Coexistence probability in the last passage percolation model is $6-8\log 2$

David Coupier; Philippe Heinrich

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Displaying similar documents to “Coexistence probability in the last passage percolation model is $6 - 8 log 2$ ”

Odd cutsets and the hard-core model on $ℤ^{d}$

Ron Peled, Wojciech Samotij (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the hard-core lattice gas model on $ℤ^{d}$ and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds $C d^{- 1 / 3} {(log d)}^{2}$ , the model exhibits multiple hard-core measures, thus improving the previous bound of $C d^{- 1 / 4} {(log d)}^{3 / 4}$ given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in $ℤ^{d}$ , the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined...

Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

Boban Karapetrović (2018)

Czechoslovak Mathematical Journal

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We show that if $α > 1$ , then the logarithmically weighted Bergman space $A_{{log}^{α}}^{2}$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{α - 1}}^{2}$ , while if $α > 2$ and $0 < ε \leq α - 2$ , then the Hilbert matrix operator $H$ maps $A_{{log}^{α}}^{2}$ into $A_{{log}^{α - 2 - ε}}^{2}$ .We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space $ℬ_{{log}^{α}}$ , $α \in ℝ$ , into itself, while $H$ maps $ℬ_{{log}^{α}}$ into $ℬ_{{log}^{α + 1}}$ .In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space $ℬ_{{log}^{α}}^{1}$ , $α > 0$ , into $ℬ_{{log}^{α - 1}}^{1}$ . We show that this result is sharp. We also show that $H$ maps $ℬ_{{log}^{α}}^{1}$ , $α \geq 0$ ,...

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

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For the real quadratic map $P_{a} (x) = x^{2} + a$ and a given $ϵ > 0$ a point $x$ has good expansion properties if any interval containing $x$ also contains a neighborhood $J$ of $x$ with $P_{a}^{n} |_{J}$ univalent, with bounded distortion and $B (0, ϵ) \subseteq P_{a}^{n} (J)$ for some $n \in ℕ$ . The $ϵ$ -weakly expanding set is the set of points which do not have good expansion properties. Let $α$ denote the negative fixed point and $M$ the first return time of the critical orbit to $[α, - α]$ . We show there is a set $ℛ$ of parameters with positive Lebesgue measure for which the Hausdorff...

Inequalities for Taylor series involving the divisor function

Horst Alzer, Man Kam Kwong (2022)

Czechoslovak Mathematical Journal

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Let $T (q) = \sum_{k = 1}^{\infty} d (k) q^{k}, | q | < 1,$ where $d (k)$ denotes the number of positive divisors of the natural number $k$ . We present monotonicity properties of functions defined in terms of $T$ . More specifically, we prove that $H (q) = T (q) - \frac{log (1 - q)}{log (q)}$ is strictly increasing on $(0, 1)$ , while $F (q) = \frac{1 - q}{q} H (q)$ is strictly decreasing on $(0, 1)$ . These results are then applied to obtain various inequalities, one of which states that the double inequality $α \frac{q}{1 - q} + \frac{log (1 - q)}{log (q)} < T (q) < β \frac{q}{1 - q} + \frac{log (1 - q)}{log (q)}, 0 < q < 1,$ holds with the best possible constant factors $α = γ$ and $β = 1$ . Here, $γ$ denotes Euler’s constant. This refines a result of Salem, who...

Equilibrium states for interval maps: the potential $- t log | D f |$

Henk Bruin, Mike Todd (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $f : I \to I$ be a $C^{2}$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $φ_{t} : x \mapsto - t log | D f (x) |$ for $t$ close to $1$ , and also that the pressure function $t \mapsto P (φ_{t})$ is analytic on an appropriate interval near $t = 1$ .

Some characterizations of the class $_{m} (Ω)$ and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

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We give some characterizations of the class $_{m} (Ω)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

On sum-product representations in $ℤ_{q}$

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

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The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$ , by performing sum and product set operations starting from a given subset $A$ of $ℤ_{q}$ . We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $| A | \sim log q$ ). Roughly speaking we show that all residue classes are obtained from a $k$ -fold sum of an $r$ -fold product set of $A$ , where $r ≪ log q$ and $log k ≪ log q$ , provided the...

