numerical results on a hypercubic cell spin glass model the authors study a spin glass hypercubic cell in d dimensions i e hypercubic lattice with l 2 where l is the lattice size for different values of d by means of monte carlo simulation in the limit d to infinity this model converges to the sk model as well as for hypercubic lattices they confirm the usual interpretation of the spin glass phase since this model is more similar to hypercubic lattices than to the infinite range model these results suggest that broken replica symmetry is useful to study finite dimensional hypercubic lattices above a certain critical dimension

critical finite size corrections for the sherrington kirkpatrick spin glass starting from the effective lagrangian of the sherrington kirkpatrick spin glass near the critical temperature the authors investigate the finite size effects for the free energy and the propagator above and at tc 1 they perform the calculation going up to the three loop diagrams for t to tc the cubic vertex only gives the leading term the inclusion of the quartic vertex gives the following terms the numerical coefficients in these terms are established by the different methods of the partial summation of perturbation series the comparison with numerical simulations is done

graphical representations for ising systems in external fields a graphical representation based on duplication is developed that is suitable for the study of ising systems in external fields two independent replicas of the ising system in the same field are treated as a single four state ashkin teller model bonds in the graphical representation connect the ashkin teller spins for ferromagnetic systems it is proved that ordering is characterized by percolation in this representation the representation leads immediately to cluster algorithms some applications along these lines are discussed

on the gibbs states of the noncritical potts model on z 2 we prove that all gibbs states of the q state nearest neighbor potts model on mathbb z 2 below the critical temperature are convex combinations of the q pure phases in particular they are all translation invariant to achieve this goal we consider such models in large finite boxes with arbitrary boundary condition and prove that the center of the box lies deeply inside a pure phase with high probability our estimate of the finite volume error term is of essentially optimal order which stems from the brownian scaling of fluctuating interfaces the results hold at any supercritical value of the inverse temperature beta beta c q log left 1 sqrt q right

multiple solutions for 2mth order sturm liouville boundary value problems growth conditions are imposed on f such that the following boundary value problem 1 my 2m f t y i 1y 2i 0 i 1y 2i 1 0 i 1y 2i 1 i 1y 2i 1 1 0 0 i m 1 has an arbitrary number of positive solutions

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

monte carlo simulations of the ising spin glass on lattices with finite connectivity the authors have simulated the ising spin glass model on a random lattice with a finite average coordination number and also on the bethe lattice with various different boundary conditions in particular they have calculated the overlap function p q for two independent samples for the random lattice the results are consistent with a spin glass transition above which p q converges to a dirac delta function for large n number of sites and below which p q has in addition a long tail similar to previous results obtained for the infinite range model for the bethe lattice they obtain results in the interior by discarding the two outer shells of the cayley tree when calculating the thermal averages for fixed uncorrelated boundary conditions p q seems to converge to a delta function even below the spin glass transition whereas on a closed lattice correlated boundary conditions p q has a long tail similar to its behaviour in the random lattice case

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

an intermediate phase with slow decay of correlations in one dimensional 1 x y 2 percolation ising and potts models we rigorously establish the existence of an intermediate ordered phase in one dimensional 1 x y 2 percolation ising and potts models the ising model truncated two point function has a power law decay exponent which ranges from its low and high temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary similar results are obtained for percolation and potts models

ising models on hyperbolic graphs we consider ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph v f where v is the number of neighbors of each vertex and f is the number of sides of each face let tc be the critical temperature and t c sup t tc f 2 where f is the free boundary condition b c gibbs state is the plus b c gibbs state and is the minus b c gibbs state we prove that if the hyperbolic graph is self dual i e v f or if v is sufficiently large how large depends on f e g v 35 suffices for any f 3 and v 17 suffices for any f 17 then 0 t c tc in contrast with that t c tc for ising models on the hypercubic lattice zd with d 2 a result due to lebowitz 22 while whenever t t c f 2 the last result is an improvement in comparison with the analogous statement in refs 28 and 33 in which it was only proved that f 2 when t t c and it remains to show in both papers that f 2 whenever t t c therefore t c and tc divide 0 into three intervals 0 t c t c tc and tc in which but f 2 and f 2 and respectively

