Random Interchange Research Articles

On the algebraic connectivity of some token graphs

The k-token graph Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of Fk(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_k(G)$$\\end{document} equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.

Critical parameters for loop and Bernoulli percolation

We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively, larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with loops of length 1 and as $\beta$ crosses a critical value $\beta_c$, infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with $p=1-e^{-\beta}$ on the same graph to obtain a lower bound on $\beta_c$. For those graphs of diverging vertex degree where $\beta_c$ and the critical parameter for percolation have been calculated explicitly, that inequality has been found to be an equality. In contrast, we show in this paper that for graphs of bounded degree the inequality is strict, i.e. we show existence of an interval of values of $\beta$ where there are no infinite loops, but infinite percolation clusters almost surely.

Split-and-Merge in Stationary Random Stirring on Lattice Torus

We show that in any dimension dge 1, the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson–Dirichlet law mathsf {PD(1)}, as the size of the system grows to infinity. In the case of transient dimensions, dge 3, the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.

Scaling Limits in Models of Statistical Mechanics

The emphasis of the workshop was on the deep relations between, on the one hand, recent advances in probabilistic investigation of statistical mechanical models and spatial stochastic processes and, on the other hand, rigorous field-theoretic and analytic methods of mathematical physics. There were 52 participants, including 6 postdocs and graduate students, working in diverse intertwining areas of probability, statistical mechanics and field theory. Specific topics addressed during the 24 talks include: Universality and critical phenomena, disordered models, Gaussian free field (GFF), stochastic representation of classical and quantum-mechanical models and related random interchange and permutation processes, random planar graphs and unimodular planar maps, random walks on critical graphs and the Alexander-Orbach conjecture, reinforced random walks and non-linear -models, metastability, aging, equilibrium and dynamics for continuum particles with hard core interactions, non-equilibrium dynamics and Toom’s interfaces.

The random interchange process on the hypercube

We prove the occurrence of a phase transition accompanied by the emergence of cycles of diverging lengths in the random interchange process on the hypercube.

A numerical study of the 3D random interchange and random loop models

We have studied numerically the random interchange model and related loop models on the three-dimensional cubic lattice. We have determined the transition time for the occurrence of long loops. The joint distribution of the lengths of long loops is Poisson–Dirichlet with parameter 1 or .

VEPs elicited by local correlations and global symmetry: Characteristics and interactions

Psychophysical and fMRI studies have indicated that visual processing of global symmetry has distinctive scaling properties, and proceeds more slowly than analysis of contrast, spatial frequency, and texture. We therefore undertook a visual evoked potential (VEP) study to directly compare the dynamics of symmetry and texture processing, and to determine the extent to which they interact. Stimuli consisted of interchange between structured and random black-and-white checkerboard stimuli. For symmetry, structured stimuli were colored with 2-fold symmetry (horizontal or vertical mirror), 4-fold symmetry (both mirror axes), and 8-fold symmetry (oblique mirror axes added). For texture, structured stimuli were colored according to the “even” isodipole texture [Julesz, B., Gilbert, E. N., & Victor, J. D. (1978). Visual discrimination of textures with identical third-order statistics. Biological Cybernetics, 31, 137–140]. Thus, all stimuli had the same contrast, and check size, but differed substantially in correlation structure. To separate components of the VEP related to symmetry and texture from components that could be generated by local luminance and contrast changes, we extracted the odd-harmonic components of the VEP (recorded at C z – O z , C z – O 1, C z – O 2, C z – P z ) elicited by structured–random interchange. Responses to symmetry were largest for the 8-fold patterns, and progressively smaller for 4-fold, vertical, and horizontal symmetry patterns. Eightfold patterns were therefore used in the remainder of the study. The symmetry response is shifted to larger checks and lower temporal frequencies compared to the response to texture, and its temporal tuning is broader. Processing of symmetry makes use of neural mechanisms with larger receptive fields, and slower, more sustained temporal tuning characteristics than those involved in the analysis of texture. Sparse stimuli were used to dissociate check size and check density. VEP responses to sparse symmetry stimuli showed that there is no difference between first- and second-order symmetry for densities less than 12.5%. We discuss these findings in relation to local and global visual processes.

