Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice

We present exact calculations of the partition function of the $q$-state Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the honeycomb (brick) lattice of width $L_y=2$ and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the $q$ plane for fixed temperature and in the complex temperature plane for fixed $q$ values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) $L_y=3$, cyclic, (v) $L_y=3$, M\"obius, (vi) $L_y=4$, cylindrical, and (vii) $L_y=4$, open. In the infinite-length limit we calculate $W(q)$ and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the $L_y=4$ strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in $10^5$ for moderate $q$ values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).Comment: 48 pages, latex, with encapsulated postscript figure

Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice

We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii) (FBC_y,TPBC_x)= M\"obius, (iii) (PBC_y,PBC_x)= toroidal, and (iv) (PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the L_y=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the {\mathbb C}^2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.Comment: latex, with encapsulated postscript figure

Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets

We study the asymptotic limiting function $W({G},q) = \lim_{n \to \infty}P(G,q)^{1/n}$, where $P(G,q)$ is the chromatic polynomial for a graph $G$ with $n$ vertices. We first discuss a subtlety in the definition of $W({G},q)$ resulting from the fact that at certain special points $q_s$, the following limits do not commute: $\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne \lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}$. We then present exact calculations of $W({G},q)$ and determine the corresponding analytic structure in the complex $q$ plane for a number of families of graphs ${G}$, including circuits, wheels, biwheels, bipyramids, and (cyclic and twisted) ladders. We study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs, all but a finite number of the zeros lie exactly on a unit circle, whose position depends on the family. Using the connection of $P(G,q)$ with the zero-temperature Potts antiferromagnet, we derive a theorem concerning the maximal finite real point of non-analyticity in $W({G},q)$, denoted $q_c$ and apply this theorem to deduce that $q_c(sq)=3$ and $q_c(hc) = (3+\sqrt{5})/2$ for the square and honeycomb lattices. Finally, numerical calculations of $W(hc,q)$ and $W(sq,q)$ are presented and compared with series expansions and bounds.Comment: 33 pages, Latex, 5 postscript figures, published version; includes further comments on large-q serie

Exact Potts Model Partition Functions on Ladder Graphs

We present exact calculations of the partition function $Z$ of the $q$-state Potts model and its generalization to real $q$, the random cluster model, for arbitrary temperature on $n$-vertex ladder graphs with free, cyclic, and M\"obius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed.Comment: 73 pages, Latex, 20 postscript figures, Physica A, in pres

Association Between Bilirubin, Atazanavir, and Cardiovascular Disease Events Among People Living With HIV Across the United States

Objective: Bilirubin is an antioxidant that may suppress lipid oxidation. Elevated bilirubin is associated with decreased cardiovascular events in HIV-uninfected populations. We examined these associations in people living with HIV (PLWH). Methods: Potential myocardial infarctions (MIs) and strokes were centrally adjudicated. We examined MI types: type 1 MI (T1MI) from atherosclerotic plaque instability and type 2 MI (T2MI) in the setting of oxygen demand/supply mismatch such as sepsis. We used multivariable Cox regression analyses to determine associations between total bilirubin levels and outcomes adjusting for traditional and HIV-specific risk factors. To minimize confounding by hepatobiliary disease, we conducted analyses limited to bilirubin values <2.1 mg/dL; among those with fibrosis-4 values <3.25; and among everyone. We repeated analyses stratified by hepatitis C status and time-updated atazanavir use. Results: Among 25,816 PLWH, there were 392 T1MI and 356 T2MI during follow-up. Adjusted hazard ratios for the association of higher bilirubin levels with T1MI were not significant. Higher bilirubin levels were associated with T2MI. By contrast, among PLWH on atazanavir, higher bilirubin levels were associated with fewer T2MI (hazard ratio 0.56:0.33-1.00). Higher bilirubin levels among those on atazanavir were associated with fewer T1MI combined with ischemic stroke. Limitations: Analyses were conducted with total rather than unconjugated bilirubin. Conclusions: Among PLWH, higher bilirubin levels were associated with T2MI among some subgroups. However, among those on atazanavir, there was a protective association between bilirubin and T2MI. These findings demonstrate different associations between outcomes and elevated bilirubin due to diverse causes and the importance of distinguishing MI types

Types of Stroke among People Living with HIV in the United States

Background: Most studies of stroke in people living with HIV (PLWH) do not use verified stroke diagnoses, are small, and/or do not differentiate stroke types and subtypes.Setting: CNICS, a U.S. multisite clinical cohort of PLWH in care.Methods: We implemented a centralized adjudication stroke protocol to identify stroke type, subtype, and precipitating conditions identified as direct causes including infection and illicit drug use in a large diverse HIV cohort.Results: Among 26,514 PLWH, there were 401 strokes, 75% of which were ischemic. Precipitating factors such as sepsis or same-day cocaine use were identified in 40% of ischemic strokes. Those with precipitating factors were younger, had more severe HIV disease, and fewer traditional stroke risk factors such as diabetes and hypertension. Ischemic stroke subtypes included cardioembolic (20%), large vessel atherosclerosis (13%), and small vessel (24%) ischemic strokes. Individuals with small vessel strokes were older, were more likely to have a higher current CD4 cell count than those with cardioembolic strokes and had the highest mean blood pressure of the ischemic stroke subtypes.Conclusion: Ischemic stroke, particularly small vessel and cardioembolic subtypes, were the most common strokes among PLWH. Traditional and HIV-related risk factors differed by stroke type/subtype. Precipitating factors including infections and drug use were common. These results suggest that there may be different biological phenomena occurring among PLWH and that understanding HIV-related and traditional risk factors and in particular precipitating factors for each type/subtype may be key to understanding, and therefore preventing, strokes among PLWH