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| Self-avoiding walk | |
|---|---|
| Name | Self-avoiding walk |
| Field | Probability theory, Statistical mechanics, Mathematical physics |
| Introduced | Early 20th century |
| Prominent | Paul Flory, Benoît Mandelbrot, Michael Fisher |
Self-avoiding walk is a combinatorial model of a path on a lattice that does not visit the same vertex more than once, studied in probability theory, statistical mechanics, and mathematical physics. It serves as an idealized model of linear polymers and interfaces and connects to deep problems in complex analysis, conformal field theory, and enumerative combinatorics. Research has involved contributions from figures associated with Royal Society, Institute for Advanced Study, Princeton University, and journals such as Annals of Mathematics and Communications in Mathematical Physics.
A self-avoiding walk on a graph such as the hypercubic lattice or the square lattice is a sequence of distinct vertices connected by edges, introduced in early polymer theory by Paul Flory and formalized in probabilistic contexts by researchers linked to Cambridge University and University of Oxford. Basic invariants include the number of n-step walks c_n, the connective constant μ, and the mean-square end-to-end distance ⟨R_n^2⟩; these quantities relate to limits studied by scholars affiliated with Imperial College London and Courant Institute of Mathematical Sciences. Symmetries of lattices like the triangular lattice or the hexagonal lattice affect c_n and μ, and rigorous bounds have been pursued using techniques associated with Institute Henri Poincaré and Max Planck Institute.
Variants of the model include the interacting self-avoiding walk studied by groups at University of Cambridge and University of Chicago, the weakly self-avoiding walk linked to work from University of British Columbia, and the self-avoiding polygon studied at ETH Zurich and University of Tokyo. Lattice choices include the square lattice, cubic lattice, and hexagonal lattice, with continuum limits investigated in collaboration with researchers from Massachusetts Institute of Technology and California Institute of Technology. Related constructs such as the self-avoiding trail, prudent walk, and loop-erased random walk have been developed in research from Rutgers University, Yale University, and École Normale Supérieure.
Enumeration of c_n has been a major computational effort with data produced by teams at University of Melbourne, University of Warwick, and University of Bristol, and theoretical analysis by figures connected to Harvard University and Stanford University. Critical exponents ν and γ describe scaling of ⟨R_n^2⟩ ∼ n^{2ν} and c_n ∼ μ^n n^{γ-1}, topics of work by scholars from Princeton University and University of Geneva. Field-theoretic predictions from groups around Cornell University and University of Pennsylvania use renormalization methods originating with Kenneth Wilson and relate to conformal predictions advanced by researchers at University of Cambridge and Trinity College Dublin.
Rigorous results include exact determination of the connective constant for the hexagonal lattice by researchers associated with University of Geneva and Institut des Hautes Études Scientifiques, while conjectures about scaling limits as Schramm–Loewner evolution (SLE) link to work at University of Bonn and University of Chicago. The conjecture that the two-dimensional scaling limit is SLE_{8/3} has been a focal point for teams at Princeton University, University of Oxford, and University of California, Berkeley. High-dimensional behavior approaching mean-field predictions connects to investigations at University of Toronto and University of British Columbia, influenced by methods from Ecole Polytechnique and University of Cambridge.
Analytical techniques include lace expansion developed by scholars at Rutgers University and University of British Columbia, renormalization group methods pioneered by Institut des Hautes Études Scientifiques affiliates, and combinatorial bijections explored at University of Paris-Saclay and University of Warwick. Complex analysis and discrete holomorphic observables used in the hexagonal lattice proof draw on traditions from École Normale Supérieure and University of Geneva. Probabilistic coupling, subadditivity, and diagrammatic estimates have been advanced in work from Technion – Israel Institute of Technology and University of Oxford.
Applications span modeling of linear polymers in chemistry groups at Max Planck Society and Bell Labs, interfaces and percolation connections explored by teams at University of Cambridge and University of Amsterdam, and relations to fractal geometry investigated by Benoît Mandelbrot-associated centers and Institute for Advanced Study researchers. Connections to Schramm–Loewner evolution research groups link to predictions in conformal field theory and studies of critical phenomena in contexts tied to National Institute of Standards and Technology and Los Alamos National Laboratory. Computational biology groups at Salk Institute and Wellcome Trust Sanger Institute have used self-avoiding walk ideas in coarse-grained polymer and protein modeling.
Enumeration and Monte Carlo studies have been performed using pivot algorithms developed by researchers at Brown University and University of Edinburgh, transfer-matrix methods advanced by teams at University of Oxford and University of Manchester, and exact enumeration techniques from University of New South Wales and University of Melbourne. High-precision estimates combine parallel computing resources from Lawrence Berkeley National Laboratory and Argonne National Laboratory with algorithmic advances inspired by work at Google Research and Microsoft Research. Statistical analysis of simulation outputs has been supported by collaborators at Carnegie Mellon University and University of Washington.
Category:Mathematical models