Modeling Stacking Faults in the Layered Molecular-Based Magnets AMIIFe(C2O4)3 {MII=Mn, Fe; A=Organic Cation}

The influence of stacking faults in the crystal structures of molecular-based magnets N(n-CnH2n+1)4MIIFe′III(C2O4)3 {MII=Mn, Fe; n=3–5} is discussed on the basis of X-ray powder diffraction profiles. Polycrystalline samples of N(n-C5H11)4 compounds are monophasic with orthorhombic unit cells very similar to those of N(n-C5H11)4MnIIFeIII(C2O4)3 determined by single crystal data. However, polycrystalline N(n-C3H7)4 and N(n-C4H9)4 compounds are biphasic, containing varying contributions from R3c and P6(3) phases. Structures containing stacking faults were modeled by assuming random interchange between R3c and P6(3) stacking within the crystallites. X-ray diffraction profiles calculated from the model structures exhibit broadening similar to that seen in the experimental profiles. The presence of stacking faults explains the difficulty in obtaining macroscopic single crystals of the N(n-C3H7)4 and N(n-C4H9)4 compounds.

Ab-initio calculations of the GMR-effect in Fe/V multilayers

In a self-consistent semi-empirical numerical approach based on ab-initio calculations for small samples, we evaluate the GMR effect for disordered (0 0 1)-(3-Fe/3-V) ∞ multilayers by means of a Kubo formalism. We consider four different types of disorder arrangements: In case (i) and (ii), the disorder consists in the random interchange of some Fe and V atoms, respectively, at interface layers; in case (iii) in the formation of small groups of three substitutional Fe atoms in a V interface layer and a similar V group in a Fe layer at a different interface; and for case (iv) in the substitution of some V atoms in the innermost V layers by Fe. For cases (i) and (ii), depending on the distribution of the impurities, the GMR effect is enhanced or reduced by increasing disorder; in case (iii) the GMR effect is highest, whereas finally, in case (iv), a negative GMR is obtained (`inverse GMR').

X-ray Analysis of the Kinetics of Transesterification in Blends of Wholly Aromatic Thermotropic Copolyesters

The kinetics of the transesterification reaction occurring in blends of wholly aromatic copolyesters have been investigated by simulation of the changes that occur in the X-ray scattering data as a result of the interchange. The X-ray fiber diagrams of copolyesters prepared from p-hydroxybenzoic acid (HBA) and 2-hydroxy-6-naphthoic acid (HNA) contain nonperiodic diffraction maxima along the chain axis direction that vary in position (scattering angle) with the monomer composition and are due to the structural correlations in chains of completely random comonomer sequence. Melt blends of two different compositions initially show the scattering maxima of both components, but these shift as the specimen is held in the melt, until they merge to give the diffraction data characteristic of the intermediate composition. We have simulated the changes in these data by modeling the changes in the correlation function that occur as a result of the transesterification reaction. We find that the data are best reproduced using a completely random interchange model for a completely compatible blend. Simulations based on preferential reactivities of one or other of the monomers or partial segregation of one phase gave significantly inferior agreement with the positions of the observed diffraction maxima. The statistical modeling allows us to derive kinetic parameters for the reaction, for which we obtain an activation energy of 142 kJ/mol and a rate constant of 2.3 × 10-4 s-1 at 315 °C. Compared to the equivalent data for isotropic polyesters, the activation energy compares very well, but the rate constant is 1 order of magnitude lower, perhaps due to the ordering in the thermotropic melt.

Electrostatic partitioning in slit pores by Gibbs ensemble Monte Carlo simulation

AbstractEquilibrium partitioning of spherical solutes between slit pores and bulk solution is investigated by the Gibbs ensemble Monte Carlo method. Two types of perturbatins are performed in this simulation: a random displacement of solutes that ensures equilibrium within both bulk and pore regions, and random interchanges of solutes that equalize the interaction potentials between the two regions. To study the effects of electrostatic interactions, interaction energies between the solutes and pore walls and between pairs of solutes are evaluated by using a singularity method. Partition coefficients calculated for neutral solutes, which experience purely steric interactions, increase with increasing solute concentration and agree well with existing theoretical results. For pores and solutes of like charge, results for the limit of infinitely dilute solute concentration show a sharp decline in partition coefficient with decreasing ionic strength of solution. As the solute concentration increases, the interplay of solute–wall and solute–solute interactions becomes increasingly important, and the partition coefficient increases accordingly. The density profiles indicate unambiguously that, whether solutes and proes are uncharged or of like charge, solute–solute interaction promotes enhanced concentrations near the wall, causing the partition coefficient to increase. Even at solute concentrations as low as 5%, effects of solute–solute interactions caused by electrostatic charge can more than compensate for sphere–wall repulsive interactions, indicating that concentration effects should be considered at least as important as electrostatic effects in partitioning phenomena.

