The critical Ising model via Kac-Ward matrices

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer \bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version

A generalized Kac-Ward formula

The Kac-Ward formula allows to compute the Ising partition function on a planar graph G with straight edges from the determinant of a matrix of size 2N, where N denotes the number of edges of G. In this paper, we extend this formula to any finite graph: the partition function can be written as an alternating sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of an orientable surface in which G embeds. We give two proofs of this generalized formula. The first one is purely combinatorial, while the second relies on the Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on geometric techniques. As a consequence of this second proof, we also obtain the following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the Ising model are one and the same.Comment: 23 pages, 8 figures; minor corrections in v2; to appear in J. Stat. Mech. Theory Ex

Oral Fluid–Based Biomarkers of Alveolar Bone Loss in Periodontitis

Periodontal disease is a bacteria-induced chronic inflammatory disease affecting the soft and hard supporting structures encompassing the teeth. When left untreated, the ultimate outcome is alveolar bone loss and exfoliation of the involved teeth. Traditional periodontal diagnostic methods include assessment of clinical parameters and radiographs. Though efficient, these conventional techniques are inherently limited in that only a historical perspective, not current appraisal, of disease status can be determined. Advances in the use of oral fluids as possible biological samples for objective measures of current disease state, treatment monitoring, and prognostic indicators have boosted saliva and other oral-based fluids to the forefront of technology. Oral fluids contain locally and systemically derived mediators of periodontal disease, including microbial, host-response, and bone-specific resorptive markers. Although most biomarkers in oral fluids represent inflammatory mediators, several specific collagen degradation and bone turnover-related molecules have emerged as possible measures of periodontal disease activity. Pyridinoline cross-linked carboxyterminal telopeptide (ICTP), for example, has been highly correlated with clinical features of the disease and decreases in response to intervention therapies, and has been shown to possess predictive properties for possible future disease activity. One foreseeable benefit of an oral fluid–based periodontal diagnostic would be identification of highly susceptible individuals prior to overt disease. Timely detection and diagnosis of disease may significantly affect the clinical management of periodontal patients by offering earlier, less invasive, and more cost-effective treatment therapies.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73247/1/annals.1384.028.pd