We consider a special case of the n-component cubic model on the square lattice, for which an expansion exists in Ising-type graphs. We construct a transfer matrix and perform a finite-size-scaling analysis to determinethe critical points for several values of n. Furthermore we determine several universal quantities, includingthree critical exponents. For n﹤2, these results agree well with the theoretical predictions for the criticalO (n) branch. This model is also a special case of the (Nα, Nβ) model of Domany and Riedel. It appears that theself-dual plane of the latter model contains the exactly known critical points of the n=1 and 2 cubic models.For this reason we have checked whether this is also the case for 1﹤n﹤2. However, this possibility isexcluded by our numerical results.

We explore the phase diagram of an O(n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O(n) model, so that n is not restricted to positive integers. For low activities of thevacancies, we observe critical points of the known universality class.At high activities the transition becomes first order. For n=0 themodel includes an exactly known theta point, used to describe a collapsingpolymer in two dimensions. When we vary n from 0 to 1, weobserve a tricritical point which interpolates between the universalityclasses of the theta point and the Ising tricritical point.

We investigate tricritical behavior of the O(n) model in two dimensions by means of transfer-matrix andfinite-size scaling methods. For this purpose we consider an O(n) symmetric spin model on the honeycomb latticewith vacancies; the tricritical behavior is associated with the percolation threshold of the vacancies. The vacancies are represented by face variables on the elementary hexagons of the lattice. We apply a mapping of the spin degreesof freedom model on a non-intersecting-loop model, in which the number n of spin components assumes the role of acontinuously variable parameter. This loop model serves as a suitable basis for the construction of the transfer matrix.Our results reveal the existence of a tricritical line, parametrized by n, which connects the known universality classes ofthe tricritical Ising model and the theta point describing the collapse of a polymer. On the other side of the Ising point,the tricritical line extends to the n = 2 point describing a tricritical O(2) model.

We propose exact expressions for the conformal anomaly and for three critical exponents of thetricritical O(n)loop model as a function of n in the range-2 ≤n≤3/2. These findings are based on ananalogy with known relations between Potts and O (n)models and on an exact solution of a "ritricritical" Potts model described in the literature. We verify the exact expressions for the tricritical O(n)model by means of a finite-size scaling analysis based on numerical transfer-matrix calculations.

A critical dilute O(n) model on the kagome lattice is investigated analytically and numerically. We employa number of exact equivalences which, in a few steps, link the critical O(n) spin model on the kagome latticetothe exactly solvable critical q-state Potts model on the honeycomb lattice with q= (n+1) 2. The intermediatesteps involve the random-cluster model on the honeycomb lattice and a fully packed loop model with loopweight n'=q and a dilute loop model with loop weight n, both on the kagome lattice. This mapping enablesthe determination ofa branch of critical points of the dilute O（n）model, as well as some of its criticalproperties. These properties differ from those of the generic O（n）critical points. For n=0, our model reproducesthe known universal properties of the point describing the collapse of a polymer. For n≠0 it displaysa line of multicritical points, with the same universal behavior as a branch of critical points that was foundearlier in a dilute O（n）model on the square lattice. These findings are supported by a finite-size-scalinganalysis in combination with transfer-matrix calculations.