Abstract
The percolation renormalization group (PRG) method is reviewed and appliedto the Potts model, the hard-square model, self-avoiding walks, andlattice animals respectively.The Potts model is transformed ot a bond-correlated percolation model, andits critical behavior is studied by the PRG method. The critical point,critical exponents, free energy and magnetization are calculated. Theresults show smooth convergence as the size of cells increases. A similarprocedure can be used to treat the hard-core particle model. In the caseof hard-square model, the result suggests that this model belongs to theuniversality class of the lsing model.For self-avoiding walks on the Manhattan lattice and the L lattice, wedefine several PRG weight functions and find that the ordinarily usedweight functions do not lead to expected values. There is the same problemin the PRG application to bond lattice animals on the square lattice. Wetherefore suggest a modified PRG scheme for SAWs and lattice animals.At last we discuss some features and possible further application of thePRG method.本篇論文討論展透重整群(Percolation Renormalization Group,PRG)在相變統模型上的應用。過去展透重整群的應用侷限於隨機展透模型及聚合物模型，現在我們利用它來研究Potts 模型以及硬核模型(Hard-core particle model)。對於Potts 模型，我們根據它與鍵相關展透模型的關連，使用PRG 方法計算它的臨界點，臨界指數，自由能，及磁化強度。我們也以類似的方法計算硬核模型的臨界性質。我們發現隨著晶胞(cell)的擴大，計算結果都有很好的收歛性。採用適當的外插方法後，這些結果將很高的精確度。此外，我們還利用PRG 方法來研究兩種聚合物模型：鍵晶格動物(Bond latticeanimals)以及Manhattan 晶格和L 晶格上的不重複行走(self-avoiding walks) 。這部分的工作是為了檢驗PRG 方法的收歛性質。PRG 方法的應用範圍可望拓展至空位自旋模型(dilute spin models), o(n)模型，Diffusion limit aggregation,量子展透模型(quantum percolation model) 等等。在觀念上，PRG 方法有簡潔明確的優點，而且它可以系統化的改進其準確度，是目前存在的少數幾種高精確度重整群方法之一。因此我們認為值得進一步干鞏固展透重整群的理論基礎，並研究它和其他方法之間的關係。