We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\mathbb{Z}^4$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order $p$ (for any $p>0$) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction.
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 lim…
A self-avoiding walk with small attractive interactions is described here. The existence of the connective constant is established, and the diffusive behavior is proved using the method of the lace expansion.
We show that 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. We test these predictions by Monte Carlo simulations and find excellent agreement, thus providing evidence that the SAW is conformally invariant in three dimensions.
Let $μ$ be the self-avoiding walk connective constant on $\ZZ^d$. We show that the asymptotic expansion for $β_c=1/μ$ in powers of $1/(2d)$ satisfies Borel type bounds. This supports the conjecture that the expansion is Borel summable.
We prove that the scaling limit of the weakly self-avoiding walk on a $d$-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is $o(V^{1/2})$ where $V$ is the volume (number of points) of the torus and if $d>4$. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on $\mathbb{Z}^d$. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the toru…
The self-avoiding walk, and lattice spin systems such as the $φ^4$ model, are models of interest both in mathematics and in physics. Many of their important mathematical problems remain unsolved, particularly those involving critical exponents. We survey some of these problems, and report on recent advances in their mathematical understanding via a rigorous nonperturbative renormalization group method.
Following similar analysis to that in Lacoin (PTRF 159, 777-808, 2014), we can show that the quenched critical point for self-avoiding walk on random conductors on the d-dimensional integer lattice is almost surely a constant, which does not depend on the location of the reference point. We provide its upper and lower bounds that are valid for all dimensions.
We consider a long-range version of self-avoiding walk in dimension $d > 2(α\wedge 2)$, where $d$ denotes dimension and $α$ the power-law decay exponent of the coupling function. Under appropriate scaling we prove convergence to Brownian motion for $α\ge 2$, and to $α$-stable Lévy motion for $α< 2$. This complements results by Slade (1988), who proves convergence to Brownian motion for nearest-neighbor self-avoiding walk in high dimension.
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.
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 integra…
We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-α}$ with $α>0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{\mathrm{c}}\equiv2(α\wedge2)$, we prove large-$t$ asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length $t$ or the average spatial size of an oriente…
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.
There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on…
The main result of this paper is a general central limit theorem for distributions defined by certain renewal type equations. We apply this to weakly self-avoiding random walks. We give good error estimates and Gaussian tail estimates which have not been obtained by other methods. We use the lace expansion and at the same time develop a new perspective on this method: We work with a fixed point argument directly in the x-space without using Laplace or Fourier transformation.
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)
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.
We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $β\rightarrow β_c-$ of the probability measure on all finite length walks $ω$ with the probability of $ω$ proportional to $β_c^{|ω|}$ where $|ω|$ is the number of steps in $ω$. The self-avoiding walk in a strip $\{z : 0<\Im(z)<y\}$ is defined by considering all self-avoiding walks $ω$ in the strip which start at the origin and end somewhere on the top boundary with probability proportional to $β_c^{|ω|}$ We pro…
We prove that for the $d$-regular tessellations of the hyperbolic plane by $k$-gons, there are exponentially more self-avoiding walks of length $n$ than there are self-avoiding polygons of length $n$. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed $k$, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion $d-1-O(1/d)$ as $d\to \infty$; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all …
We consider the two-dimensional self-avoiding walk (SAW) in a simply connected domain that contains the origin. The SAW starts at the origin and ends somewhere on the boundary. The distribution of the endpoint along the boundary is expected to differ from the SLE partition function prediction for this distribution because of lattice effects that persist in the scaling limit. We give a precise conjecture for how to compute this lattice effect correction and support our conjecture with simulations. We also give a precise conjecture for the lattice corrections that persist in the scaling limit of…
We give an overview of results on critical phenomena in 4 dimensions, obtained recently using a rigorous renormalisation group method. In particular, for the $n$-component $|φ|^4$ spin model in dimension 4, with small coupling constant, we prove that the susceptibility diverges with a logarithmic correction to the mean-field behaviour with exponent $(n+2)/(n+8)$. This result extends rigorously to $n=0$, interpreted as a supersymmetric version of the model that represents exactly the continuous-time weakly self-avoiding walk. We also analyse the critical two-point function of the weakly self-av…
We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length $n$ is comparable to the $n$th power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models…
We prove that self-avoiding walk on Z^d is sub-ballistic in any dimension d at least two. That is, writing ||u|| for the Euclidean norm of u \in Z^d, and SAW_n for the uniform measure on self-avoiding walks gamma:{0,...,n} \to Z^d for which gamma_0 = 0, we show that, for each v > 0, there exists c > 0 such that, for each positive integer n, SAW_n (max {|| gamma_k || : k \in {0,...,n}} > v n) < e^{- c n}.