one dimensional models with 1 r2 interactions the critical behaviour of a general discrete one dimensional model with inverse square interactions is discussed using renormalisation group methods

the approximate inverse in action ii convergence and stability the approximate inverse is a scheme for constructing stable inversion formulas for operator equations originally it is defined on l2 spaces in the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations indeed we show convergence if the discretization parameter tends to zero further we prove stability that is we show the regularization property finally we apply the results to the filtered backprojection algorithm in 2d tomography to obtain convergence rates

localization in the ground state of the ising model with a random transverse field we study the zero temperature behavior of the ising model in the presence of a random transverse field the hamiltonian is given by h j sum limits left langle x y right rangle sigma 3 x sigma 3 y sum limits x h x sigma 1 x wherej 0 x y zd 1 3 are the usual pauli spin 1 2 matrices andh h x x zd are independent identically distributed random variables we consider the ground state correlation function 3 x 3 y and prove 1 letd be arbitrary for anym 0 andj sufficiently small we have for almost every choice of the random transverse fieldh and everyx zd that left langle sigma 3 x sigma 3 y right rangle leqq c x h e m left x y right for ally zd withcxh 2 letd 2 ifj is sufficiently large then for almost every choice of the random transverse fieldh the model exhibits long range order i e mathop overline lim limits left y right to infty left langle sigma 3 x sigma 3 y right rangle 0 for anyx zd

existence of multiple symmetric positive solutions of higher order lidstone problems this paper gives the sufficient conditions of the existence of at least three symmetric positive solutions for one type of higher order autonomous lidstone problem by applying the five functionals fixed point theorem the analogous result for higher order nonautonomous singular lidstone problem is also proved here

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

long range one dimensional potts models a cluster mean field and extrapolation approach we obtain approximations for the critical temperature tc of q state ferromagnetic potts models on one dimensional lattices with algebraically decaying ferromagnetic pair interactions i e decaying as 1 r with 1 2 initially we use a cluster mean field method to get approximations to the critical temperature and we look at increasingly large clusters this gives us a sequence of ever more accurate approximate values which can be used as input to various extrapolation algorithms we see that as the interaction decreases more slowly and the value of q increases we obtain more and more accurate tc estimates our best estimates we believe to be correct to four figure accuracy

a finite dimensional realization of the mollifier method for compact operator equations we introduce and analyze a stable procedure for the approximation of f if where f is the least residual norm solution of the minimal norm of the ill posed equation af g with compact operator a x y between hilbert spaces and x has some smoothness assumption our method is based on a finite number of singular values of a and some finite rank operators our results are in a more general setting than the one considered by rieder and schuster 2000 and nair and lal 2003 with special reference to the mollifier method and it is also applicable under fewer smoothness assumptions on

one dimensional 1 j i s percolation models the existence of a transition for s 2 consider a one dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi j j 1 ifpj is fixed forj 2 and mathop lim limits j to infty j2pj 1 then unoriented percolation occurs forp1 sufficiently close to 1 this result analogous to the existence of spontaneous magnetization in long range one dimensional ising models is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size oriented percolation is shown to occur forp1 close to 1 if mathop lim limits j to infty jspj 0 for somes 2 analogous results are valid for one dimensional site bond percolation models

positive solutions for higher order lidstone boundary value problems a new approach via sperner s lemma abstract consider the higher order nonlinear scalar differential equation where associated to the lidstone boundary conditions existence of a solution of boundary value problems bvp 1 2 such that are given under superlinear or sublinear growth in f similarly existence for the bvp 1 3 under the same assumptions is proved such that we further prove analogous results for the case when i e derivatives of the obtaining solution satisfy inverse inequalities the approach is based on an analysis of the corresponding vector field on the face plane and the well known from combinatorial analysis knaster kuratowski mazurkiewicz s principle or as it is known sperner s lemma

random cluster analysis of a class of binary lattice gases we introduce a class of binary lattice gases which can be viewed as a lattice analogue of the continuum widom rowlinson model and which also is related to the beach model of burton and steif this new model is shown to exhibit phase transition for large particle intensities stochastic monotonicity results of varying strength are derived in various parts of the parameter space the main tool is a random cluster representation of the model analogous to the fortuin kasteleyn representation of the potts model

