Criticality and conformality in the random dimer model

Abstract

In critical systems, the e ect of a localized perturbation a ects points that are arbitrarily far away from the perturbation location. In this paper, we study the e ect of localized perturbations on the solution of the random dimer problem in 2𝐷. By means of an accurate numerical analysis, we show that a local perturbation of the optimal covering induces an excitation whose size is extensive with nite probability. We compute the fractal dimension of the excitations and scaling exponents. In particular, excitations in random dimer problems on non-bipartite lattices have the same statistical properties of domain walls in the 2𝐷 spin glass. Excitations produced in bipartite lattices, instead, are compatible with a loop-erased self-avoiding random walk process. In both cases, we nd evidences of conformal invariance of the excitations, that are compatible with SLE 𝜅 with parameter 𝜅 depending on the bipartiteness of the underlying lattice only.

Energy landscapes of some matching-problem ensembles

“…Since then, it has been adapted to e.g. random-field Ising models [59,60], the traveling salesperson problem [23], and matching [46,49]. The basic idea is to compute a ground state first and then slightly perturb the system.…”

“…For the matching problem, we use edge flips as a perturbation technique, similar to [49] or like a 'spin flip' for spin glasses. First, a maximum-weight matching M 0 with total weight W 0 is computed.…”

“…The interest of matching problems in physics also stems from the fact that other problems can be solved by mapping to suitable matching problems. This has been done for 2D spin glasses [33], dimer covering [48,49], negative-weight percolation [50] and controllability of dynamical networks [51,52].…”

“…However, later an extended analysis [54] showed that an RS ansatz is sufficient to obtain the same results as in [53] and that ergodicity is not broken in this model. Furthermore, in [49] perfect matchings, also called random dimers, on lattice graphs with L × L sites were investigated. It was observed that removing a single matched edge and recomputing the optimal matching can affect arbitrary far points.…”

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

“…Statistical mechanics predicts the properties of a macroscopic system from the laws of its microscopic dynamics [2][3][4]. In this area, a major role is played by phase transitions that regulate what is achievable in principle (information theoretical thresholds) and what is achievable in practice (algorithmic thresholds) [5][6][7][8][9][10][11][12][13]. The graphical representation (the phase diagram) of the various phases delimited by phase transitions allows researchers to predict the response of the system as a function of its own tunable parameters.…”