B2 and B32 ordering transformations of equiatomic bcc alloys with ballistic and thermal atom movements

We used Monte Carlo simulations to study the kinetics and steady states of B2 and B32 ordering in an equiatomic binary bcc alloy having both thermal and ballistic atom movements. Atom movements occurred by a vacancy mechanism, where first-neighbor atoms exchanged sites with the vacancy. In the thermodynamic case, where this jump probability was set by a Boltzmann factor alone, we found the formation of large amounts of transient B2 (B32) during disorder → B32 (B2) order transformations. The transient order developed in distinct regions, but vanished completely when equilibrium was attained. Ballistic jumps were included as random interchanges of the vacancy with one of its neighboring atoms. Even with small fractions of ballistic jumps, there were large changes in the transient states and steady states of order in the alloy. A kinetic explanation is proposed, in which the presence of ballistic jumps contributed a greater amount of internal energy to the B2 phase than to the B32 phase when there was a strong asymmetry in the second-nearest neighbor pair potentials (VAA2 ↗ VBB2). A two-phase coexistence in the steady state was explained by fluctuations in the local density of ballistic jumps.

Can frustration in a binary mixture lead to a freezing-in of concentration fluctuations?

Abstract We have examined the possibility that in a binary mixture, on the border of ordering and clustering tendencies, concentration fluctuations can ‘freeze in’ at a definite temperature, T g, independent of the cooling rate. The resulting state, a concentration glass, would be analogous to the spin‐glass state where the spin fluctuations ‘freeze in’ as a consequence of an exchange interaction which is randomly ferro–and antiferromagnetic. We develop the relevant theory on the basis of a lattice–gas model for binary mixtures with random interchange energies, using a mean–field approximation. For the very simple model considered we propose a phase diagram and discuss some of the physical manifestations of the new state. We suggest that the glass–forming ability of many metallic alloys near specific concentrations may be due to the occurrence of the above ‘freezing in’ phenomenon.

In vivo knee stability. A quantitative assessment using an instrumented clinical testing apparatus.

Twenty-eight male and twenty-one female subjects with no history of previous injury to their knees were examined using a newly developed clinical testing apparatus designed to record anterior-posterior tibial force versus displacement and varus-valgus moment versus angulation during manual manipulation of the knee. Joint stiffness and laxity were measured from test tracings made with the knee muscles relaxed and tensed. Agreement between these measurements and those made previously on thirty-five fresh cadaver knee specimens was very good. Anterior-posterior laxity averaged 3.7 millimeters in full extension, 5.5 in 20 degrees of flexion, and 4.8 millimeters in 90 degrees of flexion, while the mean varus-valgus laxity was 6.7 degrees in full extension. The common clinical assumption that normal right-left differences are negligible was found to be invalid. Individual right-left differences averaged 26 to 35 per cent for laxity and 19 to 24 per cent for stiffness. There was no discernible tendency for one knee to be more stable than the other; random interchanges of relative stability between the right and left knees were observed for each individual at different knee positions. When requested to tense the knee muscles, these subjects were able to increase their knee stiffness an average of two to four times while knee laxity was reduced to 25 to 50 per cent of the normal value.

Clusters, metastability, and nucleation: Kinetics of first-order phase transitions

We describe and interpret computer simulations of the time evolution of a binary alloy on a cubic lattice, with nearest neighbor interactions favoring like pairs of atoms. Initially the atoms are arranged at random; the time evolution proceeds by random interchanges of nearest neighbor pairs, using probabilities compatible with the equilibrium Gibbs distribution at temperatureT. For temperatures 0.59Tc, 0.81 Tc, and 0.89T c, with densityρ of A atoms equal to that in the B-rich phase at coexistence, the density C1 of clusters ofl A atoms approximately satisfies the following empirical formulas: C1 ≈w(1 −ρ)3 andC 1, ≈ (1 −ρ)4Q1w1 (2 ⩽l ⩽ 10). Herew is a parameter and we defineQ l =∑ K e −βE(K) , where the sum goes over all translationally nonequivalentl-particle clusters andE(K) is the energy of formation of the clusterK. Forl > 10,Q 1 is not known exactly; so we use an extrapolation formulaQ l ≈Aw −l l −α exp(−bl σ), wherew s is the value ofw at coexistence. The same formula (withw > w s) also fits the observed values of C, (for small values ofl) at densities greater than the coexistence density (forT=0.59Tc): When the supersaturation is small, the simulations show apparently metastable states, a theoretical estimate of whose lifetime is compatible with the observations. For higher supersaturation the system is observed to undergo a slow process of segregation into two coexisting phases (andw therefore changes slowly with time). These results may be interpreted as a more quantitative formulation (and confirmation) of ideas used in standard nucleation theory. No evidence for a “spinodal” transition is found.

