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Renormalisation Group Analysis of Weakly Self-avoiding Walk in Dimensions Four and Higher
Abstract
We outline a proof, by a rigorous renormalisation group method, that the critical two-point function for continuous-time weakly self-avoiding walk on Z^d decays as |x|^{-(d-2)} in the critical dimension d=4, and also for all d>4. Comment: 25 pages, 1 diagram, lecture at ICM 2010, Hyderabad, in section on probability and statistics, crosslisted with section on mathematical physics
... This is a statement that the critical exponent η exists and is equal to zero. The proof is based on a rigorous renormalisation group method; a summary of the method and proof is given in [15]. Early indications of the critical nature of the dimension d = 4 were given in [3,12], following proofs of triviality of φ 4 field theory above dimension 4 [2,28]. ...
... In [15], an extension of Theorem 1.1 states that the critical two-point function has decay |a−b| 2−d for all dimensions d ≥ 4, but [15] provides only a sketch of proof. Our principal interest is the critical dimension d = 4, and we provide the details of the proof for d = 4 here. ...
... In [15], an extension of Theorem 1.1 states that the critical two-point function has decay |a−b| 2−d for all dimensions d ≥ 4, but [15] provides only a sketch of proof. Our principal interest is the critical dimension d = 4, and we provide the details of the proof for d = 4 here. ...
We prove decay of the critical two-point function for the
continuous-time weakly self-avoiding walk on , in the upper
critical dimension d=4. This is a statement that the critical exponent
exists and is equal to zero. Results of this nature have been proved previously
for dimensions using the lace expansion, but the lace expansion does
not apply when d=4. The proof is based on a rigorous renormalisation group
analysis of an exact representation of the continuous-time weakly self-avoiding
walk as a supersymmetric field theory. Much of the analysis applies more widely
and has been carried out in a previous paper, where an asymptotic formula for
the susceptibility is obtained. Here, we show how observables can be
incorporated into the analysis to obtain a pointwise asymptotic formula for the
critical two-point function. This involves perturbative calculations similar to
those familiar in the physics literature, but with error terms controlled
rigorously.
... As we discuss in more detail in Section 1.4 below, this result provides an essential ingredient in a renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk [2,4], where the boundary condition lim j→∞ µ j = 0 is the appropriate boundary condition for the study of a critical trajectory. It is this application that provides our immediate motivation to study the dynamical system Φ, but we expect that the methods developed here will have further applications to dynamical systems arising in renormalisation group analyses. ...
... We think of Φ = (Φ j ) j∈N 0 as the evolution map of a discrete time-dependent dynamical system, although it is more usual in dynamical systems to have the spaces X j be identical. Our application in [2,4] requires the greater generality of j-dependent spaces. ...
... Our applications include situations in which β j approaches a positive limit as j → ∞, but also situations in which β j is approximately constant in j over a long initial interval j ≤ j Ω and then abruptly decays to zero. In this setting, j Ω plays the role of a mass scale, due to the fact that in [4] the parameters ofφ j and the maps ψ j , ρ j are continuous functions of a mass parameter m ≥ 0, and j Ω → ∞ as m ↓ 0. We are interested in continuity of the flow Φ in this massless limit, and (A4) is designed to accommodate this aspect. ...
We prove structural stability under perturbations for a class of
discrete-time dynamical systems near a non-hyperbolic fixed point. We
reformulate the stability problem in terms of the well-posedness of an
infinite-dimensional nonlinear ordinary differential equation in a Banach space
of carefully weighted sequences. Using this, we prove existence and regularity
of flows of the dynamical system which obey mixed initial and final boundary
conditions. The class of dynamical systems we study, and the boundary
conditions we impose, arose in a renormalisation group analysis of the
4-dimensional weakly self-avoiding walk.
... Finally, exact enumeration plus series analysis has been used; currently the most extensive enumerations in dimensions d ≥ 3 use the lace expansion [21], and for d = 3 walks have been enumerated to length n = 30. The exact enumeration estimates for d = 3 are µ = 4.684043 (12), γ = 1.1568 (8), ν = 0.5876(5) [21]. Monte Carlo estimates are consistent with these values: γ = 1.1575(6) [16] and ν = 0.587597(7) [20]. ...
... The hierarchical lattice is a simplification of the hypercubic lattice Z 4 which is particularly amenable to the renormalisation group approach. Recently there has been progress in the application of renormalisation group methods to a continuous-time weakly self-avoiding walk model on Z 4 itself, and in particular it has been proved in this context that the critical two-point function has |x| −2 decay [12], which is a statement that the critical exponent η is equal to 0. This is the topic of Section 7 below. 1.6.5. ...
