اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFrozen $1$-RSB structure of the symmetric Ising perceptron
51
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB structure conjectured by Krauth and Mezard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.
قيم البحث
اقرأ أيضاً
Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and normal distributions for various functionals of the process.
This is the first of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising models on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluste
r boundaries in the sum of two independent currents with free and wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilsons conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with the second paper, this completes the proof of conformal invariance.
Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin patterned ensemble
s sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a dial we can turn from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f_m show a visually stunning convergence to the semi-circle as m tends to infinity, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f_m is the product of a Gaussian and a degree 2m-2 polynomial; the formula equals that of the m x m Gaussian Unitary Ensemble (GUE). The proof is by the moments. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.
This is the second of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluste
r boundaries in the sum of two independent currents both with free or wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilsons conjecture on the XOR-Ising model. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the continuum Gaussian Free Field. In this paper, we derive crossing properties of the discrete models required to prove this characterization.
We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field, where a subset of vertices is designated as the boundary. We show that the influence of boundary conditions on any given spin is maximised when the external fi
eld is identically $0$. One corollary is that spin-spin correlations are maximised when the external field vanishes and the boundary condition is free, which proves a conjecture of Shlosman. In particular, the random field Ising model on ${mathbb Z}^d$, $dgeq 3$, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model in $dgeq 3$ satisfies the conjectured strong spatial mixing property in the entire high temperature regime.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
