Connective constant
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. 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 formula_1, may provide clues 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. Definition. The connective constant is defined as follows. Let formula_2 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 formula_3. Then by applying Fekete's lemma to the logarithm of the above relation, the limit formula_4 can be shown to exist. This number formula_5 is called the connective constant, and clearly depends on the particular lattice chosen for the walk since formula_2 does. The value of formula_5 is precisely known only for two lattices, see below. For other lattices, formula_5 has only been approximated numerically. It is conjectured that formula_9 as n goes to infinity, where formula_5 and formula_11, the critical amplitude, depend on the lattice, and the exponent formula_12, which is believed to be  universal and dependent on the dimension of the lattice, is conjectured to be formula_13. Known values. These values are taken from the 1998 Jensen–Guttmann paper and a more recent paper by Jacobsen, Scullard and Guttmann. The connective constant of the formula_14 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. Duminil-Copin–Smirnov proof. In 2010 , Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that formula_16 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. 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. 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. For a self-avoiding walk formula_12 between two mid-edges formula_18 and formula_19, we define formula_20 to be the number of vertices visited and its winding formula_21 as the total rotation of the direction in radians when formula_12 is traversed from formula_18 to formula_19. The aim of the proof is to show that the partition function converges for formula_26 and diverges for formula_27 where the critical parameter is given by formula_28 . This immediately implies that formula_29. Given a domain formula_30 in the hexagonal lattice, a starting mid-edge formula_18, and two parameters formula_32 and formula_33, we define the parafermionic observable formula_34 If formula_35 and formula_36, then for any vertex formula_37 in formula_30, we have where formula_40 are the mid-edges emanating from formula_37. 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 formula_42 with 2L cells forming the left hand side, T cells across, and upper and lower sides at an angle of formula_43. (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 formula_42 are given by We now define partition functions for self-avoiding walks starting at formula_18 and ending on different parts of the boundary. Let formula_47 denote the left hand boundary, formula_48 the  right hand boundary, formula_49 the upper boundary, and formula_50 the lower boundary. Let By summing the identity over all vertices in formula_53 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. Letting formula_55, we get a strip domain formula_56 and partition functions It was later shown that formula_58, but we do not need this for the proof. We are left with the relation From here, we can derive the inequality And arrive by induction at a strictly positive lower bound for  formula_61. Since formula_62, we have established that formula_63. 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 formula_64 and formula_65. Note that we can bound which implies formula_67. Finally, it is possible to bound the partition function by the bridge partition functions And so, we have that formula_29 as desired. Conjectures. Nienhuis argued in favor of Flory's prediction that the mean squared displacement of the self-avoiding random walk formula_70 satisfies the scaling relation formula_71, with formula_72 . The scaling exponent formula_73 and the universal constant formula_74 could be computed if the self-avoiding walk possesses a conformally invariant scaling limit, conjectured to be a Schramm–Loewner evolution with formula_75.