The Euclidean Matching Problem by Gabriele Sicuro

By Gabriele Sicuro

This thesis discusses the random Euclidean bipartite matching challenge, i.e., the matching challenge among assorted units of issues randomly generated at the Euclidean area. The presence of either randomness and Euclidean constraints makes the examine of the common homes of the answer hugely appropriate. The thesis stories a couple of recognized effects approximately either matching difficulties and Euclidean matching difficulties. It then is going directly to offer a whole and basic answer for the only dimensional challenge with regards to convex price functionals and, furthermore, discusses a possible method of the common optimum matching rate and its finite measurement corrections within the quadratic case. The correlation capabilities of the optimum matching map within the thermodynamical restrict also are analyzed. finally, utilizing a useful strategy, the thesis places ahead a common recipe for the computation of the correlation functionality of the optimum matching in any measurement and in a widespread domain.

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30) n Notably, Q αβ are not merely a set of variables. 31) is the average respect to the replicated system whose Hamiltonian func⎡ n N ⎣− H R [{σ α }α ; J, h] := Ji j σiα σ jα + h α=1 i< j ⎤ σiα ⎦ . 32) i=1 The variables Q αβ play the role of spin glass order parameters, as we will see. If we suppose that all replicas are equivalent (replica symmetric hypothesis) β Q αβ = σiα σi R = σiα R β σi = σi R 2 = q =: qEA , α = β. 33) The quantity qEA is called Edward–Anderson order parameter. We expect that for β → 0 the spins are randomly oriented, and therefore qEA = 0, whilst in the β → +∞ limit qEA > 0, having σi = 0 for each realization.

24) where we have introduced ⎛ exp ⎝β 2 z[q] := ⎞ σ α ⎠ , q := (qαβ )αβ . ,σ n Observe that z appears as a sort of partition function for a set of n coupled spins, each one associated to one of the n replica indexes. Being the exponent of the integrand proportional to N , we can use the steepest descent method to evaluate the expression. In particular, the extremality condition respect to the generic element qαβ gives qαβ = ∂ ln z[q] ≡ σ ασ β z. 26) In Eq. 26) we have adopted the notation ⎞ ⎛ O σ 1, .

42) j=0 In the previous equation, Q lαβ is simply the l-th power of the element Q αβ of the matrix Q. 43) i=1 where I[a,b] (x) is the indicator function. 44) and therefore q(x) is defined in the interval [0, 1]. At the same time, m j − m j+1 → − d x and we can write 1 n 1 k Q lαβ αβ = (m j − j=0 k→+∞ m j+1 )q lj −−−−→ − q l (x) d x. 45) 42 3 Random Optimization Problems and Statistical Mechanics From Eqs. 34), we obtain β − β f¯ ∼ − 2 2 4 1 Q 2αβ + β 2 4 α=β + 2 1 ln z[Q]. 47a) q(x) d x . 48) where the function P(x, h) satisfies the so called Parisi equation: 2 dq(x) ∂ 2 P(x, h) ∂ P(x, h) +x =− ∂x 2 dx ∂h 2 ∂ P(x, h) ∂h , P(1, h) = ln (2 cosh βh) .

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