Gordon Slade's Publications

Books
Recent papers on Renormalisation Group
Journal Articles and Conference Proceedings
Expository Writing
Collaborators




Books:

  1. G. Slade. The Lace Expansion and its Applications, Lecture Notes in Mathematics #1879, xiv + 232 pages. Springer, Berlin, (2006).
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    (Lecture notes for the XXXIVth Saint-Flour Summer School on Probability, July 8-24 2004, and for the Summer School in Probability at PIMS/UBC, June 6-30 2005.)
    Students' solutions to all the exercises in the lecture notes, edited by S. Kliem and R. Liang (November 2, 2005):
    PS file
  2. N. Madras and G. Slade, The Self-Avoiding Walk , Birkhäuser, Boston, (1993). xiv + 425 pages. Paperback edition published in 1996. Reprinted as a Modern Birkhäuser Classic 2013.

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Recent papers on Renormalisation Group:

    The following nine papers develop a general renormalisation group method and apply it to study the critical behaviour of the 4-dimensional n-component |φ|4 spin model and the 4-dimensional weakly self-avoiding walk.
    A recommended place to start is here if you are interested in applications to the |φ|4 model, or here and then here if you are interested in applications to the self-avoiding walk.
    Concerning the general renormalisation group method itself, perturbative aspects are treated here (supporting software) and the centrepiece and main innovation is here. The other four papers (I,II,IV and Structural Stability) play various foundational and supporting roles.
    The versions here are not the same as the published versions. Revised versions will appear here in the near future.

  1. R. Bauerschmidt, D.C. Brydges and G. Slade. Scaling limits and critical behaviour of the 4-dimensional n-component |φ|4 spin model. J. Stat. Phys, 157:692--742, (2014).
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  2. R. Bauerschmidt, D.C. Brydges and G. Slade. Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis. Preprint. March 21, 2014. To appear in Communications in Mathematical Physics.
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  3. R. Bauerschmidt, D.C. Brydges and G. Slade. Critical two-point function of the 4-dimensional weakly self-avoiding walk. To appear in Communications in Mathematical Physics.
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  4. D.C. Brydges and G. Slade. A renormalisation group method. I. Gaussian integration and normed algebras. Preprint. March 21, 2014. To appear in Journal of Statistical Physics.
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  5. D.C. Brydges and G. Slade. A renormalisation group method. II. Approximation by local polynomials. Preprint. March 21, 2014. To appear in Journal of Statistical Physics.
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  6. R. Bauerschmidt, D.C. Brydges and G. Slade. A renormalisation group method. III. Perturbative analysis. Preprint. March 21, 2014. To appear in Journal of Statistical Physics.
    PDF file
    Software for calculations in this paper.
  7. D.C. Brydges and G. Slade. A renormalisation group method. IV. Stability analysis. Preprint. March 21, 2014. To appear in Journal of Statistical Physics.
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  8. D.C. Brydges and G. Slade. A renormalisation group method. V. A single renormalisation group step. Preprint. March 21, 2014. To appear in Journal of Statistical Physics.
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  9. R. Bauerschmidt, D.C. Brydges and G. Slade. Structural stability of a dynamical system near a non-hyperbolic fixed point. Revised April 14, 2014. To appear in Annales Henri Poincaré.
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Journal Articles and Conference Proceedings:

