Self-attracting self-avoiding walk


This article is concerned with self-avoiding walks (SAW) on \(\mathbb {Z}^{d}\) that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions (Ueltschi in Probab Theory Relat Fields 124(2):189–203, 2002). This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in \(d\ge 5\), carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander (Random Polymers, vol. 1974. Springer-Verlag, Berlin, 2009).

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  1. 1.

    Note that our definition of \(c_{n}\) involves D, i.e., we are enumerating weighted self-avoiding walks.


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The authors would like to thank both referees for their critiques and comments, which have lead to a significantly improved article. T.H. would like to thank Gordon Slade and Remco van der Hofstad for encouraging discussions. A.H. is supported by NSF grant DMS-1512908. The majority of this work was carried out while T.H. was supported by an NSERC postdoctoral fellowship at UC Berkeley; additional support was provided by EPSRC Grant EP/P003656/1.

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Appendix A: Gaussian asymptotics

Appendix A: Gaussian asymptotics

This appendix reviews [8, Theorem 1.2], which derives Gaussian asymptotics for critical two-point functions. Our motivation is that the presentation in [8] is, at places, dependent on the particular models being studied. The proofs, however, apply essentially verbatim to other models. Our review axiomatizes sufficient assumptions for models similar to self-avoiding walk. We indicate where these assumptions are used in proofs, but omit the portions of the proofs that purely replicate [8]. We emphasise that the result and techniques are those of [8], and our presentation is primarily for the benefit of the reader who is not familiar with [8].


Let \(\mathbb {R}_{\ge 0}\) denote the non-negative reals. For \(z\in \mathbb {R}_{\ge 0}\), \(G_{z}:\mathbb {Z}^{d}\rightarrow \mathbb {R}_{\ge 0}\), \({\tilde{\Pi }}_{z}:\mathbb {Z}^{d}\rightarrow \mathbb {R}\), and D a probability distribution on \(\mathbb {Z}^{d}\), we consider the convolution equation

$$\begin{aligned} G_{z}(x)= \delta _{o,x} + {\tilde{\Pi }}_{z}(x) + (zD*(\delta + {\tilde{\Pi }}_{z})*G_{z})(x). \end{aligned}$$

We will further assume that \(G_{z}\), \({\tilde{\Pi }}_{z}\), and D are all \(\mathbb {Z}^{d}\)-symmetric, and that \(G_{z}(x)\) is a power series in z with non-negative coefficients. We will see in Section A.4 that the analysis of (A.1) also applies to the convolution equation derived for \(\kappa \)-ASAW in the main body of the text.

The critical point\(z_{c}\) is \(z_{c} = \sup \{z\in \mathbb {R}_{\ge 0} \mid \chi (z)<\infty \}\), where the susceptibility\(\chi (z)\) is defined by

$$\begin{aligned} \chi (z) :=\sum _{x\in \mathbb {Z}^{d}}G_{z}(x). \end{aligned}$$

Hypotheses and Theorem

Hypothesis A.1

Assume that D is a spread-out step distribution as defined in Definition 1.

Let \(X_{n}\) be a discrete time simple random walk with step distribution D. Let \(\sigma ^{2} = \sum _{x\in \mathbb {Z}^{d}}D(x)\Vert x\Vert ^{2}_{2}\). Note that \(\sigma ^{2}\) is comparable to the spread-out parameter \(L^{2}\). The non-interacting two-point function\(S_{\mu }\) is defined by

$$\begin{aligned} S_{\mu }(x) :=\sum _{n=0}^{\infty }\mu ^{n}\mathsf {P}_{0}\left[ X_{n}=x\right] . \end{aligned}$$

An important consequence of the form of D is the following proposition. Let \(a_{d} :=\frac{d\Gamma (d/2-1)}{2\pi ^{d/2}}\), where \(\Gamma \) is Euler’s gamma function.

Proposition A.2

([8, Prop. 1.6]) Suppose \(d>2\) and Hypothesis A.1 holds. For L sufficiently large, \(\alpha >0\), \(\mu \le 1\), and \(x\in \mathbb {Z}^{d}\),

$$\begin{aligned} S_{\mu }(x)&\le \delta _{o,x} + O\left( \frac{1}{L^{2-\alpha }\left| \left| \left| x\right| \right| \right| ^{d-2}}\right) \end{aligned}$$
$$\begin{aligned} S_{1}(x)&= \frac{a_{d}}{\sigma ^{2}}\frac{1}{\left| \left| \left| x\right| \right| \right| ^{d-2}}+ O\left( \frac{1}{\left| \left| \left| x\right| \right| \right| ^{d-\alpha }}\right) . \end{aligned}$$

The implicit constants may depend on \(\alpha \), but not on L.

