Replica symmetry of the minimum matching

Wästlund, Johan

doi:10.4007/annals.2012.175.3.2

Cited by 23 publications

(54 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Wästlund [16] explicitly left open the problem of completing the proof of the original Mézard-Parisi prediction by showing (i) that the untruncated cavity equation admits a unique solution f and (ii) that f λ → f as λ → ∞. The purpose of this short paper is to establish this conjecture.…”

Section: Introductionmentioning

confidence: 93%

“…In addition, we provide a short alternative proof of the crucial result of [16] that the truncated equation (3) admits a unique, attractive solution. Remark 1.…”

Section: Introductionmentioning

confidence: 98%

“…Since then, several alternative proofs have been found [9,13,15]. [16] circumvented this issue by considering instead the truncated equation…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Mézard-Parisi equation for matchings in pseudo-dimension $d>1$

Salez

2015

Electron. Commun. Probab.

View full text Add to dashboard Cite

We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension d > 1, as conjectured by Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and Wästlund (Annals of Mathematics, 2012). This fills the last remaining gap in the proof of the original Mézard-Parisi prediction for this problem (Journal de Physique Lettres, 1985).Keywords: recursive distributional equation; random assignment problem; mean-field combinatorial optimization; cavity method.2010 MSC: 60C05, 82B44, 90C35.

show abstract

Section: Introductionmentioning

confidence: 93%

“…In addition, we provide a short alternative proof of the crucial result of [16] that the truncated equation (3) admits a unique, attractive solution. Remark 1.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

The Mézard-Parisi equation for matchings in pseudo-dimension $d>1$

Salez

2015

Electron. Commun. Probab.

View full text Add to dashboard Cite

show abstract

“…When the density of the cost distribution at zero is either blowing up to infinity or converging to zero, the situation becomes more complicated. These cases have been investigated by Wästlund [65].…”

Section: 4mentioning

confidence: 99%

A general method for lower bounds on fluctuations of random variables

Chatterjee¹

2019

Ann. Probab.

View full text Add to dashboard Cite

There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general method for lower bounds on fluctuations. The method is used to obtain new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington-Kirkpatrick model of spin glasses, first-passage percolation and random matrices. A long list of open problems is provided at the end. Contents2010 Mathematics Subject Classification. 60E15, 60C05, 60K35, 60B20.

show abstract

“…As we shall see, the outcome of Trap is intimately tied to the properties of maximum-cardinality matchings, and draws relate to sensitivity of such matchings to boundary conditions. In [14], results about minimum weight matchings in edge-weighted graphs were derived from analysis of a related game called Exploration. We shall consider another related game, which we call Vicious Trap, in which a player, after making a move, is allowed to destroy (i.e., delete from the graph) any subset of the vertices that he could have just moved to.…”

Section: Introductionmentioning

confidence: 99%

Trapping games on random boards

Basu¹,

Holroyd²,

Martin³

et al. 2016

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

We consider the following two-player game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyse outcomes with optimal play on percolation clusters of Euclidean lattices.On Z 2 with two different percolation parameters for odd and even sites, we prove that the game has no draws provided closed sites of one parity are sufficiently rare compared with those of the other parity (thus favoring one player). We prove this also for certain d-dimensional lattices with d ≥ 3. It is an open question whether draws can occur when the two parameters are equal.On a finite ball of Z 2 , with only odd sites closed but with the external boundary consisting of even sites, we identify up to logarithmic factors a critical window for the trade-off between the size of the ball and the percolation parameter. Outside this window, one or other player has a decisive advantage.Our analysis of the game is intimately tied to the effect of boundary conditions on maximum-cardinality matchings.

show abstract

Replica symmetry of the minimum matching

Cited by 23 publications

References 46 publications

The Mézard-Parisi equation for matchings in pseudo-dimension $d>1$

The Mézard-Parisi equation for matchings in pseudo-dimension $d>1$

A general method for lower bounds on fluctuations of random variables

Trapping games on random boards

Contact Info

Product

Resources

About