Non-Local Finite-Size Effects in the Dimer Model

Abstract

Abstract. We study the finite-size corrections of the dimer model on ∞ × N square lattice with two different boundary conditions: free and periodic. We find that the finite-size corrections depend in a crucial way on the parity of N , and show that, because of certain non-local features present in the model, a change of parity of N induces a change of boundary condition. Taking a careful account of this, these unusual finite-size behaviours can be fully explained in the framework of the c = −2 logarithmic conformal field theory.

On the asymptotics of dimers on tori

“…17. Applying (23) to the fundamental domain gives (which can also be verified by direct calculation). The characteristic polynomial is…”

Jamming probabilities for a vacancy in the dimer model

“…Then we define a frustrated Laplacian matrix∆ op , equal to ∆ op except that (∆ op ) ij = +1 for all those nearest neighbour sites connected by a bond (of L even ) which crosses the defect line. Because any loop enclosing the vacancy necessarily contains an odd number of frustrated edges, it can then be shown [1,3] that the determinant of∆ op counts the number of spanning webs with the correct weight, namely 2 n if the web contains n loops,…”

“…det(∆ op + B(3) ) counts the number of even spanning webs with at most N − 1 loops in presence of six roots. Again every such spanning web gives rise to two dimer configurations on the whole lattice with fifteen vacancies, namely all the sites shown inFig.6b.…”

“…If one excepts the vacancy itself, the other fourteen sites can be covered by seven dimers in eight different ways, among which only two contain a 6-dimer loop. So the number of dimer coverings in which six dimers make a loop around the vacancy (and with the number of even loops bounded by N) is equal to 4 × lim ε→∞ det(∆ op + B(3) ). Their fraction in the set of similar dimer coverings but with no restriction around the vacancy is, in the thermodynamic limit, equal to4 D det(I + GB(3) ) det(I + GB (0) ) .…”

Exact asymptotic expansion for the resistance between the center node and a node on the cobweb network boundary

“…This was extended by Izmailian and Huang [12] to other boundary conditions. In recent decades the finitesize scaling and finite-size corrections in finite critical systems and their boundary effects have attracted much attention [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Of particular importance in such studies are exact results wherein the analysis can be carried out without numerical errors.…”