By Philip W. Anderson
Easy Notions of Condensed subject Physics is a transparent advent to a couple of the main major suggestions within the physics of condensed subject. the final rules of many-body physics and perturbation conception are emphasized, delivering supportive mathematical constitution. this is often a selection and restatement of the second one 1/2 Nobel Laureate Philip Anderson’s vintage thoughts in Solids.
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500 2000 1500 1000 Pictures depicting Rp /Rp (D), for p < 2000, D as in Table 2. ) of degree k2 in log(T /2π) (modulo terms that vanish faster than any inverse power of log T /2π as T → ∞). Unfortunately it is not easy to see directly how to combine the coefﬁcients in (91) with arithmetical information to guess the form of the coefﬁcients of the lower-order terms in the moments of the zeta function. 2 (2πi)2k N Pk ··· e2 j=1 zj −zj+k G(z1 , . . , z2k )∆2 (z1 , . . , z2k ) 2k 2k i=1 zi dz1 · · · dz2k , (92) where the contours are small circles around the origin, ∆(z1 , .
For planar graphs however, the simplicity of the matrix model solutions has ﬁnally been explained combinatorially by Schaeffer , who found various bijections between planar graphs and trees, allowing for a simple enumeration, and a precise contact with the matrix model solutions . A remarkable by-product of this approach is that one may keep track on the trees of some features of the planar graphs, such as geodesic distances between vertices or faces  , a task beyond the reach of matrix models so far.
3)). The Hard-Dimer model is therefore generated by an integral of the form (18), with only g4 and g6 non-zero, and more precisely g4 = g and g6 = 3g2 z (=three 43 2D Quantum Gravity, Matrix Models and Graph Combinatorics (a) 2 3 (b) 2 3 6 5 3 4 1 4 1 1 2 6 5 2 4 6 5 3 1 4 6 5 Figure 4. A 4-valent planar graph with hard dimers, represented by thickened edges. The corresponding graph obtained by shrinking the dimers (b) has both 4-valent and 6-valent vertices. The correspondence is three-to-one per dimer, as shown.