... In (4.13), each segment represents a SAW path, and the notation [12] indicates that SAWs 1 and 2 must be mutually avoiding, apart from one shared vertex. We apply inclusion-exclusion to (4.13), first summing over all pairs of SAWs, mutually avoiding or not, and then subtracting configurations where SAWs 1 and 2 intersect. ...
These lecture notes provide a rapid introduction to a number of rigorous
results on self-avoiding walks, with emphasis on the critical behaviour.
Following an introductory overview of the central problems, an account is given
of the Hammersley--Welsh bound on the number of self-avoiding walks and its
consequences for the growth rates of bridges and self-avoiding polygons. A
detailed proof that the connective constant on the hexagonal lattice equals
is then provided. The lace expansion for self-avoiding
walks is described, and its use in understanding the critical behaviour in
dimensions is discussed. Functional integral representations of the
self-avoiding walk model are discussed and developed, and their use in a
renormalisation group analysis in dimension 4 is sketched. Problems and
solutions from tutorials are included.
... The aim of this and of a previous paper, [Falco, 2012], is to show that the Brydges-Yau's technique is truly an effective method to deal with the BKT line of the Coulomb gas. In [Falco, 2012], building on a technical suggestion due to D. Brydges and on the general scheme of [Brydges, 2009] (see also [Dimock, 2009;Brydges and Slade, 2010]), we already showed that some difficulties of [Dimock and Hurd, 2000] can be avoided; and that a convergent series representation for the free energy along the BKT line, for z small enough, can be provided. In this paper we take up the mathematically more sophisticated and physically more interesting objective of studying the long-distance decay of fractional charge correlations (1.4), again along the BKT curve and for z small enough. ...
... Polymer gas representation. Following [Brydges and Yau, 1990;Brydges, 2009;Brydges and Slade, 2010], each Ω j can be efficiently represented as a polymer gas. Before describing this formulation, we have to introduce a multiscale decomposition of the lattice and, correspondingly, special types of lattice domains. ...
The two dimensional Coulomb gas is the prototypical model of statistical
mechanics displaying a special kind of phase transition, named after
Berezinskii, Kosterlitz and Thouless. Physicists and mathematicians proposed
several predictions about this system. Two of them, valid along the phase
transition curve and for small activity, are: a) the long-distance decay of the
``fractional charge'' correlation is power law, with a multiplicative
logarithmic correction; b) in such a decay, the exponent of the power law, as
well as the exponent of the logarithmic correction, have a certain precise
dependence upon the charge value. In this paper we provide a proof of these two
long standing conjectures.
... To have control on it, we must distinguish the irrelevant part of V j , namely the terms that, along the flow, become smaller and smaller by simple 'power counting' arguments, from the relevant part, namely the terms that require a more careful study. In order to do that, it is important to introduce some special kind of lattice domains: blocks and polymers ([5], [10]). Define |x| := max{|x 0 |, |x 1 |}. ...
... Without loss of generality, assume y 1 positive. The strategy -partially inspired to [10] -is to mimic the treatment of the continuous version of (7.3). We look ...
With a rigorous renormalization group approach, we study the pressure of the
two dimensional Coulomb Gas along a small piece of the Kosterlitz-Thouless
transition line, i.e. the boundary of the dipole region in the
activity-temperature phase-space.
... It can also be regarded as a test-bed problem for physically relevant models which have logarithmic corrections to scaling, for example the θ-transition in three dimensions is believed to be identified with a tricritical point, and has meanfield behavior with logarithmic corrections. Four-dimensional SAWs have been studied by Monte Carlo [3,4] and enumeration [5][6][7] methods, and we note that rigorous results have recently been obtained for the 4-dimensional weakly self-avoiding walk via the rigorous renormalization group [8]. ...
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is .
... It can also be regarded as a test-bed problem for physically relevant models which have logarithmic corrections to scaling, for example the θ-transition in three dimensions is believed to be identified with a tricritical point, and has meanfield behavior with logarithmic corrections. Four-dimensional SAWs have been studied by Monte Carlo [3,4] and enumeration [5][6][7] methods, and we note that rigorous results have recently been obtained for the 4-dimensional weakly self-avoiding walk via the rigorous renormalization group [8]. ...
We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is .
... It is not expected that invariance under this limited group completely determines the SAW in three dimensions, but one can still ask if the SAW is conformally invariant and if this invariance leads to any non-trivial predictions for the three-dimensional SAW. We restrict our attention to three dimensions since the scaling limit of the SAW has been proved to be Brownian motion in more than four dimensions [7,8], and substantial progress toward proving this in four dimensions has been made [1]. ...