Note that, for fixed d, the leading coefficient in (A.5) is proportional to \(L^{-2}\). The next two hypotheses deal with the critical point and behaviour of \(G_{z}\) for \(z_{0}\le z<z_{c}\), where \(z_{0}>0\) is a chosen value of the parameter z.

Hypothesis A.3

The critical point \(z_{c}\) satisfies \(z_{0}<z_{c}<\infty \). The susceptibility specified by (A.2) diverges as the critical point is approached from below: \(\lim _{z\uparrow z_{c}}\chi (z)=\infty \).

Hypothesis A.4

\(G_{z}\) is well-defined, not identically zero, and monotone increasing in z. For \(z_{0}\le z<z_{c}\) and for each \(x\in \mathbb {Z}^{d}\),

  1. (i)

    \(G_{z_{0}}(x)\le S_{1}(x)\),

  2. (ii)

    \(G_{z}(x)\) is continuous for \(z\in \left[ z_{0},z_{c}\right) \), and

  3. (iii)

    for \(t>0\) and \(z\in \left[ z_{0},z_{c}-t\right) \) there are constants \(c(t),C(t)>0\) such that

    $$\begin{aligned} G_{z}(x)\le C(t)e^{-c(t)\left| \left| \left| x\right| \right| \right| }. \end{aligned}$$

The most substantial hypothesis is the next one.

Hypothesis A.5


$$\begin{aligned} G_{z}(x)\le \beta \left| \left| \left| x\right| \right| \right| ^{-d+2}, \qquad x\ne o. \end{aligned}$$

Suppose also that \(z_{0}\le z\le 2\). If \(\beta <\beta _{0}\), there is a constant \(c=c(d)>0\) such that

$$\begin{aligned} \left| {\tilde{\Pi }}_{z}(x)\right| \le c\beta \delta _{o,x} + \frac{c\beta ^{2}}{\left| \left| \left| x\right| \right| \right| ^{3(d-2)}}. \end{aligned}$$

Theorem A.6

([8, Theorem 1.2]) Assume D, \(G_{z}\), and \({\tilde{\Pi }}_{z}\) satisfy the hypotheses of Section A.2. Choose \(0<\alpha <2\). Let \(\beta _{0}\) be the constant of Hypothesis A.5.

There is an \(L_{0}(d,\alpha ,\beta _{0})\) such that, for \(L\ge L_{0}\), the function \(G_{z_{c}}:\mathbb {Z}^{d}\rightarrow \mathbb {R}\) is well-defined, and there is an \(A>0\) such that

$$\begin{aligned} G_{z_{c}}(x) \sim \frac{a_{d}A}{\sigma ^{2}\left| \left| \left| x\right| \right| \right| ^{2-d}} \left( 1+O\left( \frac{L^{2}}{\left| \left| \left| x\right| \right| \right| ^{2-\alpha }}\right) \right) . \end{aligned}$$

The implicit constants are uniform in x and L. The values of \(z_{c}\) and A are \(1+O(L^{\alpha -2})\).


The next proposition is the heart of the analysis. In what follows we assume the hypotheses of Theorem A.6; in particular, \(\beta _{0}\) is given.

Proposition A.7

Fix \(\alpha >0\). There is an \(L_{0}=L_{0}(\beta _{0},d,\alpha ,z_{0})\) such that, for \(L\ge L_{0}\),

$$\begin{aligned} G_{z_{c}}(x) \le \frac{\mathrm {const}}{L^{2-\alpha }\left| \left| \left| x\right| \right| \right| ^{d-2}}, \qquad x\ne o, \end{aligned}$$

and \(z_{c}\le 1 + O(L^{-2+\alpha })\).

Lemma A.8

(Lemma 2.1 [8]) Let \(f:[z_{1},z_{c})\rightarrow \mathbb {R}\), and \(a\in (0,1)\). Suppose

  1. (i)

    f is continuous on \([z_{1},z_{c})\),

  2. (ii)

    \(f(z_{1})\le a\), and

  3. (iii)

    for \(z\in [z_{1},z_{c})\) the inequality \(f(z)\le 1\) implies the inequality \(f(z)\le a\).

Then \(f(z)\le a\) for all \(z\in [z_{1},z_{c})\).

