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).

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Notes

  1. 1.

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

References

  1. 1.

    Ueltschi, D.: A self-avoiding walk with attractive interactions. Probab. Theory Relat. Fields 124(2), 189–203 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    den Hollander, F.: Random Polymers. Lecture Notes in Mathematics, vol. 1974. Springer-Verlag, Berlin (2009)

    Google Scholar 

  3. 3.

    Bauerschmidt, R., Duminil-Copin, H., Goodman, J., Slade, G.: Lectures on self-avoiding walks, In: Probability and statistical physics in two and more dimensions. Clay Mathematics Proceedings, vol. 15, pp. 395–467. American Mathematical Society, Providence, RI (2012)

  4. 4.

    Slade, G.: The Lace Expansion and Its Applications. Lecture Notes in Mathematics, vol. 1879. Springer-Verlag, Berlin (2006)

  5. 5.

    Steele, J.M.: Probability theory and combinatorial optimization. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1997)

  6. 6.

    Madras, N., Slade, G.: The self-avoiding walk. Modern Birkhäuser Classics, Birkhäuser/Springer, New York (2013). Reprint of the 1993 original

    Google Scholar 

  7. 7.

    Hammersley, J.M., Welsh, D.J.: Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. 13(1), 108–110 (1962)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Hara, T., van der Hofstad, R., Slade, G.: Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31(1), 349–408 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Hammond, A.: An upper bound on the number of self-avoiding polygons via joining. Ann. Probab. 46(1), 175–206 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    van der Hofstad, R., Slade, G.: A generalised inductive approach to the lace expansion. Probab. Theory Relat. Fields 122(3), 389–430 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Hara, T., Slade, G.: On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys. 59(5–6), 1469–1510 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Sakai, A.: Lace expansion for the Ising model. Commun. Math. Phys. 272(2), 283–344 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Helmuth, T.: Loop-weighted walk. Ann. Inst. Henri Poincaré D 3(1), 55–119 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    van der Hofstad, R., Holmes, M.: An expansion for self-interacting random walks. Braz. J. Probab. Stat. 26(1), 1–55 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bauerschmidt, R., Slade, G., Wallace, B.C.: Four-dimensional weakly self-avoiding walk with contact self-attraction. J. Stat. Phys. 167(2), 317–350 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Pétrélis, N., Torri, N.: Collapse transition of the interacting prudent walk. Ann. Inst. Henri Poincaré D 5(3), 387–435 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Hardy, G. H., Ramanujan, S.: Asymptotic formulæ for the distribution of integers of various types [Proceedings of the London Mathematical Society (2) 16 (1917) pp. 112–132]. In: Collected papers of Srinivasa Ramanujan, pp. 245–261, AMS Chelsea Publising, Providence, RI (2000)

  18. 18.

    Aizenman, M.: Geometric analysis of \(\varphi ^{4}\) fields and Ising models. I, II. Commun. Math. Phys. 86(1), 1–48 (1982)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Brydges, D., Spencer, T.: Self-avoiding walk in \(5\) or more dimensions. Commun. Math. Phys. 97(1–2), 125–148 (1985)

    MathSciNet  MATH  Google Scholar 

Download references

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 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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Tyler Helmuth.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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].

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)

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 \(x\in \mathbb {Z}^{d}\),

  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

    $$\begin{aligned} G_{z}(x)\le C(t)e^{-c(t)\left| \left| \left| x\right| \right| \right| }. \end{aligned}$$
    (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 \(1+O(L^{\alpha -2})\).

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

  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.

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 \(z\uparrow z_{c}\). \(\square \)

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 \(\Pi \).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hammond, A., Helmuth, T. Self-attracting self-avoiding walk. Probab. Theory Relat. Fields 175, 677–719 (2019). https://doi.org/10.1007/s00440-018-00898-7

Download citation

Keywords

  • 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