If the three dimensional self-avoiding walk (SAW) is conformally invariant,
then one can compute the hitting densities for the SAW in a half-space and in a
sphere. The ensembles of SAW's used to define these hitting densities involve
walks of arbitrary lengths, and so these ensembles cannot be directly studied
by the pivot Monte Carlo algorithm for the SAW. We show that these mixed length
ensembles should have the same scaling limit as certain weighted ensembles that
only involve walks with a single length, thus providing a fast method for
simulating these ensembles. Preliminary simulations which found good agreement
between the predictions and Monte Carlo simulations for the SAW were reported
in [14]. In this paper we present more accurate simulations testing the
predictions and find even stronger support for the prediction that the SAW is
conformally invariant in three dimensions.
... While Equation (1) is undoubtedly true, it has not yet been proven in every dimension. It is known to hold for d ≥ 5 with ν = 1/2 (Hara and Slade (1992a, 1992b)), and there is progress towards a proof for d = 4 (see Brydges and Slade (2010)). For d = 2, there is strong theoretical evidence that ν = 3/4, while for d = 3 it seems likely that ν is close to 0.588 (see Slade (2011) for a current review). ...
For an N-step self-avoiding walk on the hypercubic lattice ℤ d , we prove that the mean-square end-to-end distance is at least N 4/(3d) times a constant. This implies that the associated critical exponent ν is at least 2/(3d), assuming that ν exists.
... Our method of proof of Theorems 1.1-1.2 is based on a rigorous renormalisation group analysis, and applies more widely. In particular, it is used in [5,15] to prove that the critical two-point function G g,νc(g) (0, x) is asymptotic to a multiple of |x| −2 as |x| → ∞ in dimension d = 4. Also, work is in progress to extend our methods to prove existence of logarithmic corrections to scaling for certain critical 4-dimensional polymer networks [10], as well as to the study of the weakly selfavoiding walk with nearest-neighbour contact attraction [4] in dimension d = 4. In [7], we apply the renormalisation group method to study the critical behaviour of the 4-dimensional n-component |ϕ| 4 spin model, for all positive integers n ≥ 1. ...
We prove that the susceptibility of the continuous-time weakly self-avoiding
walk on , in the critical dimension d=4, has a logarithmic
correction to mean-field scaling behaviour as the critical point is approached,
with exponent for the logarithm. The susceptibility has been well
understood previously for dimensions using the lace expansion, but
the lace expansion does not apply when d=4. The proof begins by rewriting the
walk two-point function as the two-point function of a supersymmetric field
theory. The field theory is then analysed via a rigorous renormalisation group
method developed in a companion series of papers. By providing a setting where
the methods of the companion papers are applied together, the proof also serves
as an example of how to assemble the various ingredients of the general
renormalisation group method in a coordinated manner.
... In dimensions d ≥ 4 the limiting curve should be scaling to a Brownian motion; this is related to works of Brydges & Spencer [6] and Hara & Slade [10,11] (see also [3,4,5] for the upper critical dimension d = 4, and the book [15] and references therein). Dimension d = 3 is the most mysterious. ...
We consider self-avoiding walk on finite graphs with large girth. A bound on
the expected length of the self-avoiding walk is given, showing that in many
cases large girth graphs do not exhibit mean-field critical behavior.
... The utility of hierarchical models is that they are a good testing ground for RG techniques. Such methodology has been successful, in a rigorous setting, for instance in the work of Gawȩdzki and Kupiainen on (∇φ) 4 lattice models (the hierarchical model testing was done in [31,32] while the real model was treated in [33,34]) or that of Brydges and Slade [12] on the weakly self-avoiding walk in four dimensions (their approach was tested on a hierarchical model in [9,10]). Another example, in a nonrigorous context, of the success of this methodology is the work of Wilson himself when he developed his RG theory in the first place. ...
We present a quick introduction to quantum field theory and Wilson's theory
of the renormalization group from the point of view of mathematical analysis.
The presentation is geared primarily towards a probability theory, harmonic
analysis and dynamical systems theory audience.
... A different method by Brydges, Yau, Slade and so on uses the idea of decomposition of the covariance of the Gaussian field, which was initiated from [BY90], and was simplified and pedagogically presented in the lecture notes [Bry09], see also [Dim09]. The latter method has achieved several important applications in other problems such as the two-dimensional Coulomb gas model [DH00,Fal12], φ 4 field theories [BDH95,BDH98,BMS03] and self-avoiding walks [BIS09,BS10,BBS12]. ...
