Connective constant

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title: "Connective constant" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["discrete-geometry"] description: "Number associated with self-avoiding walks" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Connective_constant" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Number associated with self-avoiding walks ::

In mathematics, the connective constant is a numerical quantity associated with self-avoiding walks on a lattice. It is studied in connection with the notion of universality in two-dimensional statistical physics models. |last1=Madras |first1=N. |last2=Slade |first2=G. |year=1996 |title=The Self-Avoiding Walk |publisher=Birkhäuser |isbn=978-0-8176-3891-7 |last1=Duminil-Copin |first1=Hugo |last2=Smirnov |first2=Stanislav |year=2010 |title=The connective constant of the honeycomb lattice equals \sqrt{2 + \sqrt{2}}=2\cos\frac{\pi}{8} |eprint=1007.0575 |class=math-ph

Definition

The connective constant is defined as follows. Let c_n denote the number of n-step self-avoiding walks starting from a fixed origin point in the lattice. Since every n + m step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that c_{n+m} \leq c_n c_m . Then by applying Fekete's lemma to the logarithm of the above relation, the limit \mu = \lim_{n \rightarrow \infty} c_n^{1/n} can be shown to exist. This number \mu is called the connective constant, and clearly depends on the particular lattice chosen for the walk since c_n does. The value of \mu is precisely known only for two lattices, see below. For other lattices, \mu has only been approximated numerically. It is conjectured that c_n \approx \mu^n n^{\gamma-1} as n goes to infinity, where \mu depends on the lattice, but the critical exponent \gamma is universal (it depends on dimension, but not the specific lattice). In 2-dimensions it is conjectured that \gamma = 43/32 |author= B. Nienhuis |year= 1984 |title= Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas |journal= J. Stat. Phys. |volume= 34 |issue= 5–6 |pages= 731–761 |publisher= |url= |doi= 10.1007/BF01009437 |bibcode= 1984JSP....34..731N

Known values

::data[format=table]

Lattice	Connective constant
Hexagonal	2\cos\frac{\pi}{8}=\sqrt{2 + \sqrt{2}}\simeq 1.85
Triangular	4.15079(4)
Square	2.63815853032790(3)
Kagomé	2.56062
Manhattan	1.733535(3)
L-lattice	1.5657(15)
(3.12^2) lattice	1.7110412...
(4.8^2) lattice	1.80883001(6)
::

These values are taken from the 1998 Jensen–Guttmann paper |last1=Jensen |first1=I. |last2=Guttmann |first2=A. J. |year=1998 |title=Self-avoiding walks, neighbor-avoiding walks and trails on semi-regular lattices |journal=Journal of Physics A |volume=31 |issue=40 |pages=8137–45 |url=http://www.ms.unimelb.edu.au/~tonyg/articles/polygons.pdf |doi=10.1088/0305-4470/31/40/008 |bibcode=1998JPhA...31.8137J The connective constant of the (3.12^2) lattice, since each step on the hexagonal lattice corresponds to either two or three steps in it, can be expressed exactly as the largest real root of the polynomial

: x^{12} - 4x^8 - 8x^7 - 4x^6 + 2x^4 + 8x^3 + 12x^2 + 8x + 2

given the exact expression for the hexagonal lattice connective constant. More information about these lattices can be found in the percolation threshold article.

Duminil-Copin–Smirnov proof==

In 2010, Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that \mu=\sqrt{2 + \sqrt{2}} for the hexagonal lattice. This had been conjectured by Nienhuis in 1982 as part of a larger study of O(n) models using renormalization techniques. |last1=Nienhuis |first1=Bernard |year=1982 |title=Exact critical point and critical exponents of O(n) models in two dimensions |journal=Physical Review Letters |volume=49 |issue=15 |pages=1062–1065 |bibcode=1982PhRvL..49.1062N |doi=10.1103/PhysRevLett.49.1062 |last1=Smirnov |first1=Stanislav |year=2010 |chapter=Discrete Complex Analysis and Probability |title=Proceedings of the International Congress of Mathematicians (Hyderabad, India) 2010 |pages=565–621 |arxiv=1009.6077 |bibcode=2010arXiv1009.6077S

: Z(x)=\sum_{\gamma: a\to H}x^{\ell(\gamma)}=\sum_{n=0}^{\infty}c_n x^n

converges for x and diverges for xx_c where the critical parameter is given by x_c=1/ \sqrt{2+\sqrt{2}}. This immediately implies that \mu=\sqrt{2+\sqrt{2}}.

Given a domain \Omega in the hexagonal lattice, a starting mid-edge a, and two parameters x and \sigma, we define the parafermionic observable

F(z)=\sum_{\gamma\subset\Omega:a\to z} e^{-i\sigma W_{\gamma}(a,z)}x^{\ell(\gamma)}.

If x=x_c=1/\sqrt{2 + \sqrt{2}} and \sigma=5/8, then for any vertex v in \Omega, we have

: (p-v)F(p) + (q-v)F(q) + (r-v)F(r)=0,

where p,q,r are the mid-edges emanating from v. This lemma establishes that the parafermionic observable is divergence-free. It has not been shown to be curl-free, but this would solve several open problems (see conjectures). The proof of this lemma is a clever computation that relies heavily on the geometry of the hexagonal lattice.

