Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Integrals, partitions, and cellular automata

Author(s): Alexander E. Holroyd; Thomas M. Liggett; Dan Romik
Journal: Trans. Amer. Math. Soc. 356 (2004), 3349-3368.
MSC (2000): Primary 26A06; Secondary 05A17, 60C05, 60K35
Posted: December 15, 2003
Retrieve article in: PDF DVI PostScript

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:

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

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26A06, 05A17, 60C05, 60K35

Retrieve articles in all Journals with MSC (2000): 26A06, 05A17, 60C05, 60K35

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: 10.1090/S0002-9947-03-03417-2
PII: S 0002-9947(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
Posted: 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.
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google