zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean curvature motion we consider the glauber dynamics for the ising model with boundary conditions at zero temperature or at a temperature that goes to zero with the system size hence the quotation marks in the title in dimension d 3 we prove that an initial domain of linear size l of spins disappears within a time which is at most l2 log l c and at least l2 c log l for some c 0 the proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe on large time scales the evolution of the interface between and domains the analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice our result apart from the spurious logarithmic factors is the first rigorous confirmation of the lifshitz law const l2 conjectured on heuristic grounds 8 13 in dimension d 2 can be shown to be of order l2 without logarithmic corrections the upper bound was proven in 6 and here we provide the lower bound for d 2 we also prove that the spectral gap of the generator behaves like for l large as conjectured in 2 2011 wiley periodicals inc

can extra updates delay mixing we consider glauber dynamics starting from an extremal configuration in a monotone spin system and show that interjecting extra updates cannot increase the expected hamming distance or the total variation distance to the stationary distribution we deduce that for monotone markov random fields when block dynamics contracts a hamming metric single site dynamics mixes in o n log n steps on an n vertex graph in particular our result completes work of kenyon mossel and peres concerning glauber dynamics for the ising model on trees our approach also shows that on bipartite graphs alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n vertex graph our result is especially effective in comparing block and single site dynamics it has already been used in works of martinelli toninelli sinclair mossel sly ding lubetzky and peres in various combinations

optimal strong stationary times for random walks on the chambers of a hyperplane arrangement this paper studies markov chains on the chambers of real hyperplane arrangements a model that generalizes famous examples such as the tsetlin library and riffle shuffles we discuss cutoff for the tsetlin library for general weights and we give an exact formula for the separation distance for the hyperplane arrangement walk we introduce lower bounds which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions using similar techniques we also prove a uniform lower bound for the mixing time of glauber dynamics on a monotone system

ordinal rankings on measures annihilating thin sets we assign a countable ordinal number to each probability measure which annihilates all h sets the descriptive set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non borel in addition it allows us to quantify the failure of rajchman s conjecture similar results are obtained for measures annihilating dirichlet sets

the droplet in the tube a case of phase transition in the canonical ensemble we consider the 2 dimensional ising model with ferromagnetic nearest neighbour interaction at inverse temperature lets n t be the total magnetization inside ann n square box per be the gibbs state in with periodic b c andm be the spontaneous magnetization we show the existence of the limit psi varrho mathop lim limits n to infty left frac 1 beta n right ln mu lambda per s n n varrho for m provided is large enough it turns out that the quantity is closely related to the wulf construction and the dependence of the function on is singular

principal series of finite subgroups of su 3 we attempt to give a complete description of the exceptional finite subgroups 36 3 72 3 and 216 3 of su 3 with the aim to make them amenable to model building for fermion masses and mixing the information on these groups which we derive contains conjugacy classes proper normal subgroups irreducible representations character tables and tensor products of their three dimensional irreducible representations we show that for these three exceptional groups usage of their principal series i e ascending chains of normal subgroups greatly facilitates the computations and illuminates the relationship between the groups as a preparation and testing ground for the usage of principal series we study first the dihedral like groups 27 and 54 because both are members of the principal series of the three groups discussed in the paper