In this paper we develop a new renormalization group method, which is based
on conditional expectations and harmonic extensions, to study functional
integrals related with small perturbations of Gaussian fields. In this new
method one integrates Gaussian fields inside domains at all scales conditioning
on the fields outside these domains, and by variation principle solves local
elliptic problems. It does not rely on an a priori decomposition of the
Gaussian covariance. We apply this method to the model of classical dipole gas
on the lattice, and show that the scaling limit of the generating function with
smooth test functions is the generating function of the renormalized Gaussian
free field.
... When d = 4, ν = 1/2 is also anticipated, though this case is more subtle from a rigorous standpoint. Recently, some impressive results have been achieved using a supersymmetric renormalization group approach for continuous-time weakly self-avoiding walk: see [1,3,4] and references therein. ...
We prove two results on the delocalization of the endpoint of a uniform
self-avoiding walk on Z^d for d>1. We show that the probability that a walk of
length n ends at a point x tends to 0 as n tends to infinity, uniformly in x.
Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 +
epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the
probability that self-avoiding walk is a polygon.
... SAWs are in the same universality class as polymers in a good solvent, and thus the model also has an important physical interpretation. Much is known about SAWs, particularly in dimensions d 4 [3,4], and considerable progress has been made in understanding SAWs in two dimensions [5][6][7], but an exact solution remains elusive for any dimension bar the trivial case d = 1. ...
We introduce a self-avoiding walk model for which end-effects are completely
eliminated. We enumerate the number of these walks for various lattices in
dimensions two and three, and use these enumerations to study the properties of
this model. We find that endless self-avoiding walks have the same connective
constant as self-avoiding walks, and the same Flory exponent . However,
there is no power law correction to the exponential number growth for this new
model, i.e. the critical exponent exactly. In addition, we have
convincing numerical evidence to support the hypothesis that the amplitude for
the number growth is a universal quantity, and somewhat weaker evidence which
suggests that the number growth has no analytic corrections to scaling. The
technique by which end-effects are eliminated may be generalised to other
models of polymers such as interacting self-avoiding walks.
... These assertions for the four-dimensional case are believed to be true, but no proof exists. See [7] for a discussion of this case. Bounds established 50 years ago by Hammersley and Welsh [19] have hardly been improved upon. ...
This is a rather personal review of the problem of self-avoiding walks and
polygons. After defining the problem, and outlining what is known rigorously
and what is merely conjectured, I highlight the major outstanding problems. I
then give several applications in which the I have been involved. These include
a study of surface adsorption of polymers, counting possible paths in a
telecommunication network, hitting probabilities of SAWs in a rectangle, and
the modelling of biological experiments on polymers. I hope to show that SAWs
are not only of intrinsic mathematical interest, but also have many interesting
and useful applications.
... In particular, in a generalized "noncommutative" notion of Gaussian processes that are supersymmetric, this correspondence becomes especially striking; see e.g. the review [7]. The last mentioned correspondence is the point of departure for an analysis of the critical behavior of models of self-avoiding walks in dimension four [9]. ...
We present a simple method to decompose the Green forms corresponding to a
large class of interesting symmetric Dirichlet forms into integrals over
symmetric positive semi-definite and finite range (properly supported) forms
that are smoother than the original Green form. This result gives rise to
multiscale decompositions of the associated Gaussian free fields into sums of
independent smoother Gaussian fields with spatially localized correlations. Our
method makes use of the finite propagation speed of the wave equation and
Chebyshev polynomials. It improves several existing results and also gives
simpler proofs.
... Recently some impressive results have been achieved using a supersymmetric renormalization group approach. These results concern continuous-time weakly selfavoiding walk: see [4,5,2] and references within. ...
We prove that self-avoiding walk on is sub-ballistic in any dimension d ≥ 2. That is, writing for the Euclidean norm of , and for the uniform measure on self-avoiding walks for which γ
0 = 0, we show that, for each v > 0, there exists such that, for each .
... These series converge for z < µ(g) −1 and z < µ −1 respectively. Presumably they converge also for z = µ(g) −1 and z = µ −1 but this is a delicate question that is unproven except in high dimensions (in fact, the decay of the two-point function with z = µ −1 is known in some cases [4,8,9]). The following proposition shows that the strong interaction limit of ...
The strong interaction limit of the discrete-time weakly self-avoiding walk
(or Domb--Joyce model) is trivially seen to be the usual strictly self-avoiding
walk. For the continuous-time weakly self-avoiding walk, the situation is more
delicate, and is clarified in this paper. The strong interaction limit in the
continuous-time setting depends on how the fugacity is scaled, and in one
extreme leads to the strictly self-avoiding walk, in another to simple random
walk. These two extremes are interpolated by a new model of a self-repelling
walk that we call the "quick step" model. We study the limit both for walks
taking a fixed number of steps, and for the two-point function.