  1. Y. Mejía Miranda and G. Slade. Expansion in high dimensions for the growth constants of lattice trees and lattice animals. Combinatorics, Probability and Computing 22:527--565, (2013). (CUP holds copyright.)
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  2. R. Bauerschmidt, H. Duminil-Copin, J. Goodman and G. Slade. Lectures on self-avoiding walks. In: Probability and Statistical Physics in Two and More Dimensions, Clay Mathematics Proceedings, vol. 15, Amer. Math. Soc., Providence, RI, 2012, pp. 395-467. These are lecture notes from the Clay Mathematics Institute Summer School and XIV Escola Brasileira de Probabilidade in Búzios, Brazil in 2010.
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  3. D.C. Brydges, A. Dahlqvist and G. Slade. The strong interaction limit of continuous-time weakly self-avoiding walk. In Probability in Complex Physical Systems: In Honour of Erwin Bolthausen and Jürgen Gärtner, eds. J-D. Deuschel et al., Springer Proceedings in Mathematics 11:275--287, (2012)
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  4. G. Slade. The self-avoiding walk: A brief survey. In Surveys in Stochastic Processes, pp. 181-199, eds. J. Blath, P. Imkeller, S. Roelly, European Mathematical Society, Zurich, (2011).
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  5. Y. Mejía Miranda and G. Slade. The growth constants of lattice trees and lattice animals in high dimensions. Elect. Comm. Probab. 16:129--136, (2011).
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  6. D. Brydges and G. Slade. Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher. In Proceedings of the International Congress of Mathematicians, 2010, eds. R. Bhatia et al., Volume 4, pp. 2232--2257, World Scientific, (2011).
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  7. D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys, 6:34--61, (2009).
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  8. N. Clisby, G. Slade. Polygons and the lace expansion. In Polygons, Polyominoes and Polycubes, pp. 117-142, ed. A.J. Guttmann, Lecture Notes in Physics, Vol. 775. Springer, Dordrecht (2009).
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  9. R. van der Hofstad, M. Holmes, G. Slade. An extension of the inductive approach to the lace expansion. Elect. Comm. Probab. 13:291--301, (2008).
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    More detailed proofs are available in the unpublished document: R. van der Hofstad, M. Holmes, G. Slade, Extension of the generalised inductive approach to the lace expansion: Full proof, available here.
  10. O. Angel, J. Goodman, F. den Hollander, G. Slade. Invasion percolation on regular trees. Ann. Probab. 36:420--466, (2008).
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  11. M.T. Barlow, A.A. Járai, T. Kumagai, G. Slade. Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Commun. Math. Phys. 278:385--431, (2008).
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  12. N. Clisby, R. Liang, G. Slade. Self-avoiding walk enumeration via the lace expansion. J. Phys. A: Math. Theor. 40:10973--11017, (2007).
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    More extensive tables of enumeration are available in machine readable form here, or in a more human readable form in the unpublished document: N. Clisby, R. Liang, G. Slade, Self-avoiding walk enumeration via the lace expansion: tables, available here.
  13. R. van der Hofstad, F. den Hollander, G. Slade. The survival probability for critical spread-out oriented percolation above 4+1 dimensions. II. Expansion. Ann. Inst. H. Poincaré Probab. Statist. 43:509--570, (2007).
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  14. R. van der Hofstad, F. den Hollander, G. Slade. The survival probability for critical spread-out oriented percolation above 4+1 dimensions. I. Induction. Probab. Theory Relat. Fields. 138:363--389, (2007).
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  15. Y. Chan, A.L. Owczarek, A. Rechnitzer, G. Slade. Mean unknotting times of random knots and embeddings. J. Stat. Mech. P05004, (2007).
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  16. R. van der Hofstad and G. Slade. Expansion in n-1 for percolation critical values on the n-cube and Zn: the first three terms. Combinatorics, Probability and Computing 15:695--713, (2006).
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  17. C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer. Random subgraphs of finite graphs: III. The phase transition for the n-cube. Combinatorica 26:395--410, (2006).
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  18. C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer. Random subgraphs of finite graphs: II. The lace expansion and the triangle condition. Ann. Probab. 33:1886--1944, (2005).
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  19. C. Borgs, J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer. Random subgraphs of finite graphs: I. The scaling window under the triangle condition. Random Struct. Alg. 27:137--184, (2005).
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  20. R. van der Hofstad and G. Slade. Asymptotic expansion in n-1 for percolation critical values on the n-cube and Zn. Random Struct. Alg. 27:331--357, (2005).
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  21. G. Slade. The phase transition for random subgraphs of the n-cube. Extended abstract for the 16th Annual International Conference on Formal Power Series and Algebraic Combinatorics, Vancouver 2004.
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  22. M. Holmes, A.A. Járai, A. Sakai and G. Slade.  High-dimensional graphical networks of self-avoiding walks. Canad. J. Math. 56:77--114, (2004).
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  23. R. van der Hofstad and G. Slade.  The lace expansion on a tree with application to networks of self-avoiding walks. Adv. Appl. Math. 30:471--528, (2003).