borel classes and closed games wadge type and hurewicz type results for each countable ordinal and pair ao al of disjoint analytic subsets of 2 we define a closed game j aos a1 and a complete iit subset ht of 2 such that i a winning strategy for player i constructs a 24 set separating ao from a1 and ii a winning strategy for player ii constructs a continuous map p 2 a u a1 with p l ao h applications of this construction include a proof in second order arithmetics of the statement every iit non ss set is ii complete an extension to all levels of a theorem of hurewicz about 22 sets a new proof of results of kunugui novikov bourgain and the authors on borel sets with sections of given class extensions of results of stern and kechris our results are valid in arbitrary polish spaces and for the classes in lavrentieff s and wadge s hierarchies introduction let e 2 be the usual cantor space and following the standard modern terminology of addison let e and fit denote respectively the borel subsets of e of additive and multiplicative borel class starting with e open and ti closed the aim of this paper is to characterize given a set a c 2 and one of the borel classes say r whether a is in r or not by a characterization we have in mind the finding of some mathematical object associated with a and which positively witnesses that a e r or that a 0 r let r be the class e t if the set a is in e a natural candidate for such a witness is simply one particular construction of a as a e set a list of sequences of open sets together with instructions on the operations to be performed to get a in t steps the problem of how to find such constructions which is deeply connected to the structural properties of sets in the plane with e sections has been extensively investigated in papers by novikov n and kunugui kun for 1 see also dellacherie de saint raymond sr 1 for t 2 bourgain b 1 b 2 and louveau lo 1 for t 3 louveau lo 2 for t 3 in case a is known to be borel in case a ranges over larger classes the problem is studied in papers by stern st kechris ke and louveau lo 3 received by the editors december 26 1985 and in revised form november 19 1986 these results have been presented at the annual meeting of the american mathematical society new orleans january 1986 in a special session on determinacy and large cardinals 1980 mclthemcltics subject clclssificcltion 1985 revision primary o2k30 04a15 28a05 secondary 54h05 26a21

comments on the classification of the finite subgroups of su 3 many finite subgroups of su 3 are commonly used in particle physics the classification of the finite subgroups of su 3 began with the work of h f blichfeldt at the beginning of the 20th century in blichfeldt s work the two series c and d of finite subgroups of su 3 are defined while the group series 3n2 and 6n2 which are subseries of c and d respectively have been intensively studied there is not much knowledge about the group series c and d in this work we show that c and d have the structures and respectively furthermore we show that while the c groups can be interpreted as irreducible representations of 3n2 the d groups can in general not be interpreted as irreducible representations of 6n2

how quickly can we sample a uniform domino tiling of the 2l times 2l square via glauber dynamics the prototypical problem we study here is the following given a 2l times 2l square there are approximately exp 4kl 2 pi ways to tile it with dominos i e with horizontal or vertical 2 times 1 rectangles where k approx 0 916 is catalan s constant kasteleyn in physica 27 1209 1225 1961 temperley and fisher in phil mag 6 1061 1063 1997 a conceptually simple even if computationally not the most efficient way of sampling uniformly one among so many tilings is to introduce a markov chain algorithm glauber dynamics where with rate 1 two adjacent horizontal dominos are flipped to vertical dominos or vice versa the unique invariant measure is the uniform one and a classical question levin et al in am math soc 2009 luby et al siam j comput 31 167 192 2001 wilson in ann appl probab 14 274 325 2004 is to estimate the time t mathrm mix it takes to approach equilibrium i e the running time of the algorithm luby et al in siam j comput 31 167 192 2001 and randall and tetali in j math phys 41 3 1598 1614 2000 fast mixing was proven t mathrm mix o l c for some finite c here we go much beyond and show that c l 2 le t mathrm mix le l 2 o 1 our result applies to rather general domain shapes not just the 2l times 2l square provided that the typical height function associated to the tiling is macroscopically planar in the large l limit under the uniform measure this is the case for instance for the temperley type boundary conditions considered in kenyon ann probab 28 759 795 2000 also our method extends to some other types of tilings of the plane for instance the tilings associated to dimer coverings of the hexagon or square hexagon lattices

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

convexity properties of the surface tension and equilibrium crystals we study the thermodynamic limit of the orientation dependent surface tension under general conditions which we show to hold true for a large class of lattice systems we prove that the limit exists and that it satisfies some convexity properties related to the pyramidal inequality introduced by r l dobrushin and s b shlosman 1 we discuss some consequences of these results for the equilibrium crystal shape

random sampling for the monomer dimer model on a lattice in the monomer dimer model on a graph each matching collection of non overlapping edges m has a probability proportional to lambda m where lambda 0 is the model parameter and m denotes the number of edges in m an approximate random sample from the monomer dimer distribution can be obtained by running an appropriate markov chain each step of which involves an elementary local change in the configuration sufficiently long jerrum and sinclair have shown roughly speaking that for an arbitrary graph and fixed lambda and varepsilon the maximal allowed variational distance from the desired distribution o lambda 2 e steps suffice where e is the number of edges and lambda the number of vertices of the graph for sufficiently nice subgraphs e g cubes of the d dimensional cubic lattice we give an explicit recipe to generate approximate random samples in asymptotically siginificantly fewer steps namely for fixed lambda and varepsilon o lambda ln lambda 2