... However, their properties remain poorly understood in low dimension, despite the existence of remarkable conjectures. See [23] for dimension 5 and above, and [6] for recent progresses in 4 dimensions. Figure 1. ...
We define a new family of self-avoiding walks (SAW) on the square lattice,
called weakly directed walks. These walks have a simple characterization in
terms of the irreducible bridges that compose them. We determine their
generating function. This series has a complex singularity structure and in
particular, is not D-finite. The growth constant is approximately 2.54 and is
thus larger than that of all natural families of SAW enumerated so far (but
smaller than that of general SAW, which is about 2.64). We also prove that the
end-to-end distance of weakly directed walks grows linearly. Finally, we study
a diagonal variant of this model.
This book provides a broad, interdisciplinary overview of non-Archimedean analysis and its applications. Featuring new techniques developed by leading experts in the field, it highlights the relevance and depth of this important area of mathematics, in particular its expanding reach into the physical, biological, social, and computational sciences as well as engineering and technology.
In the last forty years the connections between non-Archimedean mathematics and disciplines such as physics, biology, economics and engineering, have received considerable attention. Ultrametric spaces appear naturally in models where hierarchy plays a central role – a phenomenon known as ultrametricity. In the 80s, the idea of using ultrametric spaces to describe the states of complex systems, with a natural hierarchical structure, emerged in the works of Fraunfelder, Parisi, Stein and others. A central paradigm in the physics of certain complex systems – for instance, proteins – asserts that the dynamics of such a system can be modeled as a random walk on the energy landscape of the system. To construct mathematical models, the energy landscape is approximated by an ultrametric space (a finite rooted tree), and then the dynamics of the system is modeled as a random walk on the leaves of a finite tree. In the same decade, Volovich proposed using ultrametric spaces in physical models dealing with very short distances. This conjecture has led to a large body of research in quantum field theory and string theory. In economics, the non-Archimedean utility theory uses probability measures with values in ordered non-Archimedean fields. Ultrametric spaces are also vital in classification and clustering techniques. Currently, researchers are actively investigating the following areas: p-adic dynamical systems, p-adic techniques in cryptography, p-adic reaction-diffusion equations and biological models, p-adic models in geophysics, stochastic processes in ultrametric spaces, applications of ultrametric spaces in data processing, and more.
This contributed volume gathers the latest theoretical developments as well as state-of-the art applications of non-Archimedean analysis. It covers non-Archimedean and non-commutative geometry, renormalization, p-adic quantum field theory and p-adic quantum mechanics, as well as p-adic string theory and p-adic dynamics. Further topics include ultrametric bioinformation, cryptography and bioinformatics in p-adic settings, non-Archimedean spacetime, gravity and cosmology, p-adic methods in spin glasses, and non-Archimedean analysis of mental spaces. By doing so, it highlights new avenues of research in the mathematical sciences, biosciences and computational sciences.
We present a quick nontechnical introduction to quantum field theory and Wilson’s theory of the renormalization group from the point of view of mathematical analysis. The presentation is geared primarily towards a probability theory, harmonic analysis and dynamical systems theory audience. We also emphasize the use of p-adics in order to set up hierarchical versions of the renormalization group. The latter provide an ideal stepping stone towards the more involved Euclidean space setting.
Hermann Weyl is quoted with saying in Philosophie der Mathematik und Naturwissenschaft 1927, p. 36: As a matter of fact, it is by no means impossible to build up a consistent “non-Archimedean” theory of magnitudes in which the axiom of Eudoxus (usually named after Archimedes) does not hold. Indeed the Axiom of Eudoxus/Archimedes is the main difference between the real and p-adic/ultrametric space; however the axiom is more of a physical one which concerns the process of measurement: exchanging the real numbers field with the p-adic number field is tantamount to exchanging axiomatics in quantum physics.
This paper is the third in a series devoted to the development of a rigorous
renormalisation group method for lattice field theories involving boson fields,
fermion fields, or both. In this paper, we motivate and present a general
approach towards second-order perturbative renormalisation, and apply it to a
specific supersymmetric field theory which represents the continuous-time
weakly self-avoiding walk on . Our focus in this paper is on the
critical dimension d=4. The results include the derivation of the
perturbative flow of the coupling constants, with accompanying estimates on the
coefficients in the flow. These are essential results for subsequent
application to the 4-dimensional weakly self-avoiding walk, and also to the
4-dimensional n-component spin model, including a proof of
existence of logarithmic corrections to their critical scaling.