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  24. R. van der Hofstad and G. Slade.  Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. Ann. Inst. H. Poincaré Probab. Statist. 39:413--485, (2003). This paper won the Prix de l'Institut Henri Poincaré 2003.
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  25. T. Hara, R. van der Hofstad and G. Slade.  Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31:349--408, (2003).
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  26. R. van der Hofstad, F. den Hollander and G. Slade.  Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions. Commun. Math. Phys. 231:435--461, (2002).
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  27. R. van der Hofstad and G. Slade.  A generalised inductive approach to the lace expansion. Probab. Th. Rel. Fields. 122:389--430, (2002).
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  28. T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys., 99:1075--1168, (2000).
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  29. T. Hara and G. Slade. The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys., 41:1244--1293, (2000).
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  30. C. Borgs, J.T. Chayes, R. van der Hofstad, and G. Slade. Mean-field lattice trees. Annals of Combinatorics, 3:205--221, (1999).
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  31. G. Slade. Lattice trees, percolation and super-Brownian motion. In: Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, eds. M. Bramson and R. Durrett, Birkhäuser (Basel), pages 35--51, (1999).
    PDF file
  32. T. Hara and G. Slade, The incipient infinite cluster in high-dimensional percolation. Electron. Res. Announc. Amer. Math. Soc., 4:48--55, (1998).
  33. E. Derbez and G. Slade, The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69--104, (1998).
  34. R. van der Hofstad, F. den Hollander and G. Slade, A new inductive approach to the lace expansion for self-avoiding walks. Probab. Th. Rel. Fields, 111:253--286, (1998).
  35. E. Derbez and G. Slade, Lattice trees and super-Brownian motion. Canadian Mathematical Bulletin, 40:19--38, (1997).
  36. D.C. Brydges and G. Slade, Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation. Commun. Math. Phys., 182:485--504, (1996).
  37. D.C. Brydges and G. Slade, The diffusive phase of a model of self-interacting walks. Probability Theory and Related Fields, 103:285--315, (1995).
  38. T. Hara and G. Slade, The self-avoiding-walk and percolation critical points in high dimensions. Combinatorics, Probability and Computing, 4:197--215, (1995).
  39. G. Slade, Bounds on the self-avoiding-walk connective constant, Journal of Fourier Analysis and Applications, Special Issue: Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, June 28 -- July 3, 1993), 525--533, (1995).
  40. G. Slade, The critical behaviour of random systems. Proceedings of the International Congress of Mathematicians, August 3-11, 1994, Zürich, Volume 2, pages 1315--1324. Ed. S.D. Chatterji; Birkhäuser, Basel (1995).
  41. D.C. Brydges and G. Slade, A collapse transition for self-attracting walks. Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo, 1:363--372, (1994).
  42. T. Hara and G. Slade, Mean-field behaviour and the lace expansion. Pages 87--122 in Probability and Phase Transition, ed. G.R. Grimmett, Kluwer (Dordrecht), (1994). Proceedings of the NATO Advanced Study Institute on Probability Theory of Spatial Disorder and Phase Transition, July 1993, Isaac Newton Institute, Cambridge.
    PS file
  43. T. Hara, G. Slade and A.D. Sokal, New lower bounds for the self-avoiding-walk connective constant, J. Stat. Phys., 72:479--517, (1993). Erratum, J. Stat. Phys., 78:1187--1188, (1995).
  44. T. Hara and G. Slade, The number and size of branched polymers in high dimensions. J. Stat. Phys., 67:1009--1038, (1992).
  45. T. Hara and G. Slade, Self-avoiding walk in five or more dimensions. I. The critical behaviour, Commun. Math. Phys., 147:101--136, (1992).
  46. T. Hara and G. Slade, The lace expansion for self-avoiding walk in five or more dimensions. Reviews in Math. Phys., 4:235--327, (1992).
  47. T. Hara and G. Slade, Critical behaviour of self-avoiding walk in five or more dimensions. Bull. Amer. Math. Soc., 25: 417--423, (1991).
  48. G. Slade, The lace expansion and the upper critical dimension for percolation. Lectures in Applied Mathematics, 27:53--63, (1991). (Mathematics of Random Media, eds. W.E. Kohler and B.S. White, A.M.S., Providence. Proceedings of the AMS-SIAM Summer Seminar on Mathematics of Random Media, Blacksburg, June 1989.)
  49. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128:333--391, (1990).
  50. T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469--1510, (1990).
  51. G. Slade, The scaling limit of self-avoiding random walk in high dimensions. Ann. Probab., 17:91--107, (1989).
  52. T. Hara and G. Slade, The triangle condition for percolation, Bull. Amer. Math. Soc., 21, 269--273, (1989).
  53. T. Hara and G. Slade, The mean-field critical behaviour of percolation in high dimensions. Proceedings of the IXth International Congress on Mathematical Physics, Swansea, July 1988, pages 450--453. Eds. B. Simon, A. Truman, I.M. Davies; Adam Hilger, Bristol and New York, (1989).
  54. G. Slade, Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A: Math. Gen., 21:L417--L420 (1988).
  55. G. Slade, The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys., 110:661--683, (1987).
  56. G. Slade, The effective potential as an energy density: the one phase region, Commun. Math. Phys., 104:573--580, (1986).
  57. G. Slade, The loop expansion for the effective potential in the P(φ)2 quantum field theory, Commun. Math. Phys., 102:425--462, (1985).

