current fluctuations of the stationary asep and six vertex model our results in this article are twofold first we consider current fluctuations of the stationary asymmetric simple exclusion process asep run for some long time t and show that they are of order t1 3 along a characteristic line upon scaling by t1 3 we establish that these fluctuations converge to the long time height fluctuations of the stationary kardar parisi zhang kpz equation that is to the baik rains distribution this result has long been predicted under the context of kpz universality and in particular extends upon a number of results in the field including the work of ferrari and spohn from 2005 when they established the same result for the tasep and the work of balazs and seppalainen from 2010 when they established the t1 3 scaling for the general asep second we introduce a class of translation invariant gibbs measures that characterizes a one parameter family of slopes for an arbitrary ferroelectric symmetric six vertex model this family of slopes corresponds to what is known as the conical singularity or tricritical point in the free energy profile for the ferroelectric six vertex model we consider fluctuations of the height function of this model on a large grid of size t and show that they too are of order t1 3 along a certain characteristic line this confirms a prediction of bukman and shore from 1995 stating that the ferroelectric six vertex model should exhibit kpz growth at the conical singularity upon scaling the height fluctuations by t1 3 we again recover the baik rains distribution in the large t limit recasting this statement in terms of the asymmetric stochastic six vertex model confirms a prediction of gwa and spohn from 1992

on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

a reexamination of diffusion estimators with applications to financial model validation time homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables recently stanton proposed drift and diffusion estimators based on a higher order approximation scheme and kernel regression method he claimed that higher order approximations must outperform lower order approximations and concluded nonlinearity in the instantaneous return function of short term interest rates to examine the impact of higher order approximations we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators we show that these estimators will reduce the numerical approximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximation simulation studies also confirm our asymptotic results this variance inflation problem arises not only from nonparametric fitting but also from parametric fitting stanton s work also postulates the interesting qu

the arctic circle boundary and the airy process we prove that the appropriately rescaled boundary of the north polar region in the aztec diamond converges to the airy process the proof uses certain determinantal point processes given by the extended krawtchouk kernel we also prove a version of propp s conjecture concerning the structure of the tiling at the center of the aztec diamond 1 introduction and results domino tilings of the aztec diamond were introduced in 8 9 asymptotic properties of random domino tilings of the aztec diamond have been studied in 5 12 15 in particular in 12 the existence of the so called arctic circle was proved the arctic circle is the asymptotic boundary of the disordered so called temperate region of the tiling outside this boundary the tiling forms a completely regular brick wall pattern the methods in 12 combined with the results in 13 show that the fluctuations of the point of intersection of the boundary of the temperate region with a line converge to the tracy widom distribution of random matrix theory in this paper we extend this result to show that the fluctuations of the boundary around the arctic circle converges to the airy process introduced in 23 the paper is a continuation of the approach used in 14 and 15 where certain point processes with determinantal correlation functions 24 and the krawtchouk ensemble were used we will use the general techniques developed in 16 and investigate an extended point process which also has determinantal correlation functions given by a kernel which we call the extended krawtchouk kernel the aztec diamond an of order n is the union of all lattice squares m m 1 l l 1 m l 2 z that lie inside the region x1 y1 x1 y1

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

limit theorems for functionals of mixing processes with applications to u statistics and dimension estimation in this paper we develop a general approach for investigating the asymptotic distribution of functional xn f zn k k z of absolutely regular stochastic processes zn n z such functional occur naturally as orbits of chaotic dynamical systems and thus our results can be used to study probabilistic aspects of dynamical systems we first prove some moment inequalities that are analogous to those for mixing sequences with their help several limit theorems can be proved in a rather straightforward manner we illustrate this by re proving a central limit theorem of ibragimov and linnik then we apply our techniques to u statistics matrix equation with symmetric kernel h r r r we prove a law of large numbers extending results of aaronson burton dehling gilat hill and weiss for absolutely regular processes we also prove a central limit theorem under a different set of conditions than the known results of denker and keller as our main application we establish an invariance principle for u processes un h h indexed by some class of functions we finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system