Simple random walk is well understood. However, if we condition a random walk not to intersect itself, so that it is a self-avoiding walk, then it is much more di-- cult to analyse and many of the important mathematical problems remain unsolved. This paper provides an overview of some of what is known about the self-avoiding walk, includ- ing some old and some more recent results, using methods that touch on combinatorics, probability, and statistical mechanics.
In J. Stat. Phys. 115, 415-449 (2004) Brydges, Guadagni and Mitter proved the
existence of multiscale expansions of a class of lattice Green's functions as
sums of positive definite finite range functions (called fluctuation
covariances). The lattice Green's functions in the class considered are
integral kernels of inverses of second order positive self adjoint operators
with constant coefficients and fractional powers thereof. The fluctuation
coefficients satisfy uniform bounds and the sequence converges in appropriate
norms to a smooth, positive definite, finite range continuum function. In this
note we prove that the convergence is actually exponentially fast.
We consider random self-avoiding walks between two points on the boundary of
a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma
is proportional to mu^{-length(gamma)}). We show that the random trajectory
becomes space-filling in the scaling limit when the parameter mu is
supercritical.
The modern formulation of the renormalization group is explained for both critical phenomena in classical statistical mechanics and quantum field theory. The expansion in ϵ=4-d is explained [d is the dimension of space (statistical mechanics) or space time (quantum field theory)). The emphasis is on principles, not particular applications. Sections 1- 8 provide a self-contained introduction at a fairly elementary level to the statistical mechanical theory. No background is required except for some prior experience with diagrams. In particular, a diagrammatic approximation to an exact renormalization group equation is presented in sections 4 and 5; sections 6---8 include the approximate renormalization group recursion formula and the Feynman graph method for calculating exponents. Sections 10- 13 go deeper into renormalization group theory (section 9 presents a calculation of anomalous dimensions). The equivalence of quantum field theory and classical statistical mechanics near the critical point is established in section 10 ; sections 11- 13 concern problems common to both subjects. Specific field theoretic references assume some background in quantum field theory . An exact renormalization group equation is presented in section 11; sections 12 and 13 concern fundamental topological questions.
This is the second of two papers on the end-to-end distance of a weakly self-repelling walk on a four dimensional hierarchical lattice. It completes the proof that the expected value grows as a constant times
Ö{T}log\frac18T (1+O(\fracloglogTlogT) ){{\sqrt{{T}}\log^{{\frac{{1}}{{8}}}}T {{\left({{1+O{{\left({{\frac{{\log\log T}}{{\log T}}}}\right)}} }}\right)}}}}, which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice 4. Apart from completing the program in the first paper, the main result is that the Green's function is almost equal to the Green's function for the Markov process with no self-repulsion, but at a different value of the killing rate which can be accurately calculated when the interaction is small. Furthermore, the Green's function is analytic in in a sector in the complex plane with opening angle greater than .
The self-repelling Brownian polymer model (SRBP) initiated by Durrett and Rogers in [Durrett-Rogers (1992)] is the continuous space-time counterpart of the myopic (or 'true') self-avoiding walk model (MSAW) introduced in the physics literature by Amit, Parisi and Peliti in [Amit-Parisi-Peliti (1983)]. In both cases, a random motion in space is pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of SRBP in the non-recurrent dimensions. First, extending 1-dimensional results from [Tarres-Toth-Valko (2009)], we identify a natural stationary (in time) and ergodic distribution of the environment (essentially, smeared-out occupation time measure of the process), as seen from the moving particle. As main result we prove that in three and more dimensions, in this stationary (and ergodic) regime, the displacement of the moving particle scales diffusively and its finite dimensional distributions converge to those of a Wiener process. This result settles part of the conjectures (based on non-rigorous renormalization group arguments) in [Amit-Parisi-Peliti (1983)]. The main tool is the non-reversible version of the Kipnis--Varadhan-type CLT for additive functionals of ergodic Markov processes and the graded sector condition of [Sethuraman-Varadhan-Yau (2000)]. Comment: 24 pages
We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms. Comment: 28 pages. Published at http://www.i-journals.org/ps/viewarticle.php?id=152 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org/)
We define a Levy process on a d-dimensional hierarchical lattice. By construction the Green's function for this process decays as . For , we prove that the introduction of a sufficiently weak self-avoidance interaction does not change this decay provided the mass "killing" rate is chosen in a special way, so that the process is critical.
