Gordon Slade's Publications

Preprints:

1. Y. Liu, J. Park and G. Slade. Universal finite-size scaling in high-dimensional critical phenomena. 13 pages. Revised October 10, 2025.
   arXiv

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Books:

2. R. Bauerschmidt, D.C. Brydges and G. Slade, Introduction to a Renormalisation Group Method, Lecture Notes in Mathematics #2242, xii + 281 pages. Springer, Singapore, (2019).
   arXiv Corrections
3. G. Slade. The Lace Expansion and its Applications, Lecture Notes in Mathematics #1879, xiv + 232 pages. Springer, Berlin, (2006).
   pdf
   Students' solutions to all the exercises in the lecture notes, edited by S. Kliem and R. Liang (November 2, 2005): PS file
4. N. Madras and G. Slade, The Self-Avoiding Walk , Birkhäuser, Boston, (1993). xiv + 425 pages. Paperback edition 1996. Reprinted as a Modern Birkhäuser Classic 2013.

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Edited book:

5. M.T. Barlow and G. Slade (editors), Random Graphs, Phase Transitions, and the Gaussian Free Field, PIMS-CRM Summer School in Probability, Vancouver, Canada, June 5-30, 2017, Proceedings in Mathematics & Statistics (PROMS) volume 304, xvii + 407 pages. Springer, Switzerland, (2020).

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Journal Articles and Conference Proceedings:

6. Y. Liu and G. Slade. Gaussian deconvolution and the lace expansion for spread-out models. To appear in Ann. Inst. H. Poincaré Probab. Statist.
   arXiv
7. Y. Liu and G. Slade. Critical scaling profile for trees and connected subgraphs on the complete graph. Canad. Math. Bull., (2025).
   arXiv
8. Y. Liu, R. Panis and G. Slade. The torus plateau for the high-dimensional Ising model. Commun. Math. Phys., 406:article 159, (2025).
   arXiv
9. J. Park and G. Slade. Boundary conditions and the two-point function plateau for the hierarchical |φ|⁴ model in dimensions 4 and higher. Ann. Henri Poincaré, (2025).
   arXiv
10. E. Michta, J. Park and G. Slade. Boundary conditions and universal finite-size scaling for the hierarchical |φ|⁴ model in dimensions 4 and higher. Commun. Pure Appl. Math., 78:2001--2118, (2025)
    arXiv
11. Y. Liu and G. Slade. Near-critical and finite-size scaling for high-dimensional lattice trees and animals. J. Stat. Phys., 192:article 23, (2025).
    arXiv
12. Y. Liu and G. Slade. Gaussian deconvolution and the lace expansion. Probab. Theory Relat. Fields, (2024).
    arXiv
13. E. Michta and G. Slade. Weakly self-avoiding walk on a high-dimensional torus. Probab. Math. Phys., 4:331--375, (2023).
    arXiv video
14. G. Slade. The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions. Math. Phys. Anal. Geom., 26:article 6, (2023).
    arXiv
15. T. Hutchcroft, E. Michta and G. Slade. High-dimensional near-critical percolation and the torus plateau. Ann. Probab., 51:580--625, (2023).
    pdf video
16. G. Slade. Self-avoiding walk on the hypercube. Random Struct. Alg., 62:689--736, (2023).
    arXiv
17. E. Michta and G. Slade. Asymptotic behaviour of the lattice Green function. ALEA, Lat. Am. J. Probab. Math. Stat., 19:957--981, (2022)
    arXiv
18. G. Slade. A simple convergence proof for the lace expansion. Ann. Inst. H. Poincaré Probab. Statist., 58:26--33, (2022).
    arXiv
19. R. Bauerschmidt and G. Slade. Mean-field tricritical polymers. Probab. Math. Phys., 1:167--204, (2020).
    arXiv video
20. G. Slade. Self-avoiding walk on the complete graph. J. Math. Soc. Japan, 72:1189--1200, (2020).
    arXiv
21. R. Bauerschmidt, M. Lohmann and G. Slade. Three-dimensional tricritical spins and polymers. J. Math. Phys., 61:033302, (2020).
    arXiv
22. A. Sakai and G. Slade. Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees. Elect. J. Probab., 24, paper no. 65, pp.1-18, (2019).
    arXiv
23. G. Slade. Self-avoiding walk, spin systems and renormalization. Proc. R. Soc. A, 475:20181549, (2019).
    arXiv video
24. G. Slade. Critical exponents for long-range O(n) models below the upper critical dimension. Commun. Math. Phys., 358:343--436, (2018).
    arXiv
25. M. Lohmann, G. Slade and B.C. Wallace. Critical two-point function for long-range O(n) models below the upper critical dimension. J. Stat. Phys., 169:1132--1161, (2017).
    arXiv
26. R. Bauerschmidt, G. Slade and B.C. Wallace. Four-dimensional weakly self-avoiding walk with contact self-attraction. J. Stat. Phys., 167:317--350, (2017).
    pdf Corrections
27. R. Bauerschmidt, G. Slade, A. Tomberg and B.C. Wallace. Finite-order correlation length for 4-dimensional weakly self-avoiding walk and |φ|⁴ spins. Ann. Henri Poincaré, 18:375--402, (2017).
    pdf Corrections
28. G. Slade and A. Tomberg. Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component |φ|⁴ model. Commun. Math. Phys., 342:675--737, (2016).
    pdf Corrections
29. R. Bauerschmidt, D.C. Brydges and G. Slade. Renormalisation group analysis of 4D spin models and self-avoiding walk. February 12, 2016.
    For the Proceedings of the International Congress on Mathematical Physics, Santiago de Chile, 2015.
    arXiv
30. R. Bauerschmidt, D.C. Brydges and G. Slade. Critical two-point function of the 4-dimensional weakly self-avoiding walk. Commun. Math. Phys., 338:169--193, (2015).
    pdf
31. 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. Commun. Math. Phys., 337:817--877, (2015).
    pdf Corrections
32. D.C. Brydges and G. Slade. A renormalisation group method. I. Gaussian integration and normed algebras. J. Stat. Phys., 159:421--460, (2015).
    pdf Corrections
33. D.C. Brydges and G. Slade. A renormalisation group method. II. Approximation by local polynomials. J. Stat. Phys., 159:461--491, (2015).
    pdf
34. R. Bauerschmidt, D.C. Brydges and G. Slade. A renormalisation group method. III. Perturbative analysis. J. Stat. Phys., 159:492--529, (2015).
    pdf Corrections
    Software for calculations in this paper.
35. D.C. Brydges and G. Slade. A renormalisation group method. IV. Stability analysis. J. Stat. Phys., 159:530--588, (2015).
    pdf Corrections
36. D.C. Brydges and G. Slade. A renormalisation group method. V. A single renormalisation group step. J. Stat. Phys., 159:589--667, (2015).
    pdf Corrections
37. R. Bauerschmidt, D.C. Brydges and G. Slade. Structural stability of a dynamical system near a non-hyperbolic fixed point. Ann. Henri Poincaré, 16:1033--1065, (2015).
    pdf Corrections
38. 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).
    Special issue devoted to the memory of Kenneth G. Wilson.
    pdf Corrections
39. 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.)
    pdf

