type cotype and twisted sums induced by complex interpolation abstract this paper deals with extensions or twisted sums of banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension the results are presented in the context of interpolation of families of banach spaces and are applied to the study of submodules of schatten classes we also obtain nontrivial extensions of spaces without the cap which also fail the cap

pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

duality and twisted sums of banach spaces we give a negative answer to the three space problem for the banach space properties to be complemented in a dual space and to be isomorphic to a dual space solving a problem of vogt lectures held in the functional analysis seminar dusseldorf wuppertal jan feb 1987 and another posed by diaz et al in bull polish acad sci math 40 1992 221 224 precisely we construct an exact sequence 0 l2 d w 0 in which w is a separable dual and d is not isomorphic to a dual space we also show the existence of an exact sequence 0 y x z 0 where both y and z are dual spaces and x is not even complemented in its bidual to do that we perform a study of the basic questions on duality from the point of view of exact sequences of banach spaces

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

spacing between phase shifts in a simple scattering problem we prove a scattering theoretical version of the berry tabor conjecture for an almost every surface in a class of cylindrical surfaces of revolution the large energy limit of the pair correlation measure of the quantum phase shifts is poisson that is it is given by the uniform measure

lattice point problems and values of quadratic forms for d dimensional ellipsoids e with d 5 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order mathcal o r d 2 for general ellipsoids and up to an error of order o rd 2 for irrational ones the estimate refines earlier bounds of the same order for dimensions d 9 as an application a conjecture of davenport and lewis about the shrinking of gaps between large consecutive values of q m m d of positive definite irrational quadratic forms q of dimension d 5 is proved finally we provide explicit bounds for errors in terms of certain minkowski minima of convex bodies related to these quadratic forms

non perturbative criteria for gibbsian uniqueness for spin systems with an internal symmetry we provide sufficient conditions for unicity of the gibbs state and or complete analyticity by comparison to random cluster models

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

agreement percolation and phase coexistence in some gibbs systems we extend some relations between percolation and the dependence of gibbs states on boundary conditions known for ising ferromagnets to other systems and investigate their general validity percolation is defined in terms of the agreement of a configuration with one of the ground states of the system this extension is studied via examples and counterexamples including the antiferromagnetic ising and hard core models on bipartite lattices potts models and many layered ising and continuum widom rowlinson models in particular our results on the hard square lattice model make rigorous observations made by hu and mak on the basis of computer simulations moreover we observe that the naturally defined clusters of the widom rowlinson model play for the wr model itself the same role that the clusters of the fortuin kasteleyn measure play for the ferromagnetic potts models the phase transition and percolation in this system can be mapped into the corresponding liquid vapor transition of a one component fluid

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

a theory of complex interpolation for families of banach spaces abstract a detailed development is given of a theory of complex interpolation for families of banach spaces which extends the well known theory for pairs of spaces

forbidden gap argument for phase transitions proved by means of chessboard estimates chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest in this note we show that the method not only produces a point in the phase diagram where more than one gibbs states coexist but that it can also be used to rule out the existence of shift ergodic states that differ significantly from those proved to exist for models depending on a parameter say the temperature this shows that the values of the conjugate thermodynamic quantity the energy inside the transitional gap are forbidden in all shift ergodic gibbs states we point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria

diophantine approximation on rational quadrics we compute the hausdorff dimension of sets of very well approximable vectors on rational quadrics we use ubiquitous systems and the geometry of locally symmetric spaces as a byproduct we obtain the hausdorff dimension of the set of rays with a fixed maximal singular direction which move away into one end of a locally symmetric space at linear depth infinitely many times

disagreement percolation in the study of markov fields recently one of the authors van den berg has obtained a uniqueness condition for gibbs measures in terms of disagreement percolation involving two independent realizations in the present paper we study the dependence of markov fields on boundary conditions by taking a more suitable coupling this coupling leads to a new uniqueness condition which improves the one mentioned above we also compare it with the dobrushin uniqueness condition in the case of the ising model our coupling shares certain properties with the fortuin kasteleyn representation it gives an explicit expression of the boundary effect on a certain vertex in terms of percolation like probabilities