We consider a weakly self-avoiding random walk on a hierarchical lattice ind=4 dimensions. We show that for choices of the killing ratea less than the critical valuea
cthe dominant walks fill space, which corresponds to a spontaneously broken supersymmetry phase. We identify the asymptotic density to which walks fill space, ρ(a), to be a supersymmetric order parameter for this transition. We prove that ρ(a)∼(a
c−a) (−log(a
c−a))1/2 asa↗a
c, which is mean-field behavior with logarithmic corrections, as expected for a system in its upper critical dimension.
We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is the connective constant (i.e. γ=1), and that the mean square displacement is asymptotically linear in the number of steps (i.e.v=1/2). A bound is obtained forcn(x), the number ofn-step self-avoiding walks ending atx. The correlation length is shown to diverge asymptotically like (μ−−Z)1/2. The critical two-point function is shown to decay at least as fast as ⋎x⋎−2, and its Fourier transform is shown to be asymptotic to a multiple ofk−2 ask→0 (i.e. η=0). We also prove that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. The infinite self-avoiding walk is constructed. In this paper we prove these results assuming convergence of the lace expansion. The convergence of the lace expansion is proved in a companion paper.
We develop a technique for reducing the problem of the ultraviolet divergences and their removal to a free field problem. This work is an example of a problem to which a rather general method can be applied. It can be thought as an attempt towards a rigorous version (in 2 or 3 space-time dimensions) of the analysis of the structure of the functional integrals developed in [9], the underlying mechanism being essentially the same as in [11,3].
We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential. The existence and analyticity properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion. In the special case of Monomer-Dimer systems it is established that no phase transition is possible; moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Padé approximants. Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems.
In [BEI92] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green's function for the process equals . If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) βc
, the Green's function behaves like the free one. Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times , which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice ℤ4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Green's function, and requires detailed properties of the Green's function throughout a sector of the complex β plane. These estimates are derived in a companion paper [BI02].
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on Z d , having long finite-range connections, above their upper critical dimensions d = 4 (self-avoiding walk), d = 6 (percolation) and d = 8 (trees and animals). The two-point functions for these models are respectively the generating function for selfavoiding walks from the origin to x 2 Z d , the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of jxj 2Gammad as x ! 1. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoi...
We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on . The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to , the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to as , for for self-avoiding walk, for for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349--408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if ) condition under which the two-point function of a random walk on is asymptotic to as .
Imagine that you are standing at an intersection in the centre of a large city whose streets are laid out in a square grid. You choose a street at random and begin walking away from your starting point, and at each intersection you reach you choose to continue straight ahead or to turn left or right.
Let Δ be the finite difference Laplacian associated to the lattice Z
d. For dimension d≥3, a≥0, and L a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G
a≔(a−Δ)−1 can be decomposed as an infinite sum of positive semi-definite functions V
n
of finite range, V
n
(x−y)=0 for |x−y|≥O(L)n. Equivalently, the Gaussian process on the lattice with covariance G
a admits a decomposition into independent Gaussian processes with finite range covariances. For a=0, V
n
has a limiting scaling form as n→∞. As a corollary, such decompositions also exist for fractional powers (−Δ)−α/2, 0<α≤2. The results of this paper give an alternative to the block spin renormalization group on the lattice.
The consequences of the large magnitude of the bare coupling constant in
Wilson's theory of critical phenomena are examined for renormalized
field theory in 4-ɛ dimensions. The scaling behavior of the
correlation functions, the relations among critical exponents, and the
existence of a scaling equation of state, regular in the temperature
around Tc, are then obtained in this framework. Some
corrections to the scaling laws are also discussed and are shown to be
dependent on another exponent.
The Euclidean (φ4)3,ε model in R
3
corresponds to a perturbation by a φ4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0≤ε≤1. For ε=1 one recovers the covariance of a massless scalar field in R
3
. For ε=0, φ4 is a marginal interaction. For 0≤ε<1 the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for ε>0, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point.
A rigorous, non-perturbative analysis of the block spin (BS) renormalization group is performed for a large class of massless lattice models such as the dipole gas and (∇φ)4. Effective Hamiltonians are shown to be driven to the line of Gaussian fixed points by iteration of the BS-transformation. Analyticity of the presure and the dielectric constant in the (weak) coupling is proven. In particular, the perturbation expansion in activity for the dipole gas is convergent.
We investigate weak perturbations of the continuum massless Gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity.
These lectures are directed at analysts who are interested in learning some of the standard tools of theoretical physics, including functional integrals, the Feynman expan- sion, supersymmetry and the Renormalization Group. The lectures are centered on the problem of determining the asymptotics of the end-to-end distance of a self-avoiding walk on a D-dimensional simple cubic lattice as the number of steps grows. When D = 4, the end-to-end distance has been conjectured to grow as Const. n1/2 log1/8 n, where n is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the D = 4 conjecture in the simplified setting known as the "Hierarchical Lattice".