a consistent nonparametric test for serial independence abstract we propose a nonparametric test for serial independence against serial dependence of fixed order that is consistent against all such alternatives the conditions required are weak the asymptotic distribution under the null is 1 2 and the test works regardless of the underlying distribution also included are a nuisance parameter result monte carlo simulations a theoretical efficiency study an empirical example and a review of possible extensions in addition we derive a similar consistent test for lack of serial dependence of order one against serial dependence of order one

exact solution of the six vertex model with domain wall boundary conditions disordered phase the six vertex model or the square ice model with domain wall boundary conditions dwbc has been introduced and solved for finite n by korepin and izergin the solution is based on the yang baxter equations and it represents the free energy in terms of an n n hankel determinant paul zinn justin observed that the izergin korepin formula can be re expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction we use this observation to obtain the large n asymptotics of the six vertex model with dwbc in the disordered phase the solution is based on the riemann hilbert approach and the deift zhou nonlinear steepest descent method as was noticed by kuperberg the problem of enumeration of alternating sign matrices the asm problem is a special case of the six vertex model we compare the obtained exact solution of the six vertex model with known exact results for the 1 2 and 3 enumerations of asms and also with the exact solution on the so called free fermion line we prove the conjecture of zinn justin that the partition function of the six vertex model with dwbc has the asymptotics z n sim cn kappa e n 2 f as n and we find the exact value of the exponent

on u statistics and v mise statistics for weakly dependent processes some probabilistic limit theorems for hoeffding s u statistic 13 and v mises functional are established when the underlying processes are not necessarily independent we consider absolutely regular processes 24 and processes x n n 1 which are uniformly mixing 14 as well as their time reversal x n n 1 called uniformly mixing in both directions of time many authors have weakened the hypothesis of independence in statistical limit theorems and considered m dependent markov or weakly dependent processes in particular for u statistics under weak dependence sen 22 has considered mixing processes and derived a central limit theorem and a law of the iterated logarithm while yoshihara 26 proved central limit theorems and a s results in the absolutely regular and uniformly mixing case here we extend these results considerably and prove central limit theorems and their rate of convergence in the prohorov metric and a berry esseen type theorem functional central limit theorems and a s approximation by a brownian motion extensions to multisample versions and other extensions are briefly discussed

limit theorems for sums of weakly dependent banach space valued random variables we prove an estimate for the prohorov distance in the central limit theorem for strong mixing banach space valued random variables using a recent variant of an approximation theorem of berkes and philipp 1979 we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term t tfrac 1 2 gamma for strong mixing sequences we prove a strong invariance principle with error term o t log logt 1 2

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

the dilogarithmic central extension of the ptolemy thompson group via the kashaev quantization abstract quantization of universal teichmuller space provides projective representations of the ptolemy thompson group which is isomorphic to the thompson group t this yields certain central extensions of t by z called dilogarithmic central extensions we compute a presentation of the dilogarithmic central extension t kash of t resulting from the kashaev quantization and show that it corresponds to 6 times the euler class in h 2 t z meanwhile the braided ptolemy thompson groups t t of funar kapoudjian are extensions of t by the infinite braid group b and by abelianizing the kernel b one constructs central extensions t ab t ab of t by z which are of topological nature we show t kash t ab our result is analogous to that of funar and sergiescu who computed a presentation of another dilogarithmic central extension t cf of t resulting from the chekhov fock goncharov quantization and thus showed that it corresponds to 12 times the euler class and that t cf t ab in addition we suggest a natural relationship between the two quantizations in the level of projective representations

statistical regularities of self intersection counts for geodesics on negatively curved surfaces let s be a compact surface with constant negative curvature 1 from among all closed geodesics on of length t choose one at random and letnt be the number of its self intersections we prove that for a certain constant 0 the random variable nt t t has a limit distribution as t we conjecture that for surfaces of variable negative curvature the order of magnitude of typical variations is t rather than t we also prove analogous results for generic geodesics that is geodesics whose initial tangent vectors are chosen randomly according to normalized liouville measure date november 10 2011 1991 mathematics subject classification primary 57m05 secondary 53c22 37d40