Motivated by recent claims of a proof that the length scale exponent for the end-to-end distance scaling of self-avoiding walks is precisely $7/12=0.5833...$, we present results of large-scale simulations of self-avoiding walks and self-avoiding trails with repulsive contact interactions on the hypercubic lattice. We find no evidence to support this claim; our estimate $ν=0.5874(2)$ is in accord with the best previous results from simulations.
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|x|^{-(d-2)}\exp[-c|x|/ξ]$, where the correlation length $ξ$ has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $d>4$ has a "plateau." We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on…
We consider a self-avoiding walk on the dual $\mathbb{Z}^2$ lattice. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle $θ\in[\fracπ{3},\frac{2π}{3}]$. For $θ=\fracπ{3}$, this can be mapped to the self-avoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to $\sqrt{2+\sqrt{2}}$ b…
We study the 4-dimensional $n$-component $|φ|^4$ spin model for all integers $n \ge 1$, and the 4-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case $n=0$ interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order $p$, and prove the existence of a logarithmic correction to mean-field scaling, with power $\frac 12\frac{n+2}{n+8}$, for all $n \ge 0$ and $p>0$. The proof is based on an improvement of a rigorous renormalisation group method developed previously.
Simulations of the self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's Stochastic Loewner Evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance would imply that if we map the walks in the cut-plane into the half plane using the conformal map z -> sqrt(z), then the resulting walks …
We describe some ideas of John Hammersley for proving the existence of critical exponents for two-dimensional self-avoiding walks and provide numerical evidence for their correctness.
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.
We study terminally attached self-avoiding walks and bridges on the simple cubic lattice, both by series analysis and Monte Carlo methods. We provide strong numerical evidence supporting a scaling relation between self-avoiding walks, bridges, and terminally attached self-avoiding walks, and posit that a corresponding amplitude ratio is a universal quantity.
We study the variable-length ensemble of self-avoiding walks on the complete graph. We obtain the leading order asymptotics of the mean and variance of the walk length, as the number of vertices goes to infinity. Central limit theorems for the walk length are also established, in various regimes of fugacity. Particular attention is given to sequences of fugacities that converge to the critical point, and the effect of the rate of convergence of these fugacity sequences on the limiting walk length is studied in detail. Physically, this corresponds to studying the asymptotic walk length on a gen…
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 $\sqrt{2+\sqrt{2}}$ is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour …
We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent $γ$ for self-avoiding walks to unprecedented accuracy. Our final estimate for $γ$ is $1.15695300(95)$.
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 wh…
The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with $κ=8/3$ leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents, but probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE$_{8/3}$.
A prototypical problem on which techniques for exact enumeration are tested and compared is the enumeration of self-avoiding walks. Here, we show an advance in the methodology of enumeration, making the process thousands or millions of times faster. This allowed us to enumerate self-avoiding walks on the simple cubic lattice up to a length of 36 steps.
We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use seri…
The connective constant mu of a graph is the exponential growth rate of the number of n-step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB, exist …
We calculate the connective constant for self-avoiding walks on the simple cubic lattice to unprecedented accuracy, using a novel application of the pivot algorithm. We estimate that μ= 4.684 039 931(27). Our method also provides accurate estimates of the number of self-avoiding walks, even for walks with millions of steps.
