We study the three-dimensional Edwards-Anderson model with binary interactions by Monte Carlo simulations. Direct evidence of finite-size scaling is provided, and the universal finite-size scaling functions are determined. Monte Carlo data are extrapolated to infinite volume with an iterative procedure up to correlation lengths ξ ≈ 140. The infinite volume data are consistent with a conventional power law singularity at finite temperature Tc. Taking into account corrections to scaling, we find Tc = 1.156 ± 0.015, ν = 1.8 ± 0.2 and η = −0.26 ± 0.04. The data are also consistent with an exponential singularity at finite Tc, but not with an exponential singularity at zero temperature.PACS numbers: 75.10. Nr, 64.60.Fr, 75.40.Mg, 75.50.Lk The critical properties of the Ising spin glass in three dimensions are still not very well understood. Numerical simulations have led to some progress [1,2], but have been hampered by technical difficulties. Large-scale Monte Carlo (MC) simulations at correlation length ξ ≈ 10 lattice units [3][4][5] are consistent with both a continuous phase transition with power-law divergence of ξ at finite temperature T = T c , and an exponential divergence at T = 0, which is expected at the lower critical dimension. High-statistics MC simulations of smaller systems [6][7][8] give a certain evidence of a T c = 0 transition with an ordered spin glass phase below T c , but cannot exclude neither an exponential divergence at T = 0, nor a line of critical points at T ≤ T c = 0 [6,8,9], as in the KosterlitzThouless theory of the 2D XY model. Understanding whether an ordered spin glass phase exists in three dimensions is clearly an issue of major interest.In this work, we study the 3D Ising spin glass with an approach, based on finite-size scaling (FSS) and MC simulations in the paramagnetic phase, introduced in Ref.[10] (see Ref.[11] for similar methods) and so far applied to non-disordered systems. Let us summarize our main results. (i) We provide a direct test of the FSS hypothesis, independent of the nature of the divergence in the infinite system. In particular we determine, for the first time to our knowledge, the universal FSS functions.(ii) We demonstrate the effectiveness of an iterative procedure to extrapolate the MC data to infinite volume, that allows us to reach ξ ≈ 140. (iii) Exploiting the higher range of ξ, we show that an exponential divergence at T = 0 is excluded, but we still cannot decide between a power-law divergence at T c = 0 and a line of critical points terminating at T c = 0. (iv) Under the hypothesis of power-law divergence, we show that corrections to scaling are important and we estimate T c and the critical exponents.Model and FSS method -We consider the 3D Edwards-Anderson model, whose Hamiltonian iswhere σ x are Ising spins on a simple cubic lattice of linear size L with periodic boundaries, and J xy are independent random interactions taking the values ±1 with probability 1 2 . The sum runs over pairs of nearest neighbor sites. Let ξ(T, L) be a suitably defined...
We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large N limit of the average cost in dimension d = 1,2 and of the subleading correction in higher dimension. A nontrivial scaling exponent, γ(d) = d-2/d, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension d > 2.
We prove a generalization of Kirchhoff's matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a q → 0 limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N -vector model at N = −1 or, equivalently, onto the σ-model taking values in the unit supersphere in R 1|2 . It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.PACS numbers: 05.50.+q, 02.10. Ox, 11.10.Hi, 11.10.Kk Kirchhoff's matrix-tree theorem [1] and its generalizations [2], which express the generating polynomials of spanning trees and rooted spanning forests in a graph as determinants associated to the graph's Laplacian matrix, play a central role in electrical circuit theory [3] and in certain exactly-soluble models in statistical mechanics [4,5]. Like all determinants, those arising in Kirchhoff's theorem can of course be rewritten as Gaussian integrals over fermionic (Grassmann) variables.In this Letter we prove a generalization of Kirchhoff's theorem in which a large class of combinatorial objects are represented by suitable non-Gaussian Grassmann integrals. Although these integrals can no longer be calculated in closed form, our identities allow the use of field-theoretic methods to shed new light on the critical behavior of the underlying geometrical models.As a special case, we show that unrooted spanning forests, which arise as a q → 0 limit of the q-state Potts model [6], can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. Furthermore, this latter model can be mapped, to all orders in perturbation theory,It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free, in close analogy to (large classes of) two-dimensional σ-models and four-dimensional nonabelian gauge theories. Indeed, this fermionic model may, because of its great simplicity, be the most viable candidate for a rigorous nonperturbative proof of asymptotic freedom -a goal that has heretofore remained elusive in both σ-models and gauge theories.The plan of this Letter is as follows: First we prove some combinatorial identities involving Grassmann integrals, culminating in our general formula (12), and show how a special case yields unrooted spanning forests. Next we show that this latter model can be mapped onto the N -vector model at N = −1, and use this fact to deduce its renormalization-group (RG) flow at weak coupling. Finally, we conjecture the nonperturbative phase diagram in this model.Combinatorial Identities. Let G = (V, E) be a finite undirected graph with vertex set V and edge set E. Associate to each edge e a weight w e , which can be a real or complex number or, more generally, a formal algebraic variable. For i = j, let...
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