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Integrals, partitions, and cellular automata


Authors: Alexander E. Holroyd, Thomas M. Liggett and Dan Romik
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3349-3368
MSC (2000): Primary 26A06; Secondary 05A17, 60C05, 60K35
DOI: https://doi.org/10.1090/S0002-9947-03-03417-2
Published electronically: December 15, 2003
MathSciNet review: 2052953
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that

\begin{displaymath}\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab},\end{displaymath}

where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.


References [Enhancements On Off] (What's this?)

  • 1. M. Aizenman and G. Grimmett.
    Strict monotonicity for critical points in percolation and ferromagnetic models.
    Journal of Statistical Physics, 63:817-835, 1991. MR 92i:82060
  • 2. M. Aizenman and J. L. Lebowitz.
    Metastability effects in bootstrap percolation.
    J. Phys. A, 21(19):3801-3813, 1988. MR 90e:82047
  • 3. G. E. Andrews.
    The Theory of Partitions, volume 2 of Encyclopedia of Mathematics and its Applications.
    Addison-Wesley, 1976. MR 58:27738
  • 4. G. E. Andrews.
    The reasonable and unreasonable effectiveness of number theory in statistical mechanics.
    Proceedings of Symposia in Applied Mathematics, 46:21-34, 1992. MR 94c:82021
  • 5. J. Baik, P. Deift, and K. Johansson.
    On the distribution of the length of the second row of a Young diagram under Plancherel measure.
    Geom. Funct. Anal., 10(4):702-731, 2000. MR 2001m:05258a
  • 6. C. H. Brenner.
    Asymptotic analogs of the Rogers-Ramanujan identities.
    J. Comb. Theory Ser. A, 43:303-319, 1986. MR 88e:05008
  • 7. I. S. Gradshteyn and I. M. Ryzhik.
    Table of integrals, series, and products.
    Academic Press, New York, 1965. MR 81g:33001
  • 8. J. Gravner and D. Griffeath.
    First passage times for threshold growth dynamics on ${\bf Z}\sp 2$.
    Ann. Probab., 24(4):1752-1778, 1996. MR 98c:60140
  • 9. J. Gravner and D. Griffeath.
    Scaling laws for a class of critical cellular automaton growth rules.
    In Random walks (Budapest, 1998), pages 167-186. János Bolyai Math. Soc., Budapest, 1999. MR 2001a:60113
  • 10. G. H. Hardy and S. Ramanujan.
    Asymptotic formulae for the distribution of integers of various types.
    Proc. London Math. Soc., Ser. 2, 16:112-132, 1918.
    Reprinted in The Collected Papers of G. H. Hardy, vol. 1, 277-293.
  • 11. P. G. Hoel, S. C. Port, and C. J. Stone.
    Introduction to probability theory.
    Houghton Mifflin Co., Boston, Mass., 1971.
    The Houghton Mifflin Series in Statistics. MR 50:11337c
  • 12. A. E. Holroyd.
    Sharp metastability threshold for two-dimensional bootstrap percolation.
    Probability and Related Fields, 125:195-224, 2003.
  • 13. K. Johansson.
    Shape fluctuations and random matrices.
    Comm. Math. Phys., 209(2):437-476, 2000. MR 2001h:60177
  • 14. D. J. Newman.
    Analytic Number Theory.
    Number 177 in Graduate Texts in Mathematics. Springer-Verlag, 1998. MR 98m:11001

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Additional Information

Alexander E. Holroyd
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: holroyd@math.ubc.ca

Thomas M. Liggett
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, Califonia 90095-1555
Email: tml@math.ucla.edu

Dan Romik
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: romik@wisdom.weizmann.ac.il

DOI: https://doi.org/10.1090/S0002-9947-03-03417-2
Keywords: Definite integral, partition asymptotics, partition identity, combinatorial probability, threshold growth model, bootstrap percolation, cellular automaton
Received by editor(s): February 17, 2003
Received by editor(s) in revised form: May 6, 2003
Published electronically: December 15, 2003
Additional Notes: The first author’s research was funded in part by NSF Grant DMS–0072398.
The second author’s research was funded in part by NSF Grant DMS-00-70465.
Article copyright: © Copyright 2003 American Mathematical Society

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