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

simple finite non abelian flavor groups the recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with two and three dimensional irreducible representations this paper details the mathematics of the two finite simple groups with such representations the icosahedral group a5 a subgroup of so 3 and psl2 7 a subgroup of su 3

on the mixing time of the 2d stochastic ising model with plus boundary conditions at low temperature we consider the glauber dynamics for the 2d ising model in a box of side l at inverse temperature and random boundary conditions whose distribution p either stochastically dominates the extremal plus phase hence the quotation marks in the title or is stochastically dominated by the extremal minus phase a particular case is when p is concentrated on the homogeneous configuration identically equal to equal to for large enough we show that for any varepsilon 0 there exists c c beta varepsilon such that the corresponding mixing time tmix satisfies rm lim l to infty bf p left t rm mix ge rm exp cl varepsilon right 0 in the non random case or this implies that t rm mix le rm exp cl varepsilon the same bound holds when the boundary conditions are all on three sides and all on the remaining one the result although still very far from the expected lifshitz behavior tmix o l2 considerably improves upon the previous known estimates of the form t rm mix le rm exp c l frac 12 varepsilon the techniques are based on induction over length scales combined with a judicious use of the so called censoring inequality of y peres and p winkler which in a sense allows us to guide the dynamics to its equilibrium measure

on the characterization of the su 3 subgroups of type c and d we investigate the two classes of finite subgroups of su 3 that are called type c and d in the book of miller blichfeldt and dickson we present two theorems which fully determine the form of the generators in a suitable basis after exploring further properties of these groups we are able to construct a complete list of infinite series in which these groups are arranged for type c there are infinitely many series whereas for type d there are only two explicit examples of these series are presented which illustrate the general results

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

bound on the mass gap for finite volume stochastic ising models at low temperature we consider a sequence of finite volume z d d 2 reversible stochastic ising models in the low temperature regime and having invariant measures satisfying free boundary conditions we show that associated with the models are random hitting times whose expectations regarded as a function of grow exponentially in d 1 d moreover the mass gaps for the models shrink exponentially fast in d 1 d a geometrical lemma is employed in the analysis which states that if a peierls contour is sufficiently small relative to the faces of then the fraction of the contour tangent to the faces is less than a constant smaller than one

glauber dynamics on trees boundary conditions and mixing time we give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the glauber dynamics in the so called bethe approximation specifically we show that the spectral gap and the log sobolev constant of the glauber dynamics for the ising model on an n vertex regular tree with boundary are bounded below by a constant independent of n at all temperatures and all external fields this implies that the mixing time is o logn in contrast to the free boundary case where it is not bounded by any fixed polynomial at low temperatures in addition our methods yield simpler proofs and stronger results for the spectral gap and log sobolev constant in the regime where the mixing time is insensitive to the boundary condition our techniques also apply to a much wider class of models including those with hard core constraints like the antiferromagnetic potts model at zero temperature proper colorings and the hard core lattice gas independent sets

the structure of σ ideals of compact sets motivated by problems in certain areas of analysis like measure theory and harmonic analysis where ideals of compact sets are encountered very often as notions of small or exceptional sets we undertake in this paper a descriptive set theoretic study of ideals of compact sets in compact metrizable spaces in the first part we study the complexity of such ideals showing that the structural condition of being a ideal imposes severe definability restrictions a typical instance is the dichotomy theorem which states that ideals which are analytic or coanalytic must be actually either complete coanalytic or else g in the second part we discuss generators or as we call them here bases for ideals and in particular the problem of existence of borel bases for coanalytic non borel ideals we derive here a criterion for the nonexistence of such bases which has several applications finally in the third part we develop the connections of the definability properties of ideals with other structural properties like the countable chain condition etc

