Some Linear and Some Quadratic Recursion Formulas. I

Bruijn, FA Dick de; Erdös, P.

doi:10.1016/s1385-7258(51)50054-9

Cited by 32 publications

(21 citation statements)

References 1 publication

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“…Theorem 1 (de Bruijn and Erdős, Theorem 22. in [4]). Suppose that the sequence {a(n)} satisfies (10) for all integers N ≤ n ≤ m ≤ 2n.…”

Section: Sub-sequences By De Bruijn and Erdősmentioning

confidence: 98%

“…Concerning their result (Theorem 1 above) de Bruijn and Erdős [4] state, maybe somewhat carelessly, that 'It may be remarked that the inequality in (7.1) cannot be replaced by µ −1 n ≤ m ≤ µn for any µ < 2'. In their papers [3,4] they deal with many conditions and sequences, we could not really know what was in their minds, but our first new result is a strengthening of Theorem 1 for all µ > 1. We show that their condition can be weakened such that the limit exists if (10) holds only for the pairs (n, m) with n ≤ m ≤ µn for some fixed µ > 1.…”

Section: Sub-sequences By De Bruijn and Erdősmentioning

confidence: 99%

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Nearly subadditive sequences

Füredi¹,

Ruzsa²

2018

Preprint

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We show that the de Bruijn-Erdős condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence 0(1)Then, there exists a sequence {b(nsuch that the sequence of slopes {b(n)/n} n=1,2,... takes every rational number.When the series ( 1) is bounded we improve their result as follows. If there exist N and real µ > 1 such that (2) holds for all pairs (n, m) with N ≤ n ≤ m ≤ µn, then lim n b(n)/n exists.

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“…Theorem 1 (de Bruijn and Erdős, Theorem 22. in [4]). Suppose that the sequence {a(n)} satisfies (10) for all integers N ≤ n ≤ m ≤ 2n.…”

Section: Sub-sequences By De Bruijn and Erdősmentioning

confidence: 98%

Section: Sub-sequences By De Bruijn and Erdősmentioning

confidence: 99%

Nearly subadditive sequences

Füredi¹,

Ruzsa²

2018

Preprint

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“…for some c 2 > 0. The assertion now follows by setting g(n) = C n 2 2 and ϕ(n) = c 2 n 1/2 E(n) 2 for n ∈ N, and employing de Brujin-Erdös extension of Fekete's lemma (see [11,Theorem 23]).…”

Section: On the Slln For {C N } N≥0mentioning

confidence: 99%

Capacity of the range of random walks on groups

Mrazović¹,

Sandrić²,

Šebek³

2021

Preprint

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In this paper, we discuss asymptotic behavior of the capacity of the range of symmetric simple random walks on finitely generated groups. We show the corresponding strong law of large numbers and central limit theorem. Here, τ +A denotes the first return time of {S n } n≥0 to the set A, i.e. τ + A = inf{n ≥ 1 : S n ∈ A}. We are interested in the asymptotic behavior of the process {C n } n≥0 defined asObserve that when {S n } n≥0 is recurrent then C n ≡ 0.

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“…Proof of (1.2) in Theorem 1. By an approximate version of Fekete's Lemma ( [19,Theorem 23], also, see [40, Theorem 1.9.2]), since t −2 (log t) 2 dt < ∞, Proposition 5.7 implies that lim…”

Section: 2mentioning

confidence: 99%

Maximum and shape of interfaces in 3D Ising crystals

Gheissari¹,

Lubetzky²

2019

Preprint

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showed that the interface of a 3D Ising model with minus boundary conditions above the xy-plane and plus below is rigid (has O(1)-fluctuations) at every sufficiently low temperature. Since then, basic features of this interface-such as the asymptotics of its maximum-were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D Solid-On-Solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side-length n with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for Mn, its maximum: if the inverse-temperature β is large enough, then Mn/ log n → 2/α β as n → ∞, in probability, where α β is given by a large deviation rate in infinite volume.We further show that, on the large deviation event that the interface connects the origin to height h, it consists of a 1D spine that behaves like a random walk, in that it decomposes into a linear (in h) number of asymptotically-stationary weakly-dependent increments that have exponential tails. As the number T of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in T . These results generalize to every dimension d ≥ 3.

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Some Linear and Some Quadratic Recursion Formulas. I

Cited by 32 publications

References 1 publication

Nearly subadditive sequences

Nearly subadditive sequences

Capacity of the range of random walks on groups

Maximum and shape of interfaces in 3D Ising crystals

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