We consider a self-avoiding walk model (SAW) on the faces of the square lattice $\mathbb{Z}^2$. This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles $θ\in[\fracπ{3},\frac{2π}{3}]$ and satisfy the Yang-Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles. By means of the Yang-Baxter tr…
We define a new ensemble for self-avoiding walks in the upper half-plane, the fixed irredicible bridge ensemble, by considering self-avoiding walks in the upper half-plane up to their $n$-th bridge height, $Y_n$, and scaling the walk by $1/Y_n$ to obtain a curve in the unit strip, and then taking $n\to\infty$. We then conjecture a relationship between this ensemble to $\SLE$ in the unit strip from $0$ to a fixed point along the upper boundary of the strip, integrated over the conjectured exit density of self-avoiding walk spanning a strip in the scaling limit. We conjecture that there exists a…
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…
This article employs Schramm-Loewner Evolution to obtain intersection exponents for several chordal $SLE_{8/3}$ curves in a wedge. As $SLE_{8/3}$ is believed to describe the continuum limit of self-avoiding walks, these exponents correspond to those obtained by Cardy, Duplantier and Saleur for self-avoiding walks in an arbitrary wedge-shaped geometry using conformal invariance based arguments. Our approach builds on work by Werner, where the restriction property for $SLE(κ,ρ)$ processes and an absolute continuity relation allow the calculation of such exponents in the half-plane. Furthermore, …
The number of free sites next to the end of a self-avoiding walk is known as the atmosphere. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify our conjectures. Of particular interest are walks that have zero atmosphere, which are known as trapped. We demonstrate that these walks scale in the same way as the full set of self-avoiding walks, barring an overall constan…
We recently published [J. Phys A: Math. Theor. {\bf 45} 115202 (2012)] a new and more efficient implementation of a transfer-matrix algorithm for exact enumerations of self-avoiding polygons. Here we extend this work to the enumeration of self-avoiding walks on the square lattice. A detailed comparison with our previous best algorithm shows very significant improvement in the running time of the new algorithm. The new algorithm is used to extend the enumeration of self-avoiding walks to length 79 from the previous record of 71 and for metric properties, such as the average end-to-end distance,…
We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we use to obtain large samples of walks on the cubic lattice of up to $33 \times 10^6$ steps. Consequently the critical exponent $ν$ for three-dimensional self-avoiding walks is determined to great accuracy; the final estimate is $ν=0.587597(7)$. The method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum.
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 $ν$. However, there is no power law correction to the exponential number growth for this new model, i.e. the critical exponent $γ= 1$ exactly. In addition, we have convincing numerical evidence to support the hypothesis that the …
A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a $10 \times 1$ rectangle, and to evaluate the ratio of probabilities of a Brownian path hitting the short ends of the rectangle before hitting one of the long sides. For Brownian motion this probability can be calculated exactly \cite{BLWW04}. Here we consider instead the more difficult problem of a self-avoiding walk in the scaling limit, and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same ratio of probabilities.…
We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that the appropriate limiting free energy exists where there is an applied force and a surface potential term, and prove that this free energy is convex in appropriate variables. We then derive an expression for the limiting free energy in terms of the free energy without a force and the free energy with no surface interaction. Finally we show that there is a phase…
We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to $x\in{\mathbb{Z}}^d$, 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 $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$, for $d\geq 5$ for self-av…
Self-avoiding walks are a simple and well-known model of long, flexible polymers in a good solvent. Polymers being pulled away from a surface by an external agent can be modelled with self-avoiding walks in a half-space, with a Boltzmann weight $y = e^f$ associated with the pulling force. This model is known to have a critical point at a certain value $y_c$ of this Boltzmann weight, which is the location of a transition between the so-called free and ballistic phases. The value $y_c=1$ has been conjectured by several authors using numerical estimates. We provide a relatively simple proof of th…
The pivot algorithm for self-avoiding walks has been implemented in a manner which is dramatically faster than previous implementations, enabling extremely long walks to be efficiently simulated. We explicitly describe the data structures and algorithms used, and provide a heuristic argument that the mean time per attempted pivot for $N$-step self-avoiding walks is $O(1)$ for the square and simple cubic lattices. Numerical experiments conducted for self-avoiding walks with up to 268 million steps are consistent with $o(\log N)$ behavior for the square lattice and $O(\log N)$ behavior for the s…
Let $X$ be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on $X$ is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges are self-avoiding walks such that heights of interior vertices are bounded by the heights of the start- and end-vertex. The number of self-avoiding walks and the number of bridges of length $n$ starting at a vertex $o$ of $X$ grow exponentially in $n$ and the bases of these growth rates are called connective constant and bridge constant, respectively. We sho…
The Fisher transformation acts on cubic graphs by replacing each vertex by a triangle. We explore the action of the Fisher transformation on the set of self-avoiding walks of a cubic graph. Iteration of the transformation yields a sequence of graphs with common critical exponents, and with connective constants converging geometrically to the golden mean. We consider the application of the Fisher transformation to one of the two classes of vertices of a bipartite cubic graph. The connective constant of the ensuing graph may be expressed in terms of that of the initial graph. When applied to the…
In a recent paper by Beaton et al, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$. Their proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov, and confirmed a conjecture of Batchelor and Yung. We consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the lattice rotated by $π/2$. For this model there also exists a conjecture for the critical surface fugacity, made in 1998 by Ba…