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

the measure of non normal sets it is well known that almost every number in 0 1 is normal in base 2 in the sense of lebesgue measure kahane and salem asked whether the same is true with respect to any borel measure whose fourier stieltjes coefficients vanish at infinity in other words whether the set of non normal numbers is a set of uniqueness in the wide sense we show that this is not the case in fact we give best possible conditions on the rate of decay of hat mu n in order that almost every number be normal the techniques include on the one hand probability measures with respect to which the binary digits in 0 1 are independent only by blocks rather than individually and on the other hand the strong law of large numbers for weakly correlated random variables

on the parametrization of certain finite groups and their representations we discuss a convenient way of parametrizing certain finite groups they comprise groups with abelian normal subgroups and central extensions thereof they include the groups dn t o and their double groups as well as the su 3 subgroups tn 3n2 6n2 72 216 36 and their z3 extensions the method allows for a rapid calculation of representations and coupling coefficients in particular it solves the multiplicity problem of nonsimply reducible groups we illustrate the method treating the hessian group 216 and its central extension

on the finite subgroups of u 3 of order smaller than 512 we use the smallgroups library to find the finite subgroups of u 3 of order smaller than 512 which possess a faithful three dimensional irreducible representation from the resulting list of groups we extract those groups that cannot be written as direct products with cyclic groups these groups are the basic building blocks for models based on finite subgroups of u 3 all resulting finite subgroups of su 3 can be identified using the well known list of finite subgroups of su 3 derived by miller blichfeldt and dickson at the beginning of the 20th century furthermore we prove a theorem which allows us to construct infinite series of finite subgroups of u 3 from a special type of finite subgroups of u 3 this theorem is used to construct some new series of finite subgroups of u 3 the first members of these series can be found in the derived list of finite subgroups of u 3 of order smaller than 512 in the last part of this work we analyze some interesting finite subgroups of u 3 especially the group which is closely related to the important su 3 subgroup s4

generation of universal linear optics by any beam splitter universal quantum computers have proved difficult to build as one response researchers have proposed limited models of quantum computation which might be easier to realize three examples are the one clean qubit model of knill and laflamme 15 the commuting hamiltonians model of bremner jozsa and shepherd 3 and the boson sampling model of aaronson and arkhipov 1 none of these models are known or believed to be capable of universal quantum computation or depending on modeling details even universal classical computation but all of them can perform certain estimation or sampling tasks for which no polynomial time classical algorithm is known one obvious way to define a limited model of quantum computation is to restrict the set of allowed gates however almost every gate set is universal 17 and so are most natural gate sets for example controlled not together with any real one qubit gate that does not square to the identity is universal 21 as a result very few nontrivial examples of non universal gate sets are known all known non universal gate sets on o 1 qubits such as the clifford group 8 are efficiently classically simulable if the input and measurement outcomes both belong to an appropriately chosen qubit basis1 as a result it is tempting to conjecture that there does not exist such an intermediate gate set or more precisely that any gate set on o 1 qubits is either efficiently classical simulable with appropriate input and output states or else universal for quantum computing strikingly this dichotomy conjecture remains open even for the special case of 1and 2 qubit gates we regard proving or disproving the conjecture as an important open problem for quantum computing theory in this paper we prove a related conjecture in the quantum linear optics model in quantum optics the hilbert space is not built up as a tensor product of qubits instead it s built up as a direct sum of optical modes an optical gate is then just a unitary transformation that acts nontrivially ono 1 of the modes and as the identity on the rest whenever we have a k mode gate we assume that we can apply it to any subset of k modes in any order as often as desired the most common optical gates considered are beamsplitters which act on two modes and correspond to a 2 2 unitary matrix with determinant 1 2 and phaseshifters which act on one mode and simply apply a phase ei note that any unitary transformation acting on the one photon hilbert space automatically gets lifted by homomorphism to a unitary transformation acting on the hilbert space of n photons furthermore every element of the n photon linear optical group that is every n photon unitary transformation achievable using linear optics arises in this way see 1 sec iii for details of course if n 2 then there are also n photon unitaries that cannot be achieved linearoptically that is the n photon linear optical group is a proper subgroup of the full unitary group on the n photon hilbert space

