G. Slade.
The Lace Expansion and its Applications,
Lecture Notes in Mathematics #1879,
xiv + 232 pages. Springer, Berlin, (2006). PDF file
(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
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.
The following 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
Scaling limits and critical behaviour of the
4-dimensional n-component |φ|^{4} spin model
if you are interested in applications to the |φ|^{4} model, or
Logarithmic correction for the susceptibility of the
4-dimensional weakly self-avoiding walk: a renormalisation group analysis
and then
Critical two-point function of the 4-dimensional weakly self-avoiding walk
if you are interested in applications to the self-avoiding walk.
A brief overview can be found in
Renormalisation group analysis of 4D spin models and self-avoiding walk.
Both models are treated in the most recent papers,
Critical correlation functions for the 4-dimensional
weakly self-avoiding walk and n-component |φ|^{4} model,
and
Finite-order correlation length for 4-dimensional
weakly self-avoiding walk and |φ|^{4} spins.
Concerning the general renormalisation group
method itself, perturbative aspects are treated in
A renormalisation group method. III. Perturbative analysis
and the centrepiece and main innovation is
A renormalisation group method. V. A single renormalisation group step.
The other four papers (I,II,IV and Structural Stability)
play various foundational and supporting roles.
The versions here are not necessarily identical to the published versions.
R. Bauerschmidt, D.C. Brydges and G. Slade.
Renormalisation group analysis of 4D spin models and self-avoiding walk.
Preprint February 12, 2016. For the Proceedings of the International Congress on Mathematical Physics,
Santiago de Chile, 2015.
PDF file
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.
PDF file
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 file
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 file
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 file
D.C. Brydges, J.Z. Imbrie, G. Slade.
Functional integral representations for self-avoiding walk.
Probability Surveys,
6:34--61, (2009).
PDF file
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 file
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 file
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.
O. Angel, J. Goodman, F. den Hollander, G. Slade.
Invasion percolation on regular trees.
Ann. Probab. 36:420--466, (2008).
PDF file
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 file
N. Clisby, R. Liang, G. Slade.
Self-avoiding walk enumeration via the lace expansion.
J. Phys. A: Math. Theor.40:10973--11017, (2007).
PDF file
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.
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 file
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 file
R. van der Hofstad and G. Slade. Expansion in n^{-1}
for percolation critical values on the n-cube and Z^{n}:
the first three terms. Combinatorics, Probability and Computing15:695--713, (2006).
PDF file
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. Combinatorica26:395--410, (2006). PDF file
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 file
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 file
R. van der Hofstad and G. Slade. Asymptotic expansion in n^{-1}
for percolation critical values on the n-cube and Z^{n}.
Random Struct. Alg. 27:331--357, (2005). PDF file
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. PDF file
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 file
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). PDF file
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. PDF file
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 file
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 file
R. van der Hofstad and G. Slade. A generalised
inductive approach to the lace expansion. Probab. Th. Rel. Fields. 122:389--430, (2002). PDF file
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). PDF file
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 file
C. Borgs, J.T. Chayes, R. van der Hofstad, and G. Slade.
Mean-field lattice trees. Annals of Combinatorics,
3:205--221, (1999). PDF file
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
E. Derbez and G. Slade, The scaling limit of lattice
trees in high dimensions. Commun. Math. Phys., 193:69--104, (1998).
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).
E. Derbez and G. Slade, Lattice trees and super-Brownian
motion. Canadian Mathematical Bulletin, 40:19--38, (1997).
D.C. Brydges and G. Slade, Statistical mechanics of the
2-dimensional focusing nonlinear Schrödinger equation. Commun. Math. Phys.,
182:485--504, (1996).
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).
T. Hara and G. Slade, The self-avoiding-walk and
percolation critical points in high dimensions. Combinatorics,
Probability and Computing, 4:197--215, (1995).
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).
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).
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).
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. PDF file
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).
T. Hara and G. Slade, The number and size of branched
polymers in high dimensions. J. Stat. Phys., 67:1009--1038, (1992).
T. Hara and G. Slade, Self-avoiding walk in five or more dimensions. I. The critical behaviour,
Commun. Math. Phys., 147:101--136, (1992).
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).
T. Hara and G. Slade, Critical behaviour of self-avoiding walk in five or more dimensions.
Bull. Amer. Math. Soc.,25: 417--423, (1991).
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.)
T. Hara and G. Slade, Mean-field critical behaviour for
percolation in high dimensions. Commun. Math. Phys.,
128:333--391, (1990).
T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals.
J. Stat. Phys., 59:1469--1510, (1990).
G. Slade, The scaling limit of self-avoiding random walk
in high dimensions. Ann. Probab., 17:91--107, (1989).
T. Hara and G. Slade, The triangle condition for percolation,
Bull. Amer. Math. Soc.,21, 269--273, (1989).
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).
G. Slade, Convergence of self-avoiding random walk to
Brownian motion in high dimensions. J. Phys. A: Math. Gen., 21:L417--L420 (1988).
G. Slade, The diffusion of self-avoiding random walk in
high dimensions. Commun. Math. Phys., 110:661--683, (1987).
G. Slade, The effective potential as an energy density: the one phase region,
Commun. Math. Phys., 104:573--580, (1986).
G. Slade, The loop expansion for the effective potential in the P(φ)_{2}
quantum field theory, Commun. Math. Phys., 102:425--462, (1985).
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. PDF file Clarification
G. Slade. Wendelin Werner awarded Fields Medal.
PIMS Newsletter10, Issue 2: 4--5, Winter 2007.
G. Slade. Scaling limits and super-Brownian motion.
Notices Amer. Math. Soc.49, No. 9 (October):1056--1067, (2002).
G. Slade. Book review:
Random walks and random environments, by Barry D. Hughes.
Bull. Amer. Math. Soc.35:347--349, (1998).
G. Slade. Random walks. American Scientist, 84:146--153, (1996).
G. Slade. Self-avoiding walks. The Mathematical Intelligencer16:29--35, (1994). PDF file