Central limit theorem for first-passage percolation time across thin cylinders

We prove that first-passage percolation times across thin cylinders of the form [0, n] × [?h _n , h _n ] ^d-1 obey Gaussian central limit theorems as long as h _n grows slower than n ^1/(d+1). It is an open question as to what is the fastest that h _n can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of (1, 0, . . . , 0), we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.     

Surfactant-Mediated and Morphology-Controlled Nanostructured LiFePO4/Carbon Composite as a Promising Cathode Material for Li-Ion Batteries

The synthesis of morphology-controlled carbon-coated nanostructured LiFePO₄ (LFP/Carbon) cathode materials by surfactant-assisted hydrothermal method using block copolymers is reported. The resulting nanocrystalline high surface area materials were coated with carbon and designated as LFP/C123 and LFP/C311. All the materials were systematically characterized by various analytical, spectroscopic and imaging techniques. The reverse structure of the surfactant Pluronic® 31R1 (PPO-PEO-PPO) in comparison to Pluronic® P123 (PEO-PPO-PEO) played a vital role in controlling the particle size and morphology which in turn ameliorate the electrochemical performance in terms of reversible specific capacity (163 mAh g⁻¹ and 140 mAh g⁻¹ at 0.1 C for LFP/C311 and LFP/C123, respectively). In addition, LFP/C311 demonstrated excellent electrochemical performance including lower charge transfer resistance (146.3 Ω) and excellent cycling stability (95 % capacity retention at 1 C after 100 cycles) and high rate capability (163.2 mAh g⁻¹ at 0.1 C; 147.1 mAh g⁻¹ at 1 C). The better performance of the former is attributed to LFP nanoparticles (<50 nm) with a specific spindle-shaped morphology. Further, we have also evaluated the electrode performance with the use of both PVDF and CMC binders employed for the electrode fabrication.     

Metal-Stabilized [B8H8]2− Derivatives with Dodecahedral Structure in the Solid and Solution States: [(Cp2MBH3)2B8H6] (Cp=η5-C5H5; M=Zr (1-Zr) and Hf (1-Hf)).

Despite the synthesis and structural characterization of closo-hydroborate dianions, [B_nH_n]²⁻ (n=6–12) more than 50 years ago, some ambiguity remains about the structure of [B₈H₈]²⁻. Although the solid-state structure of [B₈H₈]²⁻ was established by single-crystal X-ray studies in 1969, fast rearrangements in solution at accessible temperatures prevented its detailed characterization. We therefore stabilized a derivative of [B₈H₈]²⁻ by using Cp₂MBH₃ and structurally characterized two new octaborane analogues, [(Cp₂MBH₃)₂B₈H₆] (Cp=η⁵-C₅H₅; M=Zr ( 1-Zr ) and Hf ( 1-Hf )), so that the dynamics of the B₈ skeleton were arrested. The solid-state structures of both 1-Zr and 1-Hf comprise a dodecahedron core protected by {Cp₂MBH₃} moieties on both sides of the cluster. Spectroscopic characterization (¹¹B NMR) validates the intactness of the B₈ dodecahedron core in solution as well. Theoretical calculations establish that the two exo-{Cp₂MBH₃} fragments provide structural and electronic structural stability to the B₈ core and its intact dodecahedral dianionic nature. Furthermore, we propose isodesmic equations for the formation of higher analogues of the B_n core (n>8) guarded by different group 4 transition metals. Our analysis suggests that, as we move to higher polyhedra (n>10), the formation becomes unfavourable irrespective of metal.     

Toward Electrochemically Controllable Tristable Three‐Station [2]Catenanes

Encouraged by the prospect of producing an electrochemical, color‐switchable red–green–blue (RGB) dye compound, we have designed, synthesized, and characterized two three‐station [2]catenanes. Both are composed of macrocyclic polyethers containing three π‐electron‐rich stations, which act as recognition sites for a π‐electron‐deficient tetracationic cyclophane. The molecular structures of the two three‐station [2]catenanes were characterized fully by mass spectrometry and ¹H NMR spectroscopy. To anticipate the relative occupancies of the three stations in each [2]catenane by the cyclophane, model compounds with the same constitutions in the vicinity of the stations were synthesized. The relative ground‐state populations of the three stations occupied in both [2]catenanes were estimated from the thermodynamic parameters for 1:1 complexes between all these model compounds and the cyclophane, obtained from isothermal titration calorimetry (ITC). The electrochemical and electromechanical properties of the three‐station [2]catenanes were analyzed by cyclic voltammetry (CV), differential pulse voltammetry (DPV), and spectroelectrochemistry (SEC). The first three‐station [2]catenane was found to behave like a bistable system, whereas the second can be described as a quasi‐tristable system.     

Central Limit Theorem for the Free Energy of the Random Field Ising Model

A central limit theorem is proved for the free energy of the random field Ising model with all plus or all minus boundary condition, at any temperature (including zero temperature) and any dimension. This solves a problem posed by Wehr and Aizenman (J Stat Phys 60:287–306, 1990). The proof uses a variant of Stein’s method.

     

Spin glasses and Stein’s method

We introduce some applications of Stein’s method in the high temperature analysis of spin glasses. Stein’s method allows the direct analysis of the Gibbs measure without having to eate a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless–Anderson–Palmer system of equations. Along the way, we develop Stein’s method for mixtures of two Gaussian densities.     

A phase transition behavior for Brownian motions interacting through their ranks

Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson–Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson–Dirichlet law.