Localization for a Random Walk in Slowly Decreasing Random Potential

We consider a continuous time random walk X in a random environment on ?⁺ such that its potential can be approximated by the function V:?⁺→? given by $V(x)=\sigma W(x) -\frac {b}{1-\alpha}x^{1-\alpha}$ where σW a Brownian motion with diffusion coefficient σ>0 and parameters b, α are such that b>0 and 0<α<1/2. We show that P-a.s. (where P is the averaged law) $\lim_{t\to\infty} \frac{X_{t}}{(C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}}=1$ with $C^{*}=\frac{2\alpha b}{\sigma^{2}(1-2\alpha)}$ . In fact, we prove that by showing that there is a trap located around $(C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}$ (with corrections of smaller order) where the particle typically stays up to time t. This is in sharp contrast to what happens in the “pure” Sinai’s regime, where the location of this trap is random on the scale ln² t.     

Pfaffian graphs, <Emphasis Type="Italic">T</Emphasis>-joins and crossing numbers

We characterize Pfaffian graphs in terms of their drawings in the plane. We generalize the techniques used in the proof of this characterization, and prove a theorem about the numbers of crossings in T-joins in different drawings of a fixed graph. As a corollary we give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K _2j+1 and K _2j+1,2k+1. Partially supported by NSF grants DMS-0200595 and DMS-0701033.     

Direct copolymerization of sulfonated poly(phthalazinone arylene ether)s for proton‐exchange‐membrane materials

Sulfonated poly(phthalazinone ether ketone) (SPPEK) copolymers and sulfonated poly(phthalazinone ether sulfone) (SPPES) copolymers containing pendant sodium sulfonate groups were prepared by direct copolymerization. The reaction of disodium 3,3′‐disulfonate‐4,4′‐difluorobenzophenone (SDFB‐Na), 4,4′‐difluorobenzophenone (DFB), and 4‐(4‐hydroxyphenyl)‐1(2H)‐phthalazinone (DHPZ) at 170 °C in N‐methyl‐2‐pyrrolidione containing anhydrous potassium carbonate gave SPPEKs. SPPESs were similarly obtained with 3,3′‐disulfonate‐4,4′‐difluorophenyl sulfone, 4‐fluorophenyl sulfone (DFS), and DHPZ as monomers. The sulfonic acid groups, being on deactivated positions of the polymer backbone, were expected to be hydrolytically more stable than postsulfonated polymers. Fourier transform infrared and ¹H NMR were used to characterize the structures and degrees of sulfonation of the sulfonated polymers. Membrane films of SPPEKs with SDFB‐Na/DFB molar feed ratios of up to 60/40 and SPPESs with sulfonated 4‐fluorophenyl sulfone/DFS molar feed ratios of up to 50/50 were cast from N,N‐dimethylacetamide polymer solutions. Membrane films in acid form were then obtained by the treatment of the sodium‐form membrane films in 2 N sulfuric acid at room temperature. An increase in the number of sulfonate groups in the copolymers resulted in an increased glass‐transition temperature and enhanced membrane hydrophilicity. The sodium‐form copolymers were thermally more stable than their acid forms. The proton conductivities of the acid‐form copolymers with sulfonated monomer/unsulfonated monomer molar feed ratios of 0.5 and 0.6 were higher than 10^?2 S/cm and increased with temperature; they were less temperature‐dependent than those of the postsulfonated products. SPPESH‐50 showed higher conductivity than the corresponding postsulfonated poly(phthalazinone ether sulfone). © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 2731–2742, 2003     

A variable temperature EPR study of Mn-doped NH4Cl0.9I0.1 single crystal at 170 GHz: Zero-field splitting parameter and its absolute sign

EPR measurements have been carried out on a single crystal of Mn(2+)-doped NH(4)Cl(0.9)I(0.1) at 170-GHz in the temperature range of 312-4.2K. The spectra have been analyzed (i) to estimate the spin-Hamiltonian parameters; (ii) to study the temperature variation of the zero-field splitting (ZFS) parameter; (iii) to confirm the negative absolute sign of the ZFS parameter unequivocally from the temperature-dependent relative intensities of hyperfine sextets at temperatures below 10K; and (iv) to detect the occurrence of a structural phase transition at 4.35K from the change in the structure of the EPR lines with temperature below 10K.     

On the origin of the ultrafast internal conversion of electronically excited pyrimidine bases

The ultrafast radiationless decay of photoexcited uracil and cytosine has been investigated by ab initio quantum chemical methods based on CIS and CR-EOM-CCSD(T) electronic energy calculations at optimized CIS geometries. The calculated potential energy profiles indicate that the S(1) --> S(0) internal conversion of the pyrimidine bases occurs through a barrierless state switch from the initially excited (1)pipi state to the out-of-plane deformed excited state of biradical character, which intersects the ground state at a lower energy. This three-state nonradiative decay mechanism predicts that replacement of the C5 hydrogen by fluorine introduces an energy barrier for the initial state switch, whereas replacement of the C6 hydrogen by fluorine does not. These predictions are borne out by the very different fluorescence yields of 5-fluorinated bases relative to the corresponding 6-fluorinated bases. It is concluded from these results that the origin of the ultrafast radiationless decay is the same for the two pyrimidine bases.     

A one-dimensional version of the random interlacements

We base ourselves on the construction of the two-dimensional random interlacements (Comets et al., 2016) to define the one-dimensional version of the process. For this, we consider simple random walks conditioned on never hitting the origin. We compare this process to the conditional random walk on the ring graph. Our results are the convergence of the vacant set on the ring graph to the vacant set of one-dimensional random interlacements, a central limit theorem for the interlacements’ local time and the convergence in law of the local times of the conditional walk on the ring graph to the interlacements’ local times.     

On uniform closeness of local times of Markov chains and i.i.d. sequences

In this paper we consider the field of local times of a discrete-time Markov chain on a general state space, and obtain uniform (in time) upper bounds on the total variation distance between this field and the one of a sequence of $n$ i.i.d. random variables with law given by the invariant measure of that Markov chain. The proof of this result uses a refinement of the soft local time method of Popov and Teixeira (2015).     

Survival of Branching Random Walks in Random Environment

We study survival of nearest-neighbor branching random walks in random environment (BRWRE) on ℤ. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. 2×2 random matrices.