40. 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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41. 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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42. 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).
    pdf
43. 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).
    pdf
44. 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).
    pdf
45. 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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46. 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).
    pdf
47. 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).
    pdf
    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.
48. O. Angel, J. Goodman, F. den Hollander, G. Slade. Invasion percolation on regular trees. Ann. Probab. 36:420--466, (2008).
    pdf
49. 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).
    pdf
50. N. Clisby, R. Liang, G. Slade. Self-avoiding walk enumeration via the lace expansion. J. Phys. A: Math. Theor. 40:10973--11017, (2007).
    pdf
    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.
51. 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).
    pdf
52. 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).
    pdf
53. 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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54. 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).
    pdf
55. 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).
    pdf
56. 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).
    pdf
57. 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).
    pdf
58. 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).
    pdf
59. 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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60. 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).
    pdf
61. 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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62. 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 Prize of the Annales de l'Institut Henri Poincaré 2003.
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63. 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).
    pdf
64. 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).
    pdf
65. R. van der Hofstad and G. Slade.  A generalised inductive approach to the lace expansion. Probab. Th. Rel. Fields. 122:389--430, (2002).
    pdf
66. 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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67. 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).
    pdf
68. 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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69. 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).
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70. T. Hara and G. Slade, The incipient infinite cluster in high-dimensional percolation. Electron. Res. Announc. Amer. Math. Soc., 4:48--55, (1998).
71. E. Derbez and G. Slade, The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69--104, (1998).
72. 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).
73. E. Derbez and G. Slade, Lattice trees and super-Brownian motion. Canadian Mathematical Bulletin, 40:19--38, (1997).
74. D.C. Brydges and G. Slade, Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation. Commun. Math. Phys., 182:485--504, (1996).
75. 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).
76. T. Hara and G. Slade, The self-avoiding-walk and percolation critical points in high dimensions. Combinatorics, Probability and Computing, 4:197--215, (1995). Unpublished appendix.
77. 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).
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78. 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).
79. 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).
80. 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.
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81. 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).
82. T. Hara and G. Slade, The number and size of branched polymers in high dimensions. J. Stat. Phys., 67:1009--1038, (1992).
83. T. Hara and G. Slade, Self-avoiding walk in five or more dimensions. I. The critical behaviour, Commun. Math. Phys., 147:101--136, (1992).
84. 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).
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85. T. Hara and G. Slade, Critical behaviour of self-avoiding walk in five or more dimensions. Bull. Amer. Math. Soc., 25: 417--423, (1991).
86. 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.)
87. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128:333--391, (1990).
88. T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469--1510, (1990).
89. G. Slade, The scaling limit of self-avoiding random walk in high dimensions. Ann. Probab., 17:91--107, (1989).
90. T. Hara and G. Slade, The triangle condition for percolation, Bull. Amer. Math. Soc., 21, 269--273, (1989).
91. 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).
92. G. Slade, Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A: Math. Gen., 21:L417--L420 (1988).
93. G. Slade, The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys., 110:661--683, (1987).
94. G. Slade, The effective potential as an energy density: the one phase region, Commun. Math. Phys., 104:573--580, (1986).
95. 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:

96. 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
97. G. Slade. Wendelin Werner awarded Fields Medal. PIMS Newsletter 10, Issue 2: 4--5, Winter 2007.
    
98. G. Slade. Scaling limits and super-Brownian motion. Notices Amer. Math. Soc. 49, No. 9 (October):1056--1067, (2002).
99. G. Slade. Book review: Random walks and random environments, by Barry D. Hughes. Bull. Amer. Math. Soc. 35:347--349, (1998).
100. G. Slade. Random walks. American Scientist, 84:146--153, (1996).
101. G. Slade. Self-avoiding walks. The Mathematical Intelligencer 16:29--35, (1994).
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Other:

102. G. Slade. Laudatio for David Brydges on the occasion of his receiving the Henri Poincaré Prize. July 1, 2024.
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103. G. Slade. Kotani's Theorem for the Fourier Transform. Unpublished note. June 19, 2020.
     arXiv

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Collaborators:

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