First displacement time of a tagged particle in a stochastic cluster in a simple exclusion process with random slow bonds

We consider the nearest-neighbour simple exclusion process on the one-dimensional discrete torus $\mathbb{T}_N=\mathbb{Z}/N\mathbb{Z}$ , with random rates $c_N=\{c_{x,N}\colon x \in \mathbb{T}_N\}$ defined in terms of a homogeneous Poisson process on $\mathbb{R}$ with intensity $\lambda$ . Given a realization of the Poisson process, the jump rate along the edge $\{x,x+1\}$ is 1 if there is not any Poisson mark in $ (x,x+1) $ ; otherwise, it is $\lambda/N,\, \lambda \in( 0,1]$ . The density profile of this process with initial measure associated to an initial profile $\rho_0\colon \mathbb{R} \rightarrow [0,1]$ , evolves as the solution of a bounded diffusion random equation. This result follows from an appropriate quenched hydrodynamic limit. If $\lambda=1$ then $\rho$ is discontinuous at each Poisson mark with passage through the slow bonds, otherwise the conductance at the slow bonds decreases meaning no passage through the slow bonds in the continuum. The main results are concerned with upper and lower quenched and annealed bounds of $T_j$ , where $T_j$ is the first displacement time of a tagged particle in a stochastic cluster of size j (the cluster is defined via specific macroscopic density profiles). It is possible to observe that when time t grows, then $\mathbb{P}\{T_j \geq t\}$ decays quadratically in both the upper and lower bounds, and falls as slow as the presence of more Poisson marks neighbouring the tagged particle, as expected.

Complexity of discrete Seifert foliations over a graph

In the present paper, we study the complexity of an infinite family of graphs Hn = Hn(G1, G2, ..., Gm) that are discrete Seifert foliations over a graph H on m vertices with fibers G1, G2, ..., Gm. Each fiber Gi = Cn(si,1, si,2, ..., si,ki) of this foliation is the circulant graph on n vertices with jumps si,1, si,2, ..., si,ki. The family of discrete Seifert foliations is sufficiently large. It includes the generalized Petersen graphs, I-graphs, Y-graphs, H-graphs, sandwiches of circulant graphs, discrete torus graph and others. We obtain a closed formula for the number t(n) of spanning trees in Hn in terms of Chebyshev polynomials, investigate some arithmetical properties of this function and find its asymptotics as n → ∞.

Mono- and Hexanuclear Zinc Halide Complexes with Soft Thiopyridazine Based Scorpionate Ligands

Scorpionate ligands with three soft sulfur donor sites have become very important in coordination chemistry. Despite its ability to form highly electrophilic species, electron-deficient thiopyridazines have rarely been used, whereas the chemistry of electron-rich thioheterocycles has been explored rather intensively. Here, the unusual chemical behavior of a thiopyridazine (6-tert-butylpyridazine-3-thione, HtBuPn) based scorpionate ligand towards zinc is reported. Thus, the reaction of zinc halides with tris(6-tert-butyl-3-thiopyridazinyl)borate Na[TntBu] leads to the formation of discrete torus-shaped hexameric zinc complexes [TntBuZnX]6 (X = Br, I) with uncommonly long zinc halide bonds. In contrast, reaction of the sterically more demanding ligand K[TnMe,tBu] leads to decomposition, forming Zn(HPnMe,tBu)2X2 (X = Br, I). The latter can be prepared independently by reaction of the respective zinc halides and two equiv of HPnMe,tBu. The bromide compound was used as precursor which further reacts with K[TnMe,tBu] forming the mononuclear complex [TnMe,tBu]ZnBr(HPnMe,tBu). The molecular structures of all compounds were elucidated by single-crystal X-ray diffraction analysis. Characterization in solution was performed by means of 1H, 13C and DOSY NMR spectroscopy which revealed the hexameric constitution of [TntBuZnBr]6 to be predominant. In contrast, [TnMe,tBu]ZnBr(HPnMe,tBu) was found to be dynamic in solution.

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus Td. In the discrete torus N-1TdN, we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is ej, then the exchange rate is defined as N-1 times the absolute value of the inner product between ej and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.

Hydrodynamic Limit for a Type of Exclusion Process with Slow Bonds in Dimension d ≥ 2

Let Λ be a connected closed region with smooth boundary contained in the d-dimensional continuous torus T d . In the discrete torus N -1 T d N , we consider a nearest-neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on Λ in the following way: if both sites are in Λ or Λc, the exchange rate is 1; if one site is in Λ and the other site is in Λc, and the direction of the bond connecting the sites is e j , then the exchange rate is defined as N -1 times the absolute value of the inner product between e j and the normal exterior vector to ∂Λ. We show that this exclusion-type process has a nontrivial hydrodynamical behavior under diffusive scaling and, in the continuum limit, particles are not blocked or reflected by ∂Λ. Thus, the model represents a system of particles under hard-core interaction in the presence of a permeable membrane which slows down the passage of particles between two complementary regions.