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9032 results in Discrete Mathematics, Information Theory and Coding

Page 23 of 452

  • Quasipolynomial-time algorithms for Gibbs point processes

  • Part of
  • Matthew Jenssen, Marcus Michelen, Mohan Ravichandran
  • Journal:
    Combinatorics, Probability and Computing / Volume 33 / Issue 1 / January 2024
    Published online by Cambridge University Press:
    11 August 2023, pp. 1-15
    • Article
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  • We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a stable potential. This result holds for all activities $\lambda$ for which the partition function satisfies a zero-free assumption in a neighbourhood of the interval $[0,\lambda ]$. As a corollary, for all finiterange stable potentials, we obtain a quasipolynomial-time deterministic algorithm for all $\lambda \lt 1/(e^{B + 1} \hat C_\phi )$ where $\hat C_\phi$ is a temperedness parameter and $B$ is the stability constant of $\phi$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $\lambda \lt e/\Delta _\phi$, where $\Delta _\phi$ is the potential-weighted connective constant of the potential $\phi$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.

Graph Theory and Additive Combinatorics
  • Exploring Structure and Randomness
  • Yufei Zhao
  • Published online:
    10 August 2023
    Print publication:
    31 August 2023
  • Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra.

  • On the exponential growth rates of lattice animals and interfaces

  • Part of
  • Agelos Georgakopoulos, Christoforos Panagiotis
  • Journal:
    Combinatorics, Probability and Computing / Volume 32 / Issue 6 / November 2023
    Published online by Cambridge University Press:
    31 July 2023, pp. 912-955
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  • We introduce a formula for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice versa. We exploit this in both directions. We obtain the rigorous lower bound ${\dot{p}_c}({\mathbb{Z}}^3)\gt 0.2522$ for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on $a({\mathbb{Z}}^d)$ as $d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain subspecies called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold.

    Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $p\in (0,1)$.

Page 23 of 452