quasi additive estimates on the hamiltonian for the one dimensional long range ising model in this work we study the problem of getting quasi additive bounds for the hamiltonian of the long range ising model when the two body interaction term decays proportionally to 1d2 0 1 we revisit the paper by cassandro et al j math phys 46 053305 2005 where they extend to the case 0 ln3ln2 1 the result of the existence of a phase transition by using a peierls argument given by frohlich and spencer commun math phys 84 87 101 1982 for 0 the main arguments of cassandro et al j math phys 46 053305 2005 are based in a quasi additive decomposition of the hamiltonian in terms of hierarchical structures called triangles and contours which are related to the original definition of contours introduced by frohlich and spencer commun math phys 84 87 101 1982 in this work we study the existence of a quasi additive decomposition of the hamiltonian in terms of the contours defined in the work of cassandro et al j math phys 46 053305 2005 the most relevant r

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

the approximate inverse in action with an application to computerized tomography the approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind yet in some applications the computation of a crucial ingredient the reconstruction kernel is time consuming and instable it may even happen that the kernel does not exist for a particular semidiscrete system to cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator the results are applied to the reconstruction problem in 2d computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm

translation invariant gibbs states in the q state potts model we describe the set of all translation invariant gibbs states in theq state potts model for the case ofq large enough and the other parameters to be arbitrary

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

decay of correlations in the one dimensional ising model withj ij i j 2 a low temperature expansion is constructed for the one dimensional ising model with hamiltonian h sum limits i j left i j right 2 left 1 sigma i sigma j right it is shown that the two point function i j obeys upper and lower bounds of the formf i j 2 for inverse temperature sufficiently large

a phase transition in random coin tossing suppose that a coin with bias theta is tossed at renewal times of a renewal process and a fair coin is tossed at all other times let mu theta be the distribution ofthe observed sequence of coin tosses and let u n denote the chance of a renewal at time n harris and keane showed that if sum n 1 infty u n 2 infty then mu theta and mu 0 are singular while if sum n 1 infty u n 2 infty and theta is small enough then mu theta is absolutely continuous with respect to mu 0 they conjectured that absolute continuity should not depend on theta but only on the square summability of u n we show that in fact the power law governing the decay of u n is crucial and for some renewal sequences u n there is a phase transition at a critical parameter theta c in 0 1 for theta theta c the measures mu theta and mu 0 are mutually absolutely continuous but for theta theta c they are singular we also prove that when u n o n 1 the measures mu theta for theta in 1 1 are all mutually absolutely continuous

surface order large deviations for ising potts and percolation models we derive uniform surface order large deviation estimates for the block magnetization in finite volume ising or potts models with plus or free or a combination of both boundary conditions in the phase coexistence regime ford 3 the results are valid up to a limit of slab thresholds conjectured to agree with the critical temperature our arguments are based on the renormalization of the random cluster model withq 1 andd 3 and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability the results are new even for the case of independent percolation q 1 as a byproduct of our methods we obtain further results in the fk model concerning semicontinuity inp andq of the percolation probability the second largest cluster in a box and the tail of the finite cluster size distribution

non glassy ground state in a long range antiferromagnetic frustrated model in the hypercubic cell abstract we analyzed the statistical mechanics of a long range antiferromagnetic model defined on a d dimensional hypercube both at zero and finite temperatures the associated hamiltonian is derived from a recently proposed complexity measure of boolean functions in the context of neural networks learning processes we show that depending on the value of d the system either presents a low temperature antiferromagnetic stable phase or the global antiferromagnetic order disappears at any temperature in the last case the ground state is an infinitely degenerated non glassy one composed by two equal size anti aligned antiferromagnetic domains we also present some results for the ferromagnetic version of the model

potts models and random cluster processes with many body interactions known differential inequalities for certain ferromagnetic potts models with pair interactions may be extended to potts models with many body interactions as a major application of such differential inequalities we obtain necessary and sufficient conditions on the set of interactions of such a potts model in order that its critical point be astrictly monotonic function of the strengths of interactions the method yields some ancillary information concerning the equality of certain critical exponents for potts models this amounts to a small amount of rigorous universality these results are achieved in the context of a fortuin kasteleyn representation of potts models with many body interactions for such a potts model the corresponding random cluster process is a random hypergraph

