The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n ^3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n ^3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _c  = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ ² n → ∞ with n, the mixing-time has order (n/δ) log(δ ² n), and exhibits cutoff with constant and window size n/δ. In the critical window, β = 1± δ, where δ ² n is O(1), there is no cutoff, and the mixing-time has order n ^3/2. At low temperature, β = 1 + δ for δ > 0 with δ ² n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order . Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.     

Inhibition of energy transfer between conjugated polymer chains in host/guest nanocomposites generates white photo- and electroluminescence

The generation of white light requires the combination of two or more chromophores that emit simultaneously. The observed color of a mixture of light-emitting molecules, however, originates generally only from the lowest band-gap species because of efficient energy transfer between the chromophores which is difficult to avoid. Here we report on a nanocomposite material designed to yield pure and stable white photo- and electroluminescence. In this material, red, green, and blue emitting conjugated polymers are confined within the galleries of a layered semiconducting host matrix. The host hinders polymer pi-pi interactions which are responsible for the energy transfer between polymer chains, consequently, emission from the three chromophores is observed simultaneously resulting in white photoluminescence. The efficacy of the nanocomposites is demonstrated in simple single-layer white-emitting polymer diodes. The mechanism suggested here for white light generation, supported by extensive luminescence measurements, is in contrast to that previously reported in white-emitting polymer diodes where efficient energy transfer between polymer chains was essential for obtaining white light.     

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β _c , the inverse-gap is O(1) for β < β _c , polynomial in the surface area for β = β _c and exponential in it for β > β _c . This has been proved for \mathbbZ²{\mathbb{Z}^2} except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β _c and exponential for β > β _c were established, where β _c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β _c , the inverse-gap and mixing-time are both exp[Θ((β − β _c )h)].     

Collective conformations of DNA polymers assembled on surface density gradients

To study dense double-stranded DNA (dsDNA) polymer phases, we fabricated continuous density gradients of binding sites for assembly on a photochemical interface and measured both dsDNA occupancy and extension using evanescent fluorescence. Despite the abundance of available binding sites, the dsDNA density saturates after occupation of only a fraction of the available sites along the gradient. The spatial position at which the density saturates marks the onset of collective stretching of dsDNA, a direct manifestation of balancing entropic and excluded-volume interactions. The methodology presented here offers a new means to investigate dense dsDNA compartments.     

Quasi‐polynomial mixing of critical two‐dimensional random cluster models

We study the Glauber dynamics for the random cluster (FK) model on the torus with parameters (p,q), for q ∈ (1,4] and p the critical point p_c. The dynamics is believed to undergo a critical slowdown, with its continuous‐time mixing time transitioning from for p ≠ p_c to a power‐law in n at p = p_c. This was verified at p ≠ p_c by Blanca and Sinclair, whereas at the critical p = p_c, with the exception of the special integer points q = 2,3,4 (where the model corresponds to the Ising/Potts models) the best‐known upper bound on mixing was exponential in n. Here we prove an upper bound of at p = p_c for all q ∈ (1,4], where a key ingredient is bounding the number of nested long‐range crossings at criticality.     

Matching Static and Dynamic Compliance of Small‐Diameter Arteries,with Poly(lactide‐co‐caprolactone) Copolymers: In Vitro and In Vivo Studies

Mechanical mismatch between vascular grafts and blood vessels is a major cause of smaller diameter vascular graft failure. To minimize this mismatch, several poly‐l ‐lactide‐co‐ε‐caprolactone (PLC) copolymers are evaluated as candidate materials to fabricate a small diameter graft. Using these materials, tubular prostheses of 4 mm inner diameter are fabricated by dip‐coating. In vitro static and dynamic compliance tests are conducted, using custom‐built apparatus featuring a closed flow system with water at 37 °C. Grafts of PLC monomer ratio of 50:50 are the most compliant (1.56% ± 0.31?mm Hg^?2), close to that of porcine aortic branch arteries (1.56% ± 0.43?mm Hg^?2), but underwent high continuous dilatation (87 µm min^?1). Better matching is achieved by optimizing the thickness of a tubular conduit made from 70:30 PLC grafts. In vivo implantation and function of a PLC 70:30 conduit of 150 µm wall‐thickness (WT) are tested as a rabbit aorta bypass. An implanted 150 µm WT PLC 70:30 prosthesis is observed over 3 h. The recorded angiogram shows continuous blood flow, no aneurysmal dilatation, leaks, or acute thrombosis during the in vivo test, indicating the potential for clinical applications.     

Independent sets in tensor graph powers

The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gⁿ)/|V(Gⁿ)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 ( 5 ), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory     

On the number of rectangulations of a planar point set

We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n+1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schröder number, and the total number of rectangulations is O(n²⁰/n⁴).     

Extension of Arrow's theorem to symmetric sets of tournaments

Arrow's impossibility theorem [K.J. Arrow, Social Choice and Individual Values, Wiley, New York, NY, 1951] shows that the set of acyclic tournaments is not closed to non-dictatorial Boolean aggregation. In this paper we extend the notion of aggregation to general tournaments and we show that for tournaments with four vertices or more any proper symmetric (closed to vertex permutations) subset cannot be closed to non-dictatorial monotone aggregation and to non-neutral aggregation. We also demonstrate a proper subset of tournaments that is closed to parity aggregation for an arbitrarily large number of vertices. This proves a conjecture of Kalai [Social choice without rationality, Reviewed NAJ Economics 3(4)] for the non-neutral and the non-dictatorial and monotone cases and gives a counter example for the general case.