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

central limit theorems for diophantine approximants in this paper we study certain counting functions which represent the numbers of solutions of systems of linear inequalities arising in the theory of diophantine approximation we develop a method that allows us to explain the random like behavior that these functions exhibit and prove a central limit theorem for them our approach is based on a quantitative study of higher order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of siegel transforms

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

analogs of wiener s ergodic theorems for semisimple lie groups ii let g be a connected finite center simple lie group of real rank one and let k be a maximal compact subgroup of g let a at i t c r be a 1 parameter subgroup of hyperbolic translations such that g ka k is a cartan decomposition denote by ot the probability measure on g given by ot mk 6at mk where mk is the normalized haar measure on k let it denote the probablity measure with density vol bt 1xb g where bt g d gk k t and d denotes the canonical g invariant riemannian metric on g k let x b a be a standard borel probability space on which g acts measurably and ergodically by measure preserving transformations let 7r g iso lp x be the associated isometric representation and let 7r ut and 7r 3t denote the operators canonically associated with ct and it let n g dimr g k we prove maximal and pointwise ergodic theorems for the ball averages it on simple real rank one lie groups which are analogous to n wiener s maximal and pointwise ergodic theorems for ball averages on in the general case of ball averages on semisimple groups is completed in m n s our main result is the following

differentials of complex interpolation processes for kothe function spaces we continue the study of centralizers on kothe function spaces and the commutator estimates they generate see 29 our main result is that if x is a super reflexive kothe function space then for every real centralizer on x there is a complex interpolation scale of kothe function spaces through x inducing as a derivative up to equivalence and a scalar multiple thus in a loose sense all real centralizers can be identified with derivatives of complex interpolation processes we apply our ideas in an appendix to show for example that there is a twisted sum of two hilbert spaces which fails to be a umd space

isolation equidistribution and orbit closures for the sl 2 r action on moduli space we prove results about orbit closures and equidistribution for the sl 2 r action on the moduli space of compact riemann surfaces which are analogous to the theory of unipotent ows the proofs of the main theorems rely on the measure classication theorem of the rst two authors and a certain isolation property of closed sl 2 r invariant manifolds developed in this paper

counting lattice points for a locally compact second countable group g and a lattice subgroup we give an explicit quantitative solution of the lattice point counting problem in general domains in g provided that i g has finite upper local dimension and the domains satisfy a basic regularity condition ii the mean ergodic theorem for the action of g on g holds with a rate of convergence the error term we establish matches the best current result for balls in symmetric spaces of simple higher rank lie groups but holds in much greater generality a significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap in particular the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property we discuss a number of applications including counting lattice points in general domains in semisimple s algebraic groups counting rational points on group varieties with respect to a height function and quantitative angular or conical equidistribution of lattice points in symmetric spaces and in affine symmetric varieties we note that the mean ergodic theorems which we establish are based on spectral methods including the spectral transfer principle and the kunze stein phenomenon we formulate and prove appropriate analogues of both of these results in the set up of adele groups and they constitute a necessary step in our proof of quantitative results in counting rational points

the analysis of the widom rowlinson model by stochastic geometric methods we study the continuum widom rowlinson model of interpenetrating spheres using a new geometric representation for this system we provide a simple percolation based proof of the phase transition we also use this representation to formulate the problem and prove the existence of an interfacial tension between coexisting phases finally we ascribe geometric i e probabilistic significance to the correlation functions which allows us to prove the existence of a sharp correlation length in the single phase regime

