Self-attracting self-avoiding walk

Abstract

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).

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].

1.1 Setup

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

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

(A.1)

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

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

(A.2)

1.2 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

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

(A.3)

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 $x\in \mathbb {Z}^{d}$,

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

(A.4)

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

(A.5)

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 $x\in \mathbb {Z}^{d}$,

• (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

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

(A.7)

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

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

(A.8)

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

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

(A.9)

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

1.3 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 $L\ge L_{0}$,

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

(A.10)

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 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.

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

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

(A.11)

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

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

(A.12)

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

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

(A.13)

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

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

(A.14)

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

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

(A.15)

$$\begin{aligned} \mu _{z}&= 1-\lambda _{z}\left( 1-z-z\sum _{x}{\tilde{\Pi }}_{z}(x)\right) . \end{aligned}$$

(A.16)

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 $z\uparrow z_{c}$. $\square$

1.4 Other convolution equations

Consider the equation

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

(A.17)

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

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

(A.18)

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 $\Pi$.