Proof of Proposition A.7

The proof is essentially that in [8]. We present the steps in which our hypotheses, as opposed to model-specific facts, are used.

Note that it suffices to prove that (A.10) holds for \(\alpha < \frac{1}{2}\), as the right-hand side is increasing in \(\alpha \). By Hypothesis A.4 and the monotone convergence theorem, it is enough to prove this for all \(z_{0}<z<z_{c}\).

Let K be the optimal constant for the error bound in Proposition A.2:

$$\begin{aligned} K = \sup _{L\ge 1,x\ne o} L^{2-\alpha }\left| \left| \left| x\right| \right| \right| ^{d-2}S_{1}(x), \end{aligned}$$

and note K is finite by (A.4). Define

$$\begin{aligned} g_{x}(z) = (2K)^{-1}L^{2-\alpha }\left| \left| \left| x\right| \right| \right| ^{d-2}G_{z}(x), \end{aligned}$$

and let \(g(z) = \sup _{x\ne o}g_{x}(z)\). To prove (A.10), we will use Lemma A.8 with \(f(z) = \max \{g(z),\frac{z}{2z_{0}}\}\), \(z_{1}=z_{0}\), and \(a\in \left( \frac{1}{2},1\right) \) arbitrary. The claim that \(z_{c}=1+O(L^{-2+\alpha })\) will be established in the course of the argument.


Hypothesis (i) of Lemma A.8 holds.


For \(x\in \mathbb {Z}^{d}\), \(g_{x}(z)\) is continuous on \([z_{0},z_{c})\) by Hypothesis A.4. It suffices to show \(\sup _{x\ne o}g_{x}(z)\) is continuous on \([z_{0},z_{c}-t)\) for arbitrarily small \(t>0\).

Fix \(t>0\), and let \(z\in [z_{0},z_{c}-t)\). By Hypothesis A.4, \(g_{x}(z)\) decays exponentially in \(\Vert x\Vert _{2}\) with decay rate independent of z. Therefore, \(\sum _{x\in \mathbb {Z}^{d}}g_{x}(z)\) converges exponentially fast with rate independent of z. It follows that the supremum of \(g_{x}(z)\) occurs on \(B_{R}(o)\), the ball of radius R about the origin, for some \(R=R(L)>0\). This proves \(\sup _{x\ne o}g_{x}(z)\) is a continuous function of \(z\in [z_{0},z_{c}-t)\) since the supremum of a finite set of continuous functions is continuous. \(\square \)


Hypothesis (ii) of Lemma A.8 holds.


By Hypothesis A.4 and the definition of K, \(g_{x}(z_{0})\le \frac{1}{2}\) for all x. Since \(a>\frac{1}{2}\), this proves the claim. \(\square \)


Hypothesis (iii) of Lemma A.8 holds.


Fix \(z_{0}<z<z_{c}\) and suppose \(f(z)\le 1\). Then z is at most \(2z_{0}\), and

$$\begin{aligned} G_{z}(x)\le 2z_{0}KL^{-2+\alpha }\left| \left| \left| x\right| \right| \right| ^{2-d},\qquad x\ne o. \end{aligned}$$

Let \(\beta = 2z_{0}KL^{-2+\alpha }\). By Hypothesis A.5, when \(L^{-2+\alpha }\) is sufficiently small there is a \(c>0\) such that

$$\begin{aligned} \left| {\tilde{\Pi }}_{z}(x)\right| \le c\beta \delta _{o,x} + c\beta ^{2}\left| \left| \left| x\right| \right| \right| ^{-3(d-2)} \le \frac{c\beta }{\left| \left| \left| x\right| \right| \right| ^{3(d-2)}}. \end{aligned}$$

By Hypothesis A.4, \(G_{z}\) is not identically zero. Thus \(\chi (z)>0\), and the sum of (A.1) over all \(x\in \mathbb {Z}^{d}\) can be rearranged to give

$$\begin{aligned} \chi (z) = \frac{1+\sum _{x}{\tilde{\Pi }}_{z}(x)}{1-z-z\sum _{x}{\tilde{\Pi }}_{z}(x)}>0. \end{aligned}$$

By (A.12), \(\Vert {\tilde{\Pi }}_{z}(x)\Vert _{1}<1\) for L large enough. This implies the numerator, and hence the denominator, of (A.13) is strictly positive. Since \(f(z)\le 1\), this implies that

$$\begin{aligned} z < 1-z\sum _{x\in \mathbb {Z}^{d}}{\tilde{\Pi }}_{z}(x) \le 1 + O(z_{0}L^{-2+\alpha }). \end{aligned}$$

Thus \(\frac{z}{2}\) is bounded above by a for \(a\in \left( \frac{1}{2},1\right) \), provided that L is large enough.