On the divisor function over Piatetski-Shapiro sequences

Hui Wang, Yu Zhang (2023)

Czechoslovak Mathematical Journal

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Let $[x]$ be an integer part of $x$ and $d (n)$ be the number of positive divisor of $n$ . Inspired by some results of M. Jutila (1987), we prove that for $1 < c < \frac{6}{5}$ , $\sum_{n \leq x} d ([n^{c}]) = c x log x + (2 γ - c) x + O (\frac{x}{log x}),$ where $γ$ is the Euler constant and $[n^{c}]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)

Acta Arithmetica

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Let K be an algebraic number field with non-trivial class group G and $_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_{k} (x)$ denote the number of non-zero principal ideals $a_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_{k} (x)$ behaves for x → ∞ asymptotically like $x {(l o g x)}^{1 - 1 / | G |} {(l o g l o g x)}^{_{k} (G)}$ . We prove, among other results, that $₁ (C_{n ₁} \oplus C_{n ₂}) = n ₁ + n ₂$ for all integers n₁,n₂ with 1 < n₁|n₂.

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2016)

Journal of the European Mathematical Society

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Consider the classical $(2 + 1)$ -dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $ℤ^{2}$ . The model describes a crystal surface by assigning a non-negative integer height $η_{x}$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp (- β ℋ (η))$ , where $β$ is the inverse-temperature and $ℋ (η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough...

Limits of log canonical thresholds

Tommaso de Fernex, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $𝒯_{n}$ denote the set of log canonical thresholds of pairs $(X, Y)$ , with $X$ a nonsingular variety of dimension $n$ , and $Y$ a nonempty closed subscheme of $X$ . Using non-standard methods, we show that every limit of a decreasing sequence in $𝒯_{n}$ lies in $𝒯_{n - 1}$ , proving in this setting a conjecture of Kollár. We also show that $𝒯_{n}$ is closed in $𝐑$ ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

Uniform algebras and analytic multifunctions

Zbigniew Slodkowski (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Dati due elementi $f$ e $g$ in un'algebra uniforme $A$ , sia $G=f(M_{A}/f(\partial_{A})$ . Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda\to\log\max g(f^{-1}(\lambda))$ è subarmonica su $G$ e che l’applicazione $\lambda\to g(f^{-1}(\lambda))$ è analitica nel senso di Oka.

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

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We consider random walk on a discrete torus $E$ of side-length $N$ , in sufficiently high dimension $d$ . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u N^{d}$ . We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $log N$ . Moreover, this connected component occupies a...

A generalized Kahane-Khinchin inequality

S. Favorov (1998)

Studia Mathematica

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The inequality $ʃ l o g | \sum a_{n} e^{2 π i φ_{n}} | d φ_{1} \dots d φ_{n} \geq C l o g (\sum | a_{n} {|^{2})}^{1 / 2}$ with an absolute constant C, and similar ones, are extended to the case of $a_{n}$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2 π i φ}$ .

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

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For any positive integer k and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ for all n ≥ n₀, we have $r_{1, k} (A, n) \to \infty$ as n → ∞.

Representation functions with different weights

Quan-Hui Yang (2014)

Colloquium Mathematicae

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For any given positive integer k, and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Remarks on Ramanujan's inequality concerning the prime counting function

Mehdi Hassani (2021)

Communications in Mathematics

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In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $π {(x)}^{2} < \frac{e x}{log x} π (\frac{x}{e})$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac{x}{log x}$ on its right hand side by the factor $\frac{x}{log x - h}$ for a given $h$ , and by replacing the numerical factor $e$ by a given positive $α$ . Finally, we introduce and study inequalities...

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb (2013)

Annales de l'I.H.P. Probabilités et statistiques

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An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the $L^{2}$ norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

Displaying similar documents to “Coexistence probability in the last passage percolation model is 6 - 8 log 2 ”

Displaying similar documents to “Coexistence probability in the last passage percolation model is $6 - 8 log 2$ ”