Bypassing correlation decay for matchings with an application to XORSAT

Lelarge, Marc

doi:10.1109/itw.2013.6691322

Cited by 7 publications

(26 citation statements)

References 19 publications

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“…Since the upper bound on the rank follows from [30], (2.11) implies that rk(A n )/n converges to 1 − max α∈[0,1] Φ(α) in probability, as claimed.…”

Section: Proof Of Theorem 11mentioning

confidence: 81%

“…Since adding or removing a single row can only change the rank by one, Azuma's inequality shows that it suffices to compute E[rk(A n )]. In fact, since the upper bound on the rank already follows from [30], we merely need to bound E[rk(A n )] from below, or equivalently bound E[nul(A n )] from above. We are thus tempted to write…”

Section: Proof Strategymentioning

confidence: 99%

“…However, Lelarge [30] refuted this prediction. Indeed, combining the formula for the matching number of random bipartite graphs from [8] with the Leibniz determinant formula, Lelarge obtained the upper bound 1 limsup n→∞ rk(A)…”

Section: Introductionmentioning

confidence: 98%

“…For finite fields the formula (1.3) was established recently [11]. Moreover, a simple but elegant argument due to Lelarge [30] shows that the the r.h.s. of (1.3) provides an upper bound on the rank over any field.…”

Section: Introductionmentioning

confidence: 98%

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The rank of sparse random matrices

Coja-Oghlan¹,

Ergür²,

Gao³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix A with prescribed numbers of non-zero entries in each row and column over any field F. The rank formula turns out to be independent of both the field and the distribution of the non-zero matrix entries. The proofs are based on a blend of algebraic and probabilistic methods inspired by ideas from mathematical physics.MSC: 05C80, 60B20, 94B05Amin Coja-Oghlan's research received support under DFG CO 646/3. Alperen A. Ergür partially funded by Einstein Foundation, Berlin.

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“…Since the upper bound on the rank follows from [30], (2.11) implies that rk(A n )/n converges to 1 − max α∈[0,1] Φ(α) in probability, as claimed.…”

Section: Proof Of Theorem 11mentioning

confidence: 81%

Section: Proof Strategymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

The rank of sparse random matrices

Coja-Oghlan¹,

Ergür²,

Gao³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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show abstract

“…Due to the N P -hardness of calculating ln Z in general, as discussed earlier, this can of course only be true for very specific instances of random CSPs (if we suppose P = N P ). Indeed, it turns out that the Bethe approximation by Belief Propagation, which we will call the procedure above, yields the correct value of the partition function sometimes [50,52,145] and sometimes it does not [126]. The latter is assumed to happen, if the system's solution space had undergone a phase transition at which replica symmetry breaking occurred.…”

Section: Boltzmann Marginals and The Partition Functionmentioning

confidence: 99%

Large discrete structures : statistical inference, combinatorics and limits

Hahn‐Klimroth¹

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Studying large discrete systems is of central interest in, non-exclusively, discrete mathematics, computer sciences and statistical physics. The study of phase transitions, e.g. points in the evolution of a large random system in which the behaviour of the system changes drastically, became of interest in the classical field of random graphs, the theory of spin glasses as well as in the analysis of algorithms [78,82, 121]. It turns out that ideas from the statistical physics’ point of view on spin glass systems can be used to study inherently combinatorial problems in discrete mathematics and theoretical computer sciences(for instance, satisfiability) or to analyse phase transitions occurring in inference problems (like the group testing problem) [68, 135, 168]. A mathematical flaw of this approach is that the physical methods only render mathematical conjectures as they are not known to be rigorous. In this thesis, we will discuss the results of six contributions. For instance, we will explore how the theory of diluted mean-field models for spin glasses helps studying random constraint satisfaction problems through the example of the random 2−SAT problem. We will derive a formula for the number of satisfying assignments that a random 2−SAT formula typically possesses [2]. Furthermore, we will discuss how ideas from spin glass models (more precisely, from their planted versions) can be used to facilitate inference in the group testing problem. We will answer all major open questions with respect to non-adaptive group testing if the number of infected individuals scales sublinearly in the population size and draw a complete picture of phase transitions with respect to the complexity and solubility of this inference problem [41, 46]. Subsequently, we study the group testing problem under sparsity constrains and obtain a (not fully understood) phase diagram in which only small regions stay unexplored [88]. In all those cases, we will discover that important results can be achieved if one combines the rich theory of the statistical physics’ approach towards spin glasses and inherent combinatorial properties of the underlying random graph. Furthermore, based on partial results of Coja-Oghlan, Perkins and Skubch [42] and Coja-Oghlan et al. [49], we introduce a consistent limit theory for discrete probability measures akin to the graph limit theory [31, 32, 128] in [47]. This limit theory involves the extensive study of a special variant of the cut-distance and we obtain a continuous version of a very simple algorithm, the pinning operation, which allows to decompose the phase space of an underlying system into parts such that a probability measure, restricted to this decomposition, is close to a product measure under the cut-distance. We will see that this pinning lemma can be used to rigorise predictions, at least in some special cases, based on the physical idea of a Bethe state decomposition when applied to the Boltzmann distribution. Finally, we study sufficient conditions for the existence of perfect matchings, Hamilton cycles and bounded degree trees in randomly perturbed graph models if the underlying deterministic graph is sparse [93].

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The rank of sparse random matrices

Coja-Oghlan

Ergür

Gao

et al. 2022

Random Struct Algorithms

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We determine the asymptotic normalized rank of a random matrix A over an arbitrary field with prescribed numbers of nonzero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge ( 2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.

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Bypassing correlation decay for matchings with an application to XORSAT

Cited by 7 publications

References 19 publications

The rank of sparse random matrices

The rank of sparse random matrices

Large discrete structures : statistical inference, combinatorics and limits

The rank of sparse random matrices

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