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Expository Writing:

  1. G. Slade. Probabilistic Models of Critical Phenomena. This is an essay intended for a general mathematical audience, from The Princeton Companion to Mathematics, ed. T. Gowers, assoc. eds. J. Barrow-Green and I. Leader. Princeton University Press, Princeton, N.J., (2008). Reprinted by permission of Princeton University Press. Posted November 22, 2004.
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    Clarification
  2. G. Slade. Wendelin Werner awarded Fields Medal. PIMS Newsletter 10, Issue 2: 4--5, Winter 2007.
  3. G. Slade. Scaling limits and super-Brownian motion. Notices Amer. Math. Soc. 49, No. 9 (October):1056--1067, (2002).
  4. G. Slade. Book review: Random walks and random environments, by Barry D. Hughes. Bull. Amer. Math. Soc. 35:347--349, (1998).
  5. G. Slade. Random walks. American Scientist, 84:146--153, (1996).
  6. G. Slade. Self-avoiding walks. The Mathematical Intelligencer 16:29--35, (1994).
    PDF file

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Collaborators:

Omer Angel
Martin Barlow
Roland Bauerschmidt
Christian Borgs
David Brydges
Yao-ban Chan
Jennifer Chayes
Nathan Clisby
Antoine Dahlqvist
Eric Derbez
Hugo Duminil-Copin
Jesse Goodman
Takashi Hara
Remco van der Hofstad
Frank den Hollander
Mark Holmes
John Imbrie
Antal Járai
Takashi Kumagai
Richard Liang
Neal Madras
Yuri Mejía Miranda
Aleks Owczarek
Andrew Rechnitzer
Akira Sakai
Alan Sokal
Joel Spencer

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