when is the range of a multiplier on a banach algebra closed in this paper we prove the theorem let a be a banach algebra with a bounded approximate identity bai such that every proper closed ideal of a is contained in a proper closed ideal with a bai then a multiplier t a a has a closed range iff t factors as a product of an idempotent multiplier and an invertible multiplier

glauber dynamics on the cycle is monotone we study heat bath glauber dynamics for the ferromagnetic ising model on a finite cycle a graph where every vertex has degree two we prove that the relaxation time 2 is an increasing function of any of the couplings j xy we also prove some further inequalities and obtain exact asymptotics for 2 at low temperatures

glauber dynamics for the mean field ising model cut off critical power law and metastability we study the glauber dynamics for the ising model on the complete graph also known as the curie weiss model for 1 we study metastability in particular we show that the glauber dynamics restricted to states of non negative magnetization has mixing time o n log n

the flavor group δ 6n2 many non abelian finite subgroups of su 3 have been used to explain the flavor structure of the standard model in order to systematize and classify successful models a detailed knowledge of their mathematical structure is necessary in this paper we shall therefore look closely at the series of finite non abelian groups known as 6n2 its smallest members being s3 n 1 and s4 n 2 for arbitrary n we determine the conjugacy classes the irreducible representations the kronecker products as well as the clebsch gordan coefficients

finite flavour groups of fermions we present an overview of the theory of finite groups with regard to their application as flavour symmetries in particle physics in a general part we discuss useful theorems concerning group structure conjugacy classes representations and character tables in a specialized part we attempt to give a fairly comprehensive review of finite subgroups of so 3 and su 3 in which we apply and illustrate the general theory moreover we also provide a concise description of the symmetric and alternating groups and comment on the relationship between finite subgroups of u 3 and finite subgroups of su 3 although in this review we give a detailed description of a wide range of finite groups the main focus is on the methods which allow the exploration of their different aspects

the freedman group a physical interpretation for the su 3 subgroup d 18 1 1 2 1 1 of order 648 we study a subgroup fr 162 4 of su 3 of order 648 which is an extension of d 9 1 1 2 1 1 and whose generators arise from anyonic systems we show that this group is isomorphic to a semi direct product with respect to conjugation and we give a presentation of the group we show that the group d 18 1 1 2 1 1 from the series d in the existing classification for finite su 3 subgroups is also isomorphic to a semi direct product with respect to conjugation we next exhibit the isomorphism between both groups we prove that fr 162 4 is not isomorphic to the exceptional su 3 subgroup 216 3 of the same order 648 we further prove that the only su 3 finite subgroups from the 1916 classification by blichfeldt or its extended version in which fr 162 4 may be isomorphic belong to the d series finally we show that fr 162 4 and d 18 1 1 2 1 1 are both conjugate under the orthogonal matrix which we provide

an abstract form of a theorem of helson and applications to sets of synthesis and sets of uniqueness let e be a compact perfect subset of the real line r such that the restriction of the fourier transform a a e from l1 r into c e is onto helson proved that then for m e lim y y 0 is possible only if 0 in this paper we present an abstract version of this theorem of helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness

exact thresholds for ising gibbs samplers on general graphs we establish tight results for rapid mixing of gibbs samplers for the ferromagnetic ising model on general graphs we show that if d 1 tanh 1 then there exists a constant c such that the discrete time mixing time of gibbs samplers for the ferromagnetic ising model on any graph of n vertices and maximal degree d where all interactions are bounded by and arbitrary external fields are bounded by cnlogn moreover the spectral gap is uniformly bounded away from 0 for all such graphs as well as for infinite graphs of maximal degree d we further show that when dtanh 1 with high probability over the erd s renyi random graph g n d n it holds that the mixing time of gibbs samplers is n1 1 loglogn both results are tight as it is known that the mixing time for random regular and erd s renyi random graphs is with high probability exponential in n when d 1 tanh 1 and dtanh 1 respectively to our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs moreover our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees

