Abstract
This article is concerned with selfavoiding walks (SAW) on \(\mathbb {Z}^{d}\) that are subject to a selfattraction. 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 infiniterange step distributions and weak selfattractions (Ueltschi in Probab Theory Relat Fields 124(2):189–203, 2002). This article considers the case of bounded step distributions. For weak selfattractions we show that the connective constant exists, and, in \(d\ge 5\), carry out a lace expansion analysis to prove the meanfield behaviour of the critical twopoint function, hereby addressing a problem posed by den Hollander (Random Polymers, vol. 1974. SpringerVerlag, Berlin, 2009).
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Notes
 1.
Note that our definition of \(c_{n}\) involves D, i.e., we are enumerating weighted selfavoiding walks.
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Acknowledgements
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 DMS1512908. 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 twopoint 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 selfavoiding 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].
Setup
Let \(\mathbb {R}_{\ge 0}\) denote the nonnegative 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
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 nonnegative 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
Hypotheses and Theorem
Hypothesis A.1
Assume that D is a spreadout 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 spreadout parameter \(L^{2}\). The noninteracting twopoint function\(S_{\mu }\) is defined by
An important consequence of the form of D is the following proposition. Let \(a_{d} :=\frac{d\Gamma (d/21)}{2\pi ^{d/2}}\), 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 \(x\in \mathbb {Z}^{d}\),
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 welldefined, not identically zero, and monotone increasing in z. For \(z_{0}\le z<z_{c}\) and for each \(x\in \mathbb {Z}^{d}\),

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

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

(iii)
for \(t>0\) and \(z\in \left[ z_{0},z_{c}t\right) \) there are constants \(c(t),C(t)>0\) such that
$$\begin{aligned} G_{z}(x)\le C(t)e^{c(t)\left \left \left x\right \right \right }. \end{aligned}$$(A.6)
The most substantial hypothesis is the next one.
Hypothesis A.5
Assume
Suppose also that \(z_{0}\le z\le 2\). If \(\beta <\beta _{0}\), there is a constant \(c=c(d)>0\) such that
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 welldefined, and there is an \(A>0\) such that
The implicit constants are uniform in x and L. The values of \(z_{c}\) and A are \(1+O(L^{\alpha 2})\).
Proof
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 \(L\ge L_{0}\),
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

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

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

(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 modelspecific facts, are used.
Note that it suffices to prove that (A.10) holds for \(\alpha < \frac{1}{2}\), as the righthand 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:
and note K is finite by (A.4). Define
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.
Claim
Hypothesis (i) of Lemma A.8 holds.
Proof
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. \(\square \)
Claim
Hypothesis (ii) of Lemma A.8 holds.
Proof
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. \(\square \)
Claim
Hypothesis (iii) of Lemma A.8 holds.
Proof
Fix \(z_{0}<z<z_{c}\) and suppose \(f(z)\le 1\). Then z is at most \(2z_{0}\), and
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
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
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
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
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
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
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
where \(A^{*k}\) is the kfold autoconvolution of A. Iterating this yields (A.1) with
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. Selfattracting selfavoiding walk. Probab. Theory Relat. Fields 175, 677–719 (2019). https://doi.org/10.1007/s00440018008987
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Keywords
 Selfinteracting random walk
 Selfattracting walk
 Selfavoiding walk
 Linear polymers
 Lace expansion
 Critical phenomena
 HammersleyWelsh argument
Mathematics Subject Classification
 Primary 60K35
 Secondary 60D05
 82B27