Prime Percolation

REFERENCES

• van den Berg , J. and Ermakov , A. 1996 . “A new lower bound for the critical probability of site percolation on the square lattice” . Random Structures Algorithms , 8 ( 3 ) : 199 – 212 . [van den Berg and Ermakov 1996]
   Web of Science ®Google Scholar
• Broadbent , S. R. and Hammersley , J. M. 1957 . “Percolation processes. I. Crystals and mazes” . Proc. Cambridge Philos. Soc. , 53 : 629 – 641 . [Broadbent and Hammersley 1957]
   Google Scholar
• Coleman , M. D. 1990 . “The distribution of points at which binary quadratic forms are prime” . Proc. London Math. Soc. (3) , 61 ( 3 ) : 433 – 456 . [Coleman 1990]
   Google Scholar
• Cramér , H. 1937 . “On the order of magnitude of the difference between consecutive prime numbers” . Acta Arithmetica , 2 : 23 – 46 . [Cramér 1937]
   Google Scholar
• Davenport , H. 1980 . Multiplicative number theory New York : Springer. . [Davenport 1980], 2nd ed., Graduate Texts in Mathematics 74, Revised by Hugh L. Montgomery
   Google Scholar
• Domb , C. 1972 . “A note on the series expansion method for clustering problems” . Biometrika , 59 : 209 – 211 . [Domb 1972]
   Web of Science ®Google Scholar
• Durrett , R. 1988 . “Crabgrass, measles, and gypsy moths: An introduction to modern probability” . Bull. Amer. Math. Soc. , 18 : 117 – 143 . [Durrett 1988]
   Web of Science ®Google Scholar
• Èfros , A. L. 1982 . Translated as Physics and geometry of disorder: percolation theory Moscow : Mir. . [Èfros 1982], Nauka, Moscow
   Google Scholar
• Flajolet , P. and Vardi , I. 1996 . “Zeta expansions of classical constants” [Flajolet and Vardi 1996], preprint, See www.cco.caltech.edu/~ilan/papers/zeta'constants.ps
   Google Scholar
• Gawlinski , E. T. and Stanley , H. E. 1981 . “Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for noninteracting discs” . J. Phys. A , 14 ( 8 ) : L291 – L299 . [Gawlinski and Stanley 1981]
   Google Scholar
• de Gennes , P. G. 1976 . “La percolation: un concept unificateur” . La Recherche , 7 : 919 – 927 . [de Gennes 1976]
   Google Scholar
• Gethner , E. April 1996 . “In prime territory” . Math. Horizons , : 8 – 13 . [Gethner 1996]
   Google Scholar
• Gethner , E. and Stark , H. M. 1997 . “Periodic Gaussian moats” . Experiment. Math. , 6 ( 4 ) : 289 – 292 . [Gethner and Stark 1997]
   Google Scholar
• Gethner , E. , Wagon , S. and Wick , B. 1998 . “A stroll through the Gaussian primes” . Amer. Math. Monthly , 105 ( 4 ) : 327 – 337 . [Gethner et al. 1998]
   Web of Science ®Google Scholar
• Gilbert , E. N. 1961 . “Random plane networks” . J. Soc. Indust. Appl. Math. , 9 : 533 – 543 . [Gilbert 1961]
   Google Scholar
• Granville , A. 1995 . “Harald Cramer and the distribution of prime numbers” . Scand. Actuar. J. , : 12 – 28 . [Granville 1995a]
   Google Scholar
• Granville , A. “Unexpected irregularites in the distribution of prime numbers” . Proceedings of the International Congress of Mathematicians . 1994 , Zurich . Edited by: Chatterji , S. D. pp. 388 – 399 . Basel : Birkhäuser. . [Granville 1995b]
   Google Scholar
• Grimmett , G. 1989 . Percolation New York : Springer. . [Grimmett 1989]
   Google Scholar
• Guy , R. K. 1994 . Unsolved problems in number theory, , 2nd ed. New York : Springer. . [Guy 1994], Problem Books in Mathematics
   Google Scholar
• Hahn , S. W. and Zwanzwig , R. 1977 . “Series expansions in a continuum percolation problem” . J. Phys. A , 10 : 1547 – 1555 . [Hahn and Zwanzwig 1977]
   Google Scholar
• Halberstam , H. and Richert , H.-E. 1974 . Sieve methods London and New York : Academic Press. . [Halberstam and Richert 1974], London Mathematical Society Monographs 4
   Google Scholar
• Hall , P. 1985 . “On continuum percolation” . Ann. Probab. , 13 ( 4 ) : 1250 – 1266 . [Hall 1985]