What remains is to prove \(g(z)\le a\) for \(a\in \left( \frac{1}{2},1\right) \) when L is large enough. This exactly follows the presentation in [8, p. 364], and hence we omit it. \(\square \)

By Hypothesis A.4 this proves the desired bounds, as we have proven that \(f(z)\le a\) for \(z_{0}\le z<z_{c}\). The bound on \(z_{c}\) follows from (A.14), which holds as it was derived under the hypothesis that \(f(z)\le 1\). \(\square \)

Proof of Theorem A.6

This follows [8, Theorem 1.2]. The only model specific step in the cited proof is showing that an auxiliary parameter \(\mu _{z}\) increases to \(\mu _{z_{c}}=1\) as \(z\uparrow z_{c}\). We define this parameter below and show that it takes the desired value by Hypothesis A.3.

By (A.12), \({\tilde{\Pi }}_{z}(x)\) has a finite second moment when L is large enough. It therefore makes sense to define

$$\begin{aligned} \lambda _{z}&= \frac{1}{1+z\sigma ^{-2}\sum _{x}\Vert x\Vert ^{2}_{2} {\tilde{\Pi }}_{z}(x)}, \end{aligned}$$
$$\begin{aligned} \mu _{z}&= 1-\lambda _{z}\left( 1-z-z\sum _{x}{\tilde{\Pi }}_{z}(x)\right) . \end{aligned}$$

Equation (A.12) implies \(\lambda _{z}\rightarrow 1\) as \(L\rightarrow \infty \) uniformly in \(z\in \left[ z,z_{c}\right] \). By Equation (A.13) and Hypothesis A.3, as \(z\uparrow z_{c}\), the quantity in brackets in (A.16) tends to zero. Thus, \(\mu _{z_{c}}\uparrow 1\) as \(z\uparrow z_{c}\). \(\square \)

Other convolution equations

Consider the equation

$$\begin{aligned} G_{z} = \delta + z (D*G_{z}) + (\Pi _{z}*G_{z}). \end{aligned}$$

If \(\Pi \) satisfies Hypothesis A.5, it is possible to manipulate (A.17) into the form (A.1). To see this, rewrite (A.17) as

$$\begin{aligned} G&= \delta + \Pi + zD*(\delta + \Pi )*G -\Pi *(\delta + zD*G - G) \\&= \delta + \Pi + zD*(\delta + \Pi )*G + \Pi *\Pi *G, \end{aligned}$$

where, in the second equality, we have used (A.17) to rewrite the term in parentheses, and the subscripts z have been omitted. Rewriting the last factor of G using (A.17) yields

$$\begin{aligned} G = \delta + \Pi + \Pi ^{*2} + zD*(\delta + \Pi + \Pi ^{*2})*G + \Pi ^{*3}*G, \end{aligned}$$

where \(A^{*k}\) is the k-fold autoconvolution of A. Iterating this yields (A.1) with

$$\begin{aligned} {\tilde{\Pi }}_{z} = \sum _{k\ge 1}\Pi ^{*k}, \end{aligned}$$

since \(\lim _{n\rightarrow \infty }\Pi ^{*n}=0\) under the assumption that \(\Pi \) satisfies Hypothesis A.5. Finally, [8, Proposition 1.7] implies that, if \(\Pi _{z}\) satisfies Hypothesis A.5, then \({\tilde{\Pi }}_{z}\) defined by (A.18) satisfies Hypothesis A.5, for possibly different constants. The change in constants depends only on d. See [8, Section 4.1] for a further discussion of this point. Thus to apply Theorem A.6 to the convolution equation (A.17), it suffices to verify the hypotheses of Section A.2 for \(G_{z}\), D, and \(\Pi \).

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Hammond, A., Helmuth, T. Self-attracting self-avoiding walk. Probab. Theory Relat. Fields 175, 677–719 (2019).

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  • Self-interacting random walk
  • Self-attracting walk
  • Self-avoiding walk
  • Linear polymers
  • Lace expansion
  • Critical phenomena
  • Hammersley-Welsh argument

Mathematics Subject Classification

  • Primary 60K35
  • Secondary 60D05
  • 82B27