self avoiding walks in wedges the authors consider the number of self avoiding walks confined to a subset zd f of the d dimensional hypercubic lattice zd such that the coordinates x1 x2 xd of each vertex in the walk satisfy x1 or 0 and 0 or xk or fk x1 for k 2 3 d they show that if fk x to infinity as x to infinity the connective constant of walks in zd f is identical to the convective constant of walks in zd they also explore conditions on fk which lead to a smaller connective constant for walks in zd f and in particular consider walks between two parallel d 1 dimensional hyperplanes finally they contrast some of these results with recent work by grimmett on percolation on subsets of the square lattice

absence of long range order in one dimensional spin systems for a one dimensional ising model with interaction energy e left mu right sum limits 1 leqslant i j leqslant n j j i mu iota mu j left j k geqslant 0 mu iota pm 1 right it is proved that there is no long range order at any temperature when s n sum limits k 1 n kj left k right o left log n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 right the same result is shown to hold for the corresponding plane rotator model when s n o left left log n mathord left vphantom log n log log n right kern nulldelimiterspace log log n right right

replica symmetry breaking of the ising spin glass with finite connectivity the authors consider the ising spin glass on a lattice with finite connectivity m 1 using the recent method of large connectivity expansions the free energies at finite temperatures are evaluated numerically at first step replica symmetry breaking rsb the 1 m expansion at finite temperature diverges as t0 as in the replica symmetric case but is well behaved for a larger range the 1 m expansion at zero temperature is also calculated at higher steps of rsb an expression for the free energy expansion is derived for an arbitrary step of rsb up to order 1 m explicit numerical values at second step rsb are obtained from the results they speculate that the divergence in the finite temperature expansion might disappear for the exact infinite step rsb solution they also compare their results with the simulation results of graph bipartitioning

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

the influence of oppositely classified examples on the generalization complexity of boolean functions in this paper we analyze boolean functions using a recently proposed measure of their complexity this complexity measure motivated by the aim of relating the complexity of the functions with the generalization ability that can be obtained when the functions are implemented in feed forward neural networks is the sum of a number of components we concentrate on the case in which we use the first two of these components the first is related to the average sensitivity of the function and the second is in a sense a measure of the randomness or lack of structure of the function in this paper we investigate the importance of using the second term in the complexity measure and we consider to what extent these two terms suffice as an indicator of how difficult it is to learn a boolean function we also explore the existence of very complex boolean functions considering in particular the symmetric boolean functions

a remark on the notion of robust phase transitions we point out that the high qpotts model on a regular lattice at its transition temperature provides an example of a nonrobust in the sense recently proposed by pemantle and steif phase transition

four dimensional ising spin glass scaling within the spin glass phase we investigate the nature of the spin glass phase in the four dimensional king spin glass we study he probability dishibutions of overlaps of energy overlaps and ullrmetricity for several sizes we discover the existence of finite size scaling in the fails of the first and second of these this allows us to exmct mame exponents within the spin glass phase we also perform studies on quenched thermalization of large samples our results together with previous work favour the mean field picfure of the spin glass phase

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

solutions of two point boundary value problems for even order differential equations abstract the existence of solutions of the two point boundary value problems consisting of the even order differential equations x 2 n t f t x t x t x 2 n 2 t r t 0 t 1 and the boundary value conditions i x 2 i 0 i x 2 i 1 0 0 i x 2 i 1 i x 2 i 1 1 0 i 0 1 n 1 is studied sufficient conditions for the existence of at least one solution of above bvps are established it is interesting that the nonlinearity f in the equation depends on all lower derivatives especially odd order derivatives and the growth conditions imposed on f are allowed to be super linear the degrees of phases variables are allowed to be greater than 1 if it is a polynomial the results are different from known ones since we do not apply the green s functions of the corresponding problem and the method to obtain a priori bounds of solutions is different from known ones examples that cannot be solved by known results are given to illustrate our theorems