By Alexander Barvinok
Partition features come up in combinatorics and similar difficulties of statistical physics as they encode in a succinct approach the combinatorial constitution of complex structures. the focus of the booklet is on effective how you can compute (approximate) quite a few partition features, reminiscent of permanents, hafnians and their higher-dimensional types, graph and hypergraph matching polynomials, the independence polynomial of a graph and partition capabilities enumerating 0-1 and integer issues in polyhedra, which permits one to make algorithmic advances in differently intractable problems.
The booklet unifies quite a few, frequently really contemporary, effects scattered within the literature, focusing on the 3 major ways: scaling, interpolation and correlation decay. The must haves contain reasonable quantities of actual and intricate research and linear algebra, making the booklet obtainable to complicated math and physics undergraduates.
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Extra info for Combinatorics and Complexity of Partition Functions
Tn 1 ln ti ln ti ≥ n i=1 n n 58 3 Permanents for all t1 , . . , tn with equality if and only if t1 = . . = tn . Applying it with ti = 1 − ain , we get 1 n n (1 − ain ) ln (1 − ain ) ≥ i=1 n−1 n−1 ln n n with equality if and only if ain = 1/n for i = 1, . . , n. In other words, n (1 − ain )1−ain ≥ i=1 n−1 n n−1 with equality if and only if ain = 1/n for i = 1, . . , n. /n, we must have ain = 1/n for i = 1, . . , n. Since the matrix obtained from a doubly stochastic matrix by a permutation of columns remains doubly stochastic with the same permanent, we conclude that ai j = 1/n for all i and j as desired.
An ], v = per[a1 , . . , an ] and 2 2 1 w = per[a2 , a2 , a3 , . . , an ]. 1). Indeed, if v 2 < uw then the univariate polynomial t −→ u + 2vt + wt 2 has a pair of complex conjugate roots α±βi for some β > 0. Then, for any > 0, the point z 1 = 1+i , z 2 = (α+βi)(1+i ) is a root of q(z 1 , z 2 ) and if > 0 is sufficiently small, we have z 2 = α + β > 0, which contradicts the H-stability of q. 1) to the Alexandrov - Fenchel inequality for mixed volumes is as follows. Let K 1 , . . , K n ⊂ Rn be convex bodies and let λ1 , .
Suppose further that the highest degree terms of g1 , . . , gm have the same sign. Let λ1 , . . , λm be non-negative reals, not all 0 and let m g= λk gk . k=1 Then the polynomial g interlaces f ; (2) Let f and g be real polynomials such that g interlaces f and suppose that the highest terms of f and g have the same sign. Then for any λ ∈ R the polynomial f interlaces the polynomial h(x) = (x − λ) f (x) − g(x). Fig. 3 Polynomials with Real Roots 29 Proof. Let α1 < . . < αn be the roots of f , so deg f = n.