New PDF release: A Bernstein theorem for special Lagrangian graphs

By Jost J., Xin Y. L.

We receive a Bernstein theorem for distinctive Lagrangian graphs in for arbitrary simply assuming bounded slope yet no quantitative limit.

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Initially, v is ‘unexplored’, and all other vertices are ‘new’. In step t of the exploration, we pick an unexplored vertex vt , if there is one (thus v1 = v). Note that vt may be picked in any way we like: this makes no difference; for definiteness, one often picks the first unexplored vertex in some fixed order chosen in advance. Having picked an unexplored vertex vt , we test all possible edges vt w from vt to new vertices w to see which ones are present in G(n, p). If vt w is present, we add w to our ‘unexplored’ list.

Der´enyi, Palla and Vicsek [83, 158] considered in particular the special case where G is simply G(n, p), and studied heuristically the emergence of a giant component in the derived graph Gk, . A directed version of this model was introduced by Palla, Farkas, Pollner, Der´enyi and Vicsek [159] (for an overview of these and related models, see ´ Palla, Abel, Farkas, Pollner, Der´enyi and Vicsek [157]). The thresholds for the appearance of the giant component in these models, and the asymptotic size of any such component, were found rigorously in [52].

It is easily seen that λe1−λ is at most 1, so it is convenient to consider the negative of its logarithm. Thus we set (21) δ = δ(λ) = − log λe1−λ = λ − 1 − log λ. In this notation, recalling that the approximation in Stirling’s formula is correct within 10% for all k ≥ 1, we have (22) ρk (λ) ≤ k−3/2 λ−1 e−δk for all k and λ. It is easy to check that as ε → 0, we have δ(1 ± ε) ∼ ε2 /2. If λ ∼ 1 and k → ∞, then (20) reduces to (23) ρk (λ) ∼ (2π)−1/2 k−3/2 e−δk . In the case λ < 1, summing over k we obtain a tail bound on the branching process.

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A Bernstein theorem for special Lagrangian graphs by Jost J., Xin Y. L.


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