated tetrahedron lattice. The construction was generalized for arbitrary n in [12]. The recursive construction of the graph of the fractal is shown in Fig. 2. The first order graph is a single vertex with n bonds. In general, the r-th order graph will have n vertices, and... 

Figure 5. The recursive construction of the modified rectangular lattice for d = 2. (a) The graph of first order square.(b) the (r + l)-th order graph, formed by joining 2 r-th order graphs shown as shaded rectangles here (c) graph of the 5th order rectangle.

Figure 7. Graphical representation of terms that contribute to the recursion equation for fQj. the 3-simplex lattice.

For the 5-simplex lattice, the closed loop generating function can be found in terms of variables A > and B defined as before. In this case, the recursion equations are more complicated [23]. We list these here to show how the complexity of the polynomials rises rapidly with increasing n ... 

The behavior of SAWs on the d = 2 lattice was studied in [17]. The recursion equation for a polygon is written in terms of five restricted generating functions (see Fig. 12) by constructing graphs by all possible ways [17]. This gives... 

To calculate G x,u) for the 3-simplex lattice, we define restricted partition functions B and, for walks that cross the r-th order triangle, as shown in Fig. 13. is the sum of weight of all walks that enter an r-th order triangle of the 3-simplex from one corner and leave from another corner, but do not visit the third corner. Bi is the sum of weights of walks that enter and leave the r-order triangle, and also visit the third corner of the triangle. Then it is easy to see that these weights satisfy the recursion equations... 

The collapse transition on the 4-smiplex lattice wcis first studied in [35]. We restrict the attractive interaction to nearest neighbors within the Scune 2nd order simplex. Then, the recursion equations Eq. (32-33) describe the collapse transition also, if we use the... 

The grand partition function of Eq. (85) for 5- simplex lattice is written in terms of six restricted partition functions shown in Fig 19. Out of six configurations two (A i and B >) represent the sums of weights of configurations of the polymer chain within one r-th order subgraph away from surface, and the remaining four S E and represent the surface functions. As in the case of the 4-simplex, the recursion relations for and do not include other variables. 

For the 5-simplex lattice we also need five rc.strictcd partition functions to describe the generating functions of two walks two corresponding to walk Pi, two corresponding to walk P2 and which represents the configurations in which walks Pi and P2 enter and exit the r-th order subgraph once each and may occupy neighboring sites. The recursion relation for E in this case is lengthy[60] and is therefore not reproduced here. 

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