Electron microscopic study of defects in V 2MoO 8

Planar defects in the structure of V 2MoO 8 and superstructure due to the ordering of the displacements of cations are examined by electron microscopy and electron diffraction. It is found that deficiency in stoichiometry is due to the existence of “Wadsley defects” in the crystal, and the monoclinic symmetry of the structure is confirmed by the observation of (100) twins. Finally it is found that the disordering in the 〈100〉 direction is due to the random interchange of two unit supercells with dimensions 10Å and 20Å respectively.

Scrambling of fluoro-, methoxyl, dimethylamino-, and methyl groups with chlorine atoms and of methoxyl with dimethylamino-groups on germanium

Equilibrium with respect to exchange of substituents on germanium has been studied by hydrogen nuclear magnetic resonance (n.m.r.) in the systems (1) Ge(OMe)4 with GeCl4, (2) Ge(NMe2)4 with GeCl4, (3) Ge(OMe)4 with Ge(NMe2)4, and (4) GeMe4 with GelCl4. Fluorine n.m.r. was used to study the system (5) GeF4 with GeCl4. The largest deviations from statistically random interchange of the substituents were observed in system (2), with the mixed species being extremely stable. Similar non-random behaviour, with the mixed species being preferred, was found in systems (1) and (4). Exchange of chlorine for fluorine on germanium [system (5)] showed only a small deviation from randomness. The rates of exchange varied greatly, going from a reaction period of many days at 300° for system (4) and many hours at 87° for system (5) to fractions of one second at 25° for systems (1) and (2).

Investigation of polymerization mechanisms by means of molecular weight distributions of reaction products

AbstractThe study of MWD functions gives insight into the mechanism of polymerization and polycondensation reactions. Especially all kinds of side reactions, by altering the already formed macromolecules, give characteristic changes in the MWD. Some special cases are discussed. For polycondensation products, two types of MWD are found. In the early stages of the reaction before the monomer has been consumed, broad distribution curves of the Flory shape are observed. In the later stages, after the polymer has been heated, quite narrow distributions of the Gaussian type are found. The theoretical explanation involves consideration of scission and macromolecular combination reactions, which result in random interchange of monomer units among macromolecules. For products of addition polymerization of dienes in the presence of a catalytic agent butyllithium, exceedingly narrow distribution functions are observed, the polymer being almost homogeneous. This is the result of simultaneous initiation and further growth of all the chains at the same rate during the entire reaction period without termination. For radical polymerization, sometimes simple unimodal, broad distribution curves are found and the termination constants (recombination and disproportionation rates) can be calculated from the curves. In special cases, however, the distribution functions are complicated and consist of many sharp maxima. It is suggested that secondary reactions of chain scission and macroradical recombination occur, which results in a sharpening of the MWD. Subsequent crosslinking of macromolecules from such sharpened distributions gives doubling, tripling, etc. of average molecular weights. This is the explanation for the polymodal sharpened distribution curves observed.

Random Interchange of Circuits with Applications to Counting Rate Meters and Function Generators

Two identical linear circuits are excited by generators furnishing equal and opposite voltages. Homologous portions of the two circuits are interchanged at random times with, on the average, r interchanges per unit time. Because of these interchanges, the currents in the remaining portions are irregular. A theorem is derived that permits one to calculate the statistical averages of these currents. It states that one may disregard the interchanging; instead, one merely replaces the complex frequency s by s+2r in the circuit functions [e.g., the impedances Z(s)] of the portions that were being interchanged. On the basis of this theorem one may design counting rate meters with nonlinear (e.g., logarithmic) scale, useful in reactor instrumentation, and function generators for functions of practical importance, such as the logarithmic and exponential functions, and powers with arbitrary exponent. The nature of the function depends only on linear passive circuit elements, such as resistors and capacitors.

Problems and practice in the production of wave-guide transmission systems

The magnetron, considered as an h.f. generator, exhibits peculiar forms of frequency instability which lead to specific wave-guide performance requirements. Some considerations of reflections in wave guides and of probability functions show the difficulties in the way of meeting these requirements. In order to achieve consistent success, the random choice or interchange of wave-guide sections or components must be abandoned in favour of selection or production matching when the number of sections or reflections is more than two or three. The two procedures may be complementary or alternative.The realization of these features has led to an intensive development of general wave-guide manufacturing practice and has guided the measurement of sequences of different wave-guide assemblies which give a very satisfactory performance for magnetron working.