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.
[1] While the connective constant depends on the choice of lattice so itself is not universal (similarly to other lattice-dependent quantities such as the critical probability threshold for percolation), it is nonetheless an important quantity that appears in conjectures for universal laws.
Furthermore, the mathematical techniques used to understand the connective constant, for example in the recent rigorous proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice has the precise value
, may provide clues[2] to a possible approach for attacking other important open problems in the study of self-avoiding walks, notably the conjecture that self-avoiding walks converge in the scaling limit to the Schramm–Loewner evolution.
denote the number of n-step self-avoiding walks starting from a fixed origin point in the lattice.
Then by applying Fekete's lemma to the logarithm of the above relation, the limit
is called the connective constant, and clearly depends on the particular lattice chosen for the walk since
, the critical amplitude, depend on the lattice, and the exponent
, which is believed to be universal and dependent on the dimension of the lattice, is conjectured to be
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 given the exact expression for the hexagonal lattice connective constant.
More information about these lattices can be found in the percolation threshold article.
In 2010, Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that
[2] This had been conjectured by Nienhuis in 1982 as part of a larger study of O(n) models using renormalization techniques.
[6] The rigorous proof of this fact came from a program of applying tools from complex analysis to discrete probabilistic models that has also produced impressive results about the Ising model among others.
[7] The argument relies on the existence of a parafermionic observable that satisfies half of the discrete Cauchy–Riemann equations for the hexagonal lattice.
We modify slightly the definition of a self-avoiding walk by having it start and end on mid-edges between vertices.
Let H be the set of all mid-edges of the hexagonal lattice.
The aim of the proof is to show that the partition function converges for
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.
with 2L cells forming the left hand side, T cells across, and upper and lower sides at an angle of
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.
are given by We now define partition functions for self-avoiding walks starting at
and noting that the winding is fixed depending on which part of the boundary the path terminates at, we can arrive at the relation after another clever computation.
[8] We are left with the relation From here, we can derive the inequality And arrive by induction at a strictly positive lower bound for
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
Nienhuis argued in favor of Flory's prediction that the mean squared displacement of the self-avoiding random walk
could be computed if the self-avoiding walk possesses a conformally invariant scaling limit, conjectured to be a Schramm–Loewner evolution with