Moderate deviations for longest increasing subsequences: The upper tail

Löwe, Matthias; Merkl, Franz

doi:10.1002/cpa.10010

Cited by 37 publications

(47 citation statements)

References 14 publications

(5 reference statements)

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“…The above moderate deviation results can also be motivated by estimating the functions I(y) and H(y) for the large-deviation regime. In [17], the authors proved (1.13) by refining certain estimates in [1] and using a careful summation argument. In [18], the authors utilized an analogous summation argument together with the estimate Lemma 6.3 (ii) in [1], Calculations similar to (1.16), (1.17), motivate the following moderate deviation results for G{ [jN] …”

Section: P(^v > (2 + Yn-a )Vn) = P(t N > 2y/n + (Yn^^n 1^6 )mentioning

confidence: 99%

“…In two recent papers, [18] [17], the authors have considered £N in the moderate deviation regime. More precisely, for 0 < a < |, they showed [18] that for y > 0 7V->oo y3j\fl-3a 12 V ' and [17] 1 .…”

Section: Lim ^ Logp(^ < Vn(2 -Y)) = -H(y) (110) Jv-»oo IV Lim -L]ogmentioning

confidence: 99%

“…In Section 5, we use the ^-function to analyze the RHP and eventually give the proof of Proposition 2.2. Finally, in Section 6, we use the estimate in Proposition 2.2 together with a careful summation argument as in [18] to prove the main result Theorem 1.1. kindly inviting him to Universite Paris 7, where a part of work is done, and also acknowledge that a part of work is conducted while he is visiting Korea Institute for Advanced Study for 2 weeks of August, 2001. The work of the first author was supported in part by NSF Grant # DMS 97-29992.…”

Section: Remarkmentioning

confidence: 99%

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Optimal tail estimates for directed last passage site percolation with geometric random variables

Baik¹,

Deift²,

McLaughlin³

et al. 2001

Advances in Theoretical and Mathematical Physics

123

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e-print archive: http://xxx.lanl.gov/math.PR/0112162 1208 BAIK, DEIFT, MCLAUGHLIN, MILLER, AND ZHOUIn this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.

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Section: P(^v > (2 + Yn-a )Vn) = P(t N > 2y/n + (Yn^^n 1^6 )mentioning

confidence: 99%

Section: Lim ^ Logp(^ < Vn(2 -Y)) = -H(y) (110) Jv-»oo IV Lim -L]ogmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

Optimal tail estimates for directed last passage site percolation with geometric random variables

Baik¹,

Deift²,

McLaughlin³

et al. 2001

Advances in Theoretical and Mathematical Physics

123

View full text Add to dashboard Cite

show abstract

“…In this case symmetry considerations show that the maximal curve is any curve which fixes the angle θ and calculating the length we get that the service time concentrates around √ 2 1 0 p(r)dr √ n. Since we have an infinite number of maximal curves the magnitude of the error term ∆ D,n can be somewhat larger than expected. When the distribution p is the uniform distribution, it can be shown using results from [31,35] that ∆ D,n has order of magnitude n 1/6 log(n) 2/3 . We expect that the variance will actually be smaller in this case, in analogy with extreme value distributions.…”

Section: Problem Solutionsmentioning

confidence: 99%

Discrete spacetime and its applications

Bachmat¹

2008

Contemporary Mathematics

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A discrete spacetime is a random partially ordered set. It is obtained by sampling n points from a compact domain in a spacetime manifold w.r.t. the volume form. The partial order is given by restricting the causal structure on the domain to the sampled points. These objects were introduced by physicists in an attempt to reconcile gravity with quantum mechanics, [40,45,19]. They received very little direct attention in the mathematics literature.Recently, discrete spacetimes have appeared in very diverse applications which are very far removed from their original motivation. In this self-contained survey we will present some results and numerical calculations regarding the asymptotic behavior of these objects. Some of the more basic results are stated as theorems since they are particularly useful in the applications. Numerical calculations are presented with the hope that they will incite some theoretical work, aimed at proving some related assertions. We then present a few of the applications by shamelessly following the pattern of the recent survey paper [21], which describes some related material. We will present a short list of seemingly unrelated problems and show how they fit into the discrete spacetime formalism. While this is a survey paper, it does contain some new material, namely, the statement of theorem 1.3, the (simple) solution to the dimension reconstruction problem, the numerical calculations of figure 2, the connect-the-dots application and the relation between discrete spacetime and the literature on maximal layers. A more comprehensive and extensive treatment of the subject along with full proofs of the basic results will be presented in a forthcoming publication.Our own research relating to discrete spacetime owes much to the constant encouragement which was provided by Percy Deift and it is our sincere pleasure to dedicate this paper to him on the occasion of his 60th birthday.

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“…Often even the rate function of an MDP interpolates between the logarithmic probabilities that can be expected from the central limit theorem and the large deviations rate function -even if the limit is not normal (see e.g. [7,11]); situations where this is not the case are particularly interesting [3].…”

Section: Introductionmentioning

confidence: 99%