   Web of Science ®Google Scholar
• Hall , P. 1988 . Introduction to the theory of coverage processes New York : Wiley. . [Hall 1988]
   Google Scholar
• Hardy , G. H. and Littlewood , J. E. 1922 . “Some problems of ‘Partitio Numerorum’ III: On the expression of a number as a sum of primes” . Acta Math. , 44 : 1 – 70 . [Hardy and Littlewood 1922], Reprinted as pp. 561–630 of Collected Papers of G. H. Hardy, vol. 1, Oxford, Clarendon Press
   Google Scholar
• Hardy , G. H. and Wright , E. M. 1979 . An introduction to the theory of numbers, , 5th ed. Oxford : Clarendon Press. . [Hardy and Wright 1979]
   Google Scholar
• Hecke , E. 1918 . “Eine neue Art von Zetafunktionen und ihre Bezeihungen zur Verteilung der Primzahlen I” . Math. Zeitschrift , 1 : 357 – 376 . [Hecke 1918]
   Google Scholar
• Hecke , E. 1920 . “Eine neue Art von Zetafunktionen und ihre Bezeihungen zur Verteilung der Primzahlen II” . Math. Zeitschrift , 6 : 11 – 51 . [Hecke 1920]
   Google Scholar
• Hildebrand , A. and Maier , H. 1989 . “Irregularities in the distribution of primes in short intervals” . J. Reine Angew. Math. , 397 : 162 – 193 . [Hildebrand and Maier 1989]
   Web of Science ®Google Scholar
• Holben , C. A. and Jordan , J. H. 1968 . “The twin prime problem and Goldbach's conjecture in the Gaussian integers” . Fibonacci Quart. , 6 ( 5 ) : 81 – 85 . 92 [Holben and Jordan 1968]
   Google Scholar
• Jordan , J. H. and Rabung , J. R. 1970 . “A conjecture of Paul Erdös concerning Gaussian primes” . Math. Comp. , 24 : 221 – 223 . [Jordan and Rabung 1970]
   Web of Science ®Google Scholar
• Jordan , J. H. and Rabung , J. R. 1976 . “Local distribution of Gaussian primes” . J. Number Theory , 8 ( 1 ) : 43 – 51 . [Jordan and Rabung 1976]
   Web of Science ®Google Scholar
• Kesten , H. 1982 . Percolation theory for mathematicians Boston : Birkhäuser. . [Kesten 1982], Progress in Probability and Statistics 2
   Google Scholar
• Meester , R. and Roy , R. 1996 . Continuum percolation Cambridge : Cambridge University Press. . [Meester and Roy 1996], Cambridge Tracts in Mathematics 119
   Google Scholar
• Penrose , M. D. 1991 . “On a continuum percolation model” . Adv. in Appl. Probab. , 23 ( 3 ) : 536 – 556 . [Penrose 1991]
   Web of Science ®Google Scholar
• Rademacher , H. 1923 . “Über die Anwendung der Viggo Brunschen Methode auf die Theorie der algebraischen Zahlkörper” . Sitzungsber. Preuss. Akad. Wiss. , 24 : 211 – 218 . [Rademacher 1923]
   Google Scholar
• Riesel , H. 1985 . Prime numbers and computer methods for factorization Boston : Birkhäuser. . [Riesel 1985], Progress in Mathematics 57
   Google Scholar
• Russo , L. 1981 . “On the critical percolartion probabilities” . Z. Wahrsch. Verw. Gebiete , 56 ( 2 ) : 229 – 237 . [Russo 1981]
   Google Scholar
• Stauffer , D. and Aharony , A. 1994 . Introduction to percolation theory, , 2nd ed. London : Taylor and Francis Ltd. . [Stauffer and Aharony 1994]
   Google Scholar
• Vardi , I. “Number theoretic percolation” [Vardi ≥ 1998], in preparation
   Google Scholar
• Wierman , J. C. 1995 . “Substitution method critical probability bounds for the square lattice site percolation model” . Combin. Probab. Comput. , 4 ( 2 ) : 181 – 188 . [Wierman 1995]
   Google Scholar
• Ziff , R. M. and Sapoval , B. 1986 . “The efficient determination of the percolation threshold by a frontier-generating walk in a gradient” . J. Phys A , 19 : L1169 – L1172 . [Ziff and Sapoval 1986]
   Google Scholar
• Zuev , S. A. and Sidorenko , A. F. 1985 . “Continuous models of percolation theory, I” . Teoret. Mat. Fiz. , 62 ( 1 ) : 76 – 86 . [Zuev and Sidorenko 1985], In Russian; translation in Theor. Math. Phys. 62, 51–59 (1985)
   Google Scholar