This paper has three parts. The first part is a simplified presentation of the basic ideas of the renormalization group and the ε expansion applied to critical pheno- mena, following roughly a summary exposition given in 19721. The second part is an account of the history (as I remember it) of work leading up to the papers in I971-1972 on the renormalization group. Finally, some of the developments since 197 1 will be summarized, and an assessment for the future given. II. Many Length Scales and the Renormalization Group There are a number of problems in science which have, as a common charac- teristic, that complex microscopic behavior underlies macroscopic effects. In simple cases the microscopic fluctuations average out when larger scales are considered, and the averaged quantities satisfy classical continuum equ- ations. Hydrodynamics is a standard example of this where atomic fluctuations average out and the classical hydrodynamic equations emerge. Unfortunately, there is a much more difficult class of problems where fluctuations persist out to macroscopic wavelengths, and fluctuations on all intermediate length scales are important too. In this last category are the problems of fully developed turbulent fluid flow, critical phenomena, and elementary particle physics. The problem of magnetic impurities in non-magnetic metals (the Kondo problem) turns out also to be in this category. In fully developed turbulence in the atmosphere, global air circulation becomes unstable, leading to eddies on a scale of thousands of miles. These eddies break down into smaller eddies, which in turn break down, until chaotic motions on all length scales down to millimeters have been excited. On the scale of millimeters, viscosity damps the turbulent fluctuations and no smaller scales are important until atomic scales are reached.2 In quantum field theory, "elementary "particles like electrons, photons, protons and neutrons turn out to have composite internal structure on all size scales down to 0. At least this is the prediction ofquantum field theory. It is hard to make observations of this small distance structure directly; instead the particle scattering cross sections that experimentalists measure must be interpreted
This paper is a continuation of the companion paper [14], in which it
was proved that the standard model of self-avoiding walk in five or more
dimensions has the same critical behaviour as the simple random walk,
assuming convergence of the lace expansion. In this paper we prove the
convergence of the lace expansion, an upper and lower infrared bound,
and a number of other estimates that were used in the companion paper.
The proof requires a good upper bound on the critical point (or
equivalently a lower bound on the connective constant). In an appendix,
new upper bounds on the critical point in dimensions higher than two are
obtained, using elementary methods which are independent of the lace
expansion. The proof of convergence of the lace expansion is computer
assisted. Numerical aspects of the proof, including methods for the
numerical evaluation of simple random walk quantities such as the
two-point function (or lattice Green function), are treated in an
appendix.
We develop a renormalization group method for analyzing the generating functional for charge correlations of a dilute classical dipole gas. It is based on and extends the renormalization group analysis introduced by Brydges and Yau for the dipole gas partition function. Our method leads to systematic formulas for the large-distance behavior of correlation functions of all orders. We prove that in any dimensiond2, at any value>0 of the inverse temperature, and at sufficiently small activityz, the correlation functions exhibit at large distances the same behavior as for a vacuum (z=0), but with a new dielectric constant 1+ over which we have good control. The results proved here extend existing results on the two-point correlations to all higher correlations, and constitute a general confirmation of the fact that dipoles do not screen.
Correlation functions of the Edwards model of polymers at weak coupling are defined and studied at the critical point, in dimension four, by a rigorous renormalization group method which validates, at any order, perturbative renormalization group results on their behaviour at large distances. Remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics.
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the scaling limit of the uniform measures on these objects. The conjectures are based on recent results on the stochastic Loewner evolution and non-disconnecting Brownian motions. New heuristic derivations are given for the critical exponents for SAWs and SAPs.
The Euclidean (\phi^{4})_{3,\epsilon model in corresponds to a perturbation by a interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter in the range . For one recovers the covariance of a massless scalar field in . For is a marginal interaction. For the covariance continues to be Osterwalder-Schrader and pointwise positive. After introducing cutoffs we prove that for , sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. These iterations converge to the fixed point on its stable (critical) manifold which is constructed. Comment: 49 pages, plain tex, macros included
We consider an Euclidean supersymmetric field theory in ℤ3 given by a supersymmetric Φ4 perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ3. The Green’s function depends on the Lévy-Khintchine parameter
with 0<α<2. For
the Φ4 interaction is marginal. We prove for
sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ3 is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ3.
Renormalization group analysis of hierarchical weakly selfavoiding walk in four dimensions
- T Hara
- M Ohno
T. Hara and M. Ohno. Renormalization group analysis of hierarchical weakly selfavoiding walk in four dimensions. In preparation.