Spin glasses : a challenge for mathematicians : cavity and mean field models

By Michel Talagrand. 2003

367 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/88c16b9b9a7a59a1f548c1151dd79460f2e79c9b

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Citations in the corpus

A generalization of the Lindeberg principle (2005, methodology): “e well-known “Gaussian interpolation technique” for bounding |E 0f 0(U)−E 0f 0(Z˜)|. This classical method for proving Slepian-type inequalities has been used extensively in recent years by Talagrand [19] in his eﬀorts to obtain a rigorous version of the cavity method for spin glasses. For each t∈[0”
An error bound in the Sudakov-Fernique inequality (2005, background): “A proof of this lemma can be found in the appendix of [13], for example.”
Mean-field theory of hard sphere glasses and jamming (2008, result): “In principle, the analytic continuation might not be well defined, but it has been verified explicitely in mean field models that the procedure gives the correct results; see (Franz and Tria, 2006; Mézard, 1999; Mézard and Parisi, 1999a; Mézard et al., 1987; Monasson, 1995; Talagrand, 2003) for a mo”
Quenched invariance principle for simple random walk on percolation clusters (2005, background): “, various diluted spin sys tem , random copolymers [21], spin glasses [5,22], random-graph models [4], etc.”
Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, background): “See, for example, Proposition 1.1.3 in Talagrand (2003).”
The committee machine: computational to statistical gaps in learning a two-layers neural network (2018, methodology): “Our first contribution is a rigourous proof of the corresponding free entropy formula using an interpolation method [24, 33, 34].”
Quantitative Breuer-Major Theorems (2010, methodology): “Talagrand [35].”
The mean field traveling salesman and related problems (2010, background): “The replica and cavity methods of statistical mechanics, originally developed in the study of spin glasses and other disordered systems, have led to a number of purely mathematical predictions [30, 43].”
On the energy landscape of deep networks (2015, unknown): “We consider a model for deep networks with random, sparse weights; we show that the loss function of such a network resembles the Hamiltonian of a spin glass which is a well-studied model in statistical physics (Talagrand, 2003).”
Message passing algorithms for compressed sensing: I. motivation and construction (2009, background): “Eq. (15). More than 30 years after the original paper, a complete mathematical justiﬁcation of the TAP equations remains an open problem in spin glass theory, although important partial results exist [11]. While the connection between belief propagation and Bethe-Peierls approximation stimulated a c”
Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model (2002, methodology): “The methods developed in the previous Sections can be easily extended to the Derrida p-spin model [2], [4], [3], [ 17 ].”
Estimating and understanding exponential random graph models (2011, background): “Mean-field approximations have no rigorous foundation, however, and are known to be unreliable in related models such as spin glasses [53].”