finite subgroups of su 3 we present a new class of finite subgroups of su 3 of the form zm s zn semidirect product we also apply the methods used to investigate semidirect products to the known su 3 subgroups 3n2 and 6n2 and give analytic formulae for representations characters and clebsch gordan coefficients

representations and clebsch gordan coefficients of z metacyclic groups we investigate finite groups of the form zmszn and give all irreducible representations and clebsch gordan coefficients in analytic form two subclasses are considered which seem to be inportant for applications the m metacyclic groups which are important for spin systems and the k metacyclic groups which are the smallest finite groups that have an irreducible representation of dimension p 1 where p is prime

on boolean algebras related to σ ideals generated by compact sets abstract let h g be hausdorff measures on compact metric spaces x y and let b o r x j h and b o r y j 0 g be the boolean algebras of borel sets modulo ideals of borel sets that can be covered by countably many compact sets of finite h measure or g measure null respectively we shall show that if h is not finite and one of the quotient boolean algebras embeds densely in the other then for some borel b with h b h takes on borel subsets of b only values 0 or this is a particular instance of some more general results concerning boolean algebras b o r x j where j is a ideal of borel sets generated by compact sets

exponential relaxation of glauber dynamics with some special boundary conditions we consider attractive finite range glauber dynamics and show that if a certain mixing condition is satisfied then the system evolving on arbitrary subsets of the lattice with appropriate boundary conditions converges to equilibrium exponentially fast in the uniform sense uniformly over the subsets of the lattice this result applies for instance to the ferromagnetic nearest neighbor ising model in the so called basuev region where complete analyticity is expected to fail

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

m sets and measures the measures pt to be investigated are defined on the torus t and satisfy lim suplii n i 0 lyons 8 recently proved the striking theorem that every measure t of this type must have positive mass in some closed set of type uo later lyons asked whether uo could be improved to u and provided 9 ingenious counter examples for the classical examples of u sets definitions and elementary results in harmonic analysis may be found in works cited a few lines below we prove the following

some circumstances where extra updates can delay mixing peres and winkler proved a censoring inequality for glauber dynamics on monotone spins systems such as the ising model specifically if starting from a constant spin configuration the spins are updated at some sequence of sites then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution we show by means of simple counterexamples that the analogous statements fail for glauber dynamics on proper colorings of a graph and for lazy transpositions on permutations answering two questions of peres it is not known whether the censoring property holds in other natural settings such as the potts model

some comments on finite subgroups of su 3 a recently published set of finite subgroups of su 3 is shown to contain some groups which are not subgroups of su 3 the others are subgroups of one of the dihedral like family of su 3 subgroups of order 3n2 some comments are made also on the structure of other finite subgroups previously listed

dynamics of 2 1 dimensional sos surfaces above a wall slow mixing induced by entropic repulsion we study the glauber dynamics for the 2 1 d solid on solid model above a hard wall and below a far away ceiling on an l times l box of z 2 with zero boundary conditions at large inverse temperature beta it was shown by bricmont el mellouki and frohlich 1986 that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height h asymp 1 beta log l as an essential step in understanding the effect of entropic repulsion on the glauber dynamics we determine the equilibrium height h to within an additive constant h 1 4 beta log l o 1 we then show that starting from zero initial conditions the surface rises to its final height h through a sequence of metastable transitions between consecutive levels the time for a transition from height h ah a in 0 1 to height h 1 is roughly exp c l a for some constant c 0 in particular the mixing time of the dynamics is exponentially large in l i e t mix geq e c l we also provide the matching upper bound t mix leq e c l requiring a challenging analysis of the statistics of height contours at low temperature and new coupling ideas and techniques finally to emphasize the role of entropic repulsion we show that without a floor constraint at height zero the mixing time is no longer exponentially large in l