This thesis is divided into two parts.
In Part I, we give an explicit construction for a classof lattices with effectively non-integral dimensionality. Areasonable definition of dimensionality applicable to lattice systems isproposed. The construction is illustrated by several examples. We calculate the effective dimensionality of someof these lattices. The attainable values of the dimensionality d,using our construction, are densely distributed in theinterval 1 The variation of critical exponents with dimensionalityis studied for a variety of Hamiltonians. It is shown thatthe critical exponents for the spherical model, for all d,agree with the values derived in literature using formalarguments only. We also study the critical behavior of theclassical p-vector Heisenberg model and the Fortuin-Kasteleyncluster model for lattices with d<2. Itis shown that no phase transition occurs at nonzerotemperatures. The renormalization procedure is used todetermine the exact values of the connectivity constants andthe critical exponents α, γ, v for the self-avoidingwalk problem on some multiply connected lattices with d<2.It is shown by explicit construction that the criticalexponents are not functions of dimensionality alone, butdepend on detailed connectivity properties of the lattice. In Part II, we investigate a model of the melting transition in solids. Melting is treated as a layer phenomenon, the onset of melting being characterized by the ability of layers to slip past each other. We study the variation of the root-mean-square deviation of atoms in one layer as the temperature is increased. The adjacent layers are assumed held fixed and provide an external periodic potential. The coupling between atoms within the layer is assumed to be simple harmonic. The model is thus equivalentto a lattice version of the Sine-Gordon field theory in two dimensions. Using an exact equivalence,the partitionfunction for this problem is shown to be related to the grand partition function of a two-species classical lattice Coulomb gas. We use the renormalization procedure to determine the critical behavior of the lattice Coulomb gas problem. Translating the results back to the original problem, it is shown that there exists a phase transition in the model at a finite temperature Tc. Below Tc, the rootmean square deviation of atoms in the layer is finite, andvaries as (Tc-T)-1/4 near the phase transition. Above Tc, theroot mean square deviation is infinite.The specific heat shows an essential singularity at the phase transition,varying as exp(-|Tc -T|-1/2) near Tc.