Next, we focus on a finite trapezoidal domain S_{T,L} with 2L cells forming the left hand side, T cells across, and upper and lower sides at an angle of \pm \pi/3. (Picture needed.) We embed the hexagonal lattice in the complex plane so that the edge lengths are 1 and the mid-edge in the center of the left hand side is positioned at −1/2. Then the vertices in S_{T,L} are given by

: V(S_{T,L})={ z\in V(\mathbb{H}) : 0 \leq Re(z)\leq \frac{3T+1}{2}, ; |\sqrt{3}Im(z)-Re(z)|\leq 3L}.

We now define partition functions for self-avoiding walks starting at a and ending on different parts of the boundary. Let \alpha denote the left hand boundary, \beta the right hand boundary, \epsilon the upper boundary, and \bar{\epsilon} the lower boundary. Let

: A_{T,L}^x:=\sum_{\gamma \in S_{T,L}:a\to \alpha\setminus{a}} x^{\ell(\gamma)},\quad B_{T,L}^x:=\sum_{\gamma \in S_{T,L}:a\to \beta} x^{\ell(\gamma)}, \quad E_{T,L}^x:=\sum_{\gamma \in S_{T,L}:a\to \epsilon \cup \bar{\epsilon}} x^{\ell(\gamma)}.

By summing the identity

: (p-v)F(p) + (q-v)F(q) + (r-v)F(r)=0

over all vertices in V(S_{T,L}) and noting that the winding is fixed depending on which part of the boundary the path terminates at, we can arrive at the relation

: 1=\cos(3\pi/8) A_{T,L}^{x_c} + B_{T,L}^{x_c} + \cos(\pi/4) E_{T,L}^{x_c}

after another clever computation. Letting L\to\infty, we get a strip domain S_T and partition functions

: A_{T}^x:=\sum_{\gamma \in S_{T}:a\to \alpha\setminus{a}} x^{\ell(\gamma)},\quad B_{T}^x:=\sum_{\gamma \in S_{T}:a\to \beta} x^{\ell(\gamma)}, \quad E_{T}^x:=\sum_{\gamma \in S_{T}:a\to \epsilon \cup \bar{\epsilon}} x^{\ell(\gamma)}.

It was later shown that E_{T,L}^{x_c}=0, but we do not need this for the proof. |last1=Smirnov |first1=Stanislav |year=2014 |title=The critical fugacity for surface adsorption of SAW on the honeycomb lattice is 1+\sqrt{2} |journal=Communications in Mathematical Physics |volume=326 |issue=3 |pages=727–754 |arxiv=1109.0358 |bibcode=2014CMaPh.326..727B |doi=10.1007/s00220-014-1896-1 |s2cid=54799238 We are left with the relation

: 1=\cos(3\pi/8) A_{T,L}^{x_c} + B_{T,L}^{x_c}.

From here, we can derive the inequality

: A_{T+1}^{x_c} - A_{T}^{x_c} \leq x_c (B_{T+1}^{x_c})^2

And arrive by induction at a strictly positive lower bound for B_{T}^{x_c} . Since Z(x_c)\geq\sum_{T0}B_T^{x_c}=\infty, we have established that \mu\geq\sqrt{2+\sqrt{2}}.

For the reverse inequality, for an arbitrary self avoiding walk on the honeycomb lattice, we perform a canonical decomposition due to Hammersley and Welsh of the walk into bridges of widths T_{-I} and T_0\cdots T_j. Note that we can bound

: B_T^x\leq (x/x_c)^T B_T^{x_c}\leq (x/x_c)^T

which implies \prod_{T0}(1+B_T^x). Finally, it is possible to bound the partition function by the bridge partition functions

: Z(x)\leq \sum_{T_{-I} \cdots T_j} 2 \left(\prod_{k=-I}^j B_{T_k}^x\right)=2\left(\prod_{T0}(1+B_T^x)\right)^2

And so, we have that \mu=\sqrt{2+\sqrt{2}}=2\cos\frac{\pi}{8} as desired.

Conjectures

Nienhuis argued in favor of Flory's prediction that the mean squared displacement of the self-avoiding random walk \langle |\gamma(n)|^2 \rangle satisfies the scaling relation \langle |\gamma(n)|^2 \rangle=\frac{1}{c_n} \sum_{n;\mathrm{step; SAW}}|\gamma(n)|^2=n^{2\nu +o(1)}, with \nu=3/4. The scaling exponent \nu and the universal constant 11/32 could be computed if the self-avoiding walk possesses a conformally invariant scaling limit, conjectured to be a Schramm–Loewner evolution with \kappa=8/3. |last1=Lawler |first1=Gregory F. |last2=Schramm |first2=Oded |last3=Werner |first3=Wendelin |year=2004 |chapter=On the scaling limit of planar self-avoiding walk |editor-last1=Lapidus |editor-first1=Michel L. |editor-last2=van Frankenhuijsen |editor-first2=Machiel |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2: Multifractals, Probability and Statistical Mechanics, Applications |series=Proceedings of Symposia in Pure Mathematics |volume=72 |pages=339–364 |arxiv=math/0204277 |bibcode=2002math......4277L |doi=10.1090/pspum/072.2/2112127 |isbn=9780821836385 |mr=2112127 |s2cid=16710180

References

References

1. Jesper Lykke Jacobsen, Christian R Scullard and Anthony J Guttmann, 2016 J. Phys. A: Math. Theor. 49 494004

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