values of indefinite quadratic forms at integral points and flows on spaces of lattices this mostly expository paper centers on recently proved conjectures in two areas a a conjecture of a oppenheim on the values of real indefinite quadratic forms at integral points b conjectures of dani raghunathan and margulis on closures of orbits in spaces of lattices such as sl r sl z at first sight a belongs to analytic number theory and b belongs to ergodic and lie theory and they seem to be quite unrelated they are discussed together here because of a very interesting connection between the two pointed out by m s raghunathan namely a special case of b yields a proof of a the first main goal of this talk is to describe the oppenheim conjecture and various refinements and to derive them from one statement about closures of orbits in the space of unimodular lattices in r3 see proposition 2 in 2 3 in 3 and 4 we put this statement in context and describe more general conjectures and results on orbit closures and invariant probability measures on quotients of lie groups by discrete subgroups 5 gives some brief comments on the proofs and further developments 6 7 and 8 are devoted to the so called s arithmetic setting where we consider products of real and p adic groups 6 is concerned with a generalized oppenheim conjecture 7 with a generalization of the orbit closure theorem proved by m ratner r8 and 8 with applications to quadratic forms since the subject matter of that last section has not been so far discussed elsewhere we take this opportunity to present proofs obtained jointly with g prasad finally 9 gives the proof of a lemma on symmetric simple lie algebras a special case of which is used in 8 i am glad to thank m ratner and g prasad for a number of remarks on and corrections to a preliminary version of this paper thanks to which many typos and inaccuracies have been eliminated

pointwise equidistribution with an error rate and with respect to unbounded functions consider g mathrm sl d mathbb r and gamma mathrm sl d mathbb z it was recently shown by the second named author shi pointwise equidistribution for one parameter diagonalizable group action on homogeneous space preprint arxiv 1405 2067 2014 that for some diagonal subgroups g t subset g and unipotent subgroups u subset g g t trajectories of almost all points on all u orbits on g gamma are equidistributed with respect to continuous compactly supported functions varphi on g gamma in this paper we strengthen this result in two directions by exhibiting an error rate of equidistribution when varphi is smooth and compactly supported and by proving equidistribution with respect to certain unbounded functions namely siegel transforms of riemann integrable functions on mathbb r d for the first part we use a method based on effective double equidistribution of g t translates of u orbits which generalizes the main result of kleinbock and margulis on effective equidistribution of expanding translates of certain orbits in the space of lattices number theory analysis and geometry 385 396 2012 the second part is based on schmidt s results on counting of lattice points number theoretic consequences involving spiraling of lattice approximations extending recent work of athreya et al j lond math soc 91 2 383 404 2015 are derived using the equidistribution result

effective counting on translation surfaces abstract we prove an effective version of a celebrated result of eskin and masur for any sl 2 r invariant locus l of translation surfaces there exists 0 such that for almost every translation surface in l the number of saddle connections with holonomy vector of length at most t grows like c t 2 o t 2 we also provide effective versions of counting in sectors and in ellipses

a small scale density of states formula let m g be a c compact riemann manifold with classical hamiltonian h c t m assume that the corresponding quantization p1 op h is quantum completely integrable we establish an microlocal weyl law on short spectral intervals of size 2 e e 0 for various families of operators p1u u i containing p1 both in the mean and pointwise a e for u i the microlocalization refers to a small tubular neighbourhood of a non degenerate stable periodic bicharacteristic t m 0

on uniformly finitely extensible banach spaces abstract we continue the study of uniformly finitely extensible banach spaces in short ufo initiated in moreno and plichko 2009 39 and castillo and plichko 2010 18 we show that they have the uniform approximation property of pelczynski and rosenthal and are compactly extensible we will also consider their connection with the automorphic space problem of lindenstrauss and rosenthal do there exist automorphic spaces other than c 0 i and l 2 i showing that a space all whose subspaces are ufo must be automorphic when it is hereditarily indecomposable hi and a hilbert space when it is either locally minimal or isomorphic to its square we will finally show that most hi among them the super reflexive hi space constructed by ferenczi and asymptotically l 2 spaces in the literature cannot be automorphic

