7 research outputs found
A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0-1 Matrices
We consider two ensembles of 0-1 nxn matrices. The first is the set of all
nxn matrices with entries zeroes and ones such that all column sums and all row
sums equal r, uniformly weighted. The second is the set of nxn matrices with
zero and one entries where the probability that any given entry is one is r/n,
the probabilities of the individual entries being i.i.d.'s. Calling the two
expectations E and E_B respectively, we develop a formal relation
E(perm(A))=E_B(perm(A)) exp( sum T_i ) (A1)
We use two well known approximating ensembles to E, E_1 and E_2. Replacing E
by either E_1 or E_2 we can evaluate all terms in (A1). For either E_1 or E_2
the T_i have amazing properties. We conjecture that all these properties also
hold for E. We carry through a similar development treating E(perm_m(A)), with
m proportional to n, in place of E(perm(A)).Comment: 7 page
MATCHINGS IN REGULAR GRAPHS: MINIMIZING THE PARTITION FUNCTION
For a graph G on v(G) vertices let m(k)(G) denote the number of matchings of size k, and consider the partition function M-G(lambda) = Sigma(n)(k=0)m(k)(G)lambda(k). In this paper we show that if G is a d-regular graph and 0 1/v(Kd+1) ln MKd+1(lambda).The same inequality holds true if d = 3 and lambda < 0.3575. More precise conjectures are also given