The Asymptotics of Quantum Max-Flow Min-Cut

P. Elias, A. Feinstein, and C. E Shannon, “A note on the maximum flow through a network”, Information Theory, IRE Transactions on, 2

1956

Earlier work this paper cites.

L. R Ford and D. R Fulkerson, “Maximal flow through a network”, Canadian journal of Mathematics, 8(3)

1956

Earlier work this paper cites.

Since the limiting distribution in the Gaussian orthogonal ensemble is the Wigner semi-circle, which is bounded, one can deduce that the expected trace of the k k -th moment of a matrix drawn from this ensemble is at most exponential in k k . See also E. Brezin, C. Itzykson, G. Parisi, and J. B. Zuber, “Planar Diagrams”, Commun. Math. Phys. 59

1978

Earlier work this paper cites.

E.V. Shuryak and J.J.M. Verbaarschot, Nucl. Phys. A 560

1993

Cited alongside, same era.

J.J.M. Verbaarschot, Phys. Rev. Lett. 72

1994

Cited alongside, same era.

S. X. Cui, M. H. Freedman, O. Sattath, R. Stong, and G. Minton, “Quantum Max-flow/Min-cut”, arXiv:1508.04644

Original

Cited in the paper.

For a sequence of distributions where the moments converge to a limit c ⁡ ( G , k ) c(G,k) obeying Carleman’s condition ∑ k = 1 ∞ c ​ ( G , 2 ​ k ) 1 2 ​ k = ∞ \sum_{k=1}^{\infty}c(G,2k)^{\frac{1}{2k}}=\infty , the distributions converge weakly to a limiting distribution. See http://mathoverflow.net/questions/230794/converging-to-moments-obeying-carlemans-condition . Sketch: since the second moment is bounded, the sequence is tight; by Prokhorov’s theorem, there is a subsequence which converges to a limit μ \mu ; the moments of the limiting distribution μ \mu are the limits of the moments (to show that μ \mu has the correct k k -th moment, use a bound on a higher moment to establish uniform integrability) so by Carleman’s condition, all subsequences which converge must converge to the same limit; so by Prokhorov’s theorem, the sequence converges without needing to pass to a subsequence. In this particular case, since the 2 ​ k 2k -th moments of μ N ind \mu^{\rm ind}_{N} (or μ N \mu_{N} ) are bounded by c 1 ⋅ c 2 k c_{1}\cdot c_{2}^{k} for some constants c 1 , c 2 c_{1},c_{2} , it is simpler. To show that ∫ f ⁡ ( x ) ​ d ​ μ N ind ​ ( x ) \int f(x){\rm d}\mu^{\rm ind}_{N}(x) has a limit as N → ∞ N\rightarrow\infty for bounded Lipschitz functions f ⁡ ( x ) f(x) , approximate f ⁡ ( x ) f(x) on an interval [ − c , + c ] [-c,+c] by a polynomial p ⁡ ( x ) p(x) for any c > c 2 c>c_{2} and use a bound on higher moments to bound the integral ∫ | x | > c p ⁡ ( x ) ​ d ​ μ N ind ​ ( x ) \int_{|x|>c}p(x){\rm d}\mu^{\rm ind}_{N}(x)

D. Calegari, M. Freedman, and K. Walker, “Positivity of the universal pairing in 3 dimensions”, Jour. Amer. Math. Soc. 23

2010

Later among the works it cites.

“Rainbow diagrams” for random matrix theory are another term for “planar diagrams”. This is a class of diagrams that appear in computing expectation values of traces of powers of a random matrix. See, for example, A. Zee, Quantum Field Theory in a Nutshell

2010

Later among the works it cites.