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

the asymmetric random cluster model and comparison of ising and potts models we introduce the asymmetric random cluster or arc model which is a graphical representation of the potts lattice gas and establish its basic properties the arc model allows a rich variety of comparisons in the fkg sense between models with different parameter values we give for example values h for which the 0 s configuration in the potts lattice gas is dominated by the configuration of the h ising model the potts model with possibly an external field applied to one of the spins is a special case of the potts lattice gas which allows our comparisons to yield rigorous bounds on the critical temperatures of potts models for example we obtain 0 571 1 exp c 0 600 for the 9 state potts model on the hexagonal lattice another comparison bounds the movement of the critical line when a small potts interaction is added to a lattice gas which otherwise has only interparticle attraction arc models can also be compared to related models such as the partial fk model obtained by deleting a fraction of the nonsingleton clusters from a realization of the fortuin kasteleyn random cluster model this comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the potts model

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

orbits of discrete subgroups on a symmetric space and the furstenberg boundary let x be a symmetric space of noncompact type and let be a lattice in the isometry group of x we study the distribution of orbits of acting on the symmetric space x and its geometric boundary x generalizing the main equidistribution result of margulis s thesis m theorem 6 to higher rank symmetric spaces more precisely for any y x and b x we investigate the distribution of the set y b 1 in x x it is proved in particular that the orbits of in the furstenberg boundary are equidistributed and that the orbits of in x are equidistributed in sectors defined with respect to a cartan decomposition our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces which we obtain using shah s result s corollary 1 2 based on ratner s measure classification theorem r1 theorem 1

pair correlation of sums of rationals with bounded height for each positive integer q let f q denote the farey sequence of order q we prove the existence of the pair correlation measure associated to the sum f q f q modulo 1 as q tends to infinity and compute the corresponding limiting pair correlation function

quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms a quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms is proved we also give an application to eigenvalue spacing on flat 2 tori with aharonov bohm flux

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

convergence to equilibrium of random ising models in the griffiths phase we consider glauber type dynamics for disordered ising spin systems with nearest neighbor pair interactions in the griffiths phase we prove that in a nontrivial portion of the griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1 this condition has in turn been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential and that the average over the disorder of the time autocorrelation function goes to equilibrium faster than exp k log t d d 1 we then show that for the diluted ising model these upper bounds are optimal

twisting non commutative lp spaces abstract the paper makes the first steps into the study of extensions twisted sums of noncommutative l p spaces regarded as banach modules over the underlying von neumann algebra m our approach combines kalton s description of extensions by centralizers these are certain maps which are in general neither linear nor bounded with a general principle due to rochberg and weiss saying that whenever one finds a given banach space y as an intermediate space in a complex interpolation scale one automatically gets a self extension 0 y x y 0 for semifinite algebras considering l p l p m as an interpolation space between m and its predual m one arrives at a certain self extension of l p that is a kind of noncommutative kalton peck space and carries a natural bimodule structure some interesting properties of these spaces are presented for general algebras including those of type iii the interpolation mechanism produces two rather than one extensions of one sided modules one of left modules and the other of right modules whether or not one may find nontrivial self extensions of bimodules in all cases is left open

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

correlations of eigenvalues on multi dimensional flat tori we show that for almost all k dimensional flat tori the eigenvalues of the laplacian follow a uniform distribution with respect to their 2 3 and k 2 level correlations

derivatives of analytic families of banach spaces the theory of complex interpolation of banach spaces and operators which was developed by calderon lions and s g krein and extended by us and others centers on the use of the maximum principle for analytic functions in this paper we investigate the significance in interpolation theory of estimates for derivatives of analytic functions we obtain norm estimates for certain linear and non linear commutators and obtain new classical interpolation theorems the paper has four sections in the first section we show how thorin s proof of the riesz thorin theorem can be extended to give an estimate for a non linear commutator and how analogous computations can be done in families of interpolation spaces the second section introduces an extension of the complex interpolation theory which is designed to incorporate in a systematic way the type of estimates obtained in the first section in the third section the details of the abstract theory of the second section are filled in for several standard examples including lp spaces and weighted lp spaces a brief final section contains some general comments and observations

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