exact solutions for kpz type growth processes random matrices and equilibrium shapes of crystals abstract three models from statistical physics can be analyzed by employing space time determinantal processes 1 crystal facets in particular the statistical properties of the facet edge and equivalently tilings of the plane 2 one dimensional growth processes in the kardar parisi zhang universality class and directed last passage percolation 3 random matrices multi matrix models and dyson s brownian motion we explain the method and survey results of physical interest

corner percolation on ℤ2 and the square root of 17 we consider a four vertex model introduced by balint toth a dependent bond percolation model on 2 in which every edge is present with probability 1 2 and each vertex has exactly two incident edges perpendicular to each other we prove that all components are finite cycles almost surely but the expected diameter of the cycle containing the origin is infinite moreover we derive the following critical exponents the tail probability diameter of the cycle of the origin n n and the expectation mathbb e length of a typical cycle with diameter n n with gamma 5 sqrt 17 4 0 219 ldots and delta sqrt 17 1 4 1 28 ldots the value of comes from a singular sixth order ode while the relation 3 2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive brownian motion whose level sets have hausdorff dimension 3 2 we also include many open problems for example on the conformal invariance of certain linear entropy models

consistent hypothesis testing in semiparametric and nonparametric models for econometric time series abstract in this paper we modify the general hypothesis studied by robinson 1989 for semi nonparametric time series models and present a consistent testing procedure for the modified hypothesis as examples we provide consistent tests for the portfolio conditional mean variance efficiency hypothesis for theomitted variables in a multivariate nonparametric time series regression model and for the two original examples in robinson the asymptotic distributions under the null and pitman local alternatives are established by invoking central limit theorems for hilbert valued dependent random arrays to approximate the critical values of the general test we modify the conditional monte carlo approach of hansen 1996 and the stationary bootstrap of politis and romano 1994a b and show that both work asymptotically

a combinatorial interpretation of the free fermion condition of the six vertex model the free fermion condition of the six vertex model provides a five parameter sub manifold on which the bethe ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple hence allowing explicit solutions such conditions arose originally in early field theoretic s matrix approaches here we provide a combinatorial explanation for the condition in terms of a generalized gessel viennot involution by doing so we extend the use of the gessel viennot theorem originally devised for non intersecting walks only to a special weighted type of intersecting walk and hence express the partition function of n such walks starting and finishing at fixed endpoints in terms of the single walk partition functions

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

endomorphic presentations of branch groups we introduce endomorphic presentations or l presentations group presentations whose relations are iterated under a set of substitutions on the generating set and show that a broad class of groups acting on rooted trees admit explicitly constructible finite l presentations generalising results by igor lysionok and said sidki

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

coupling functions for domino tilings of aztec diamonds the inverse kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics in this paper we consider three different weightings of domino tilings of the aztec diamond and show using recurrence relations that we can compute the inverse kasteleyn matrix these weights are the one periodic weighting where the horizontal edges have one weight and the vertical edges have another weight the qvol weighting which corresponds to multiplying the product of tile weights by q if we add a box to the height function and the two periodic weighting which exhibits a flat region with defects in the center

domino tilings and the six vertex model at its free fermion point at the free fermion point the six vertex model with domain wall boundary conditions dwbc can be related to the aztec diamond a domino tiling problem we study the mapping on the level of complete statistics for general domains and boundary conditions this is obtained by associating with both models a set of non intersecting lines in the lindstrom gessel viennot lgv scheme one of the consequences for dwbc is that the boundaries of the ordered phases are described by the airy process in the thermodynamic limit

fluctuations of an atomic ledge bordering a crystalline facet when a high symmetry facet joins the rounded part of a crystal the step line density vanishes as square root of r with r denoting the distance from the facet edge this means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation we investigate the statistical properties of the border ledge fluctuations in the scaling regime they turn out to be non gaussian and related to the edge statistics of gaussian unitary ensemble multimatrix models

directed loop monte carlo simulations of vertex models we show how the directed loop monte carlo algorithm can be applied to study vertex models the algorithm is employed to calculate the arrow polarization in the six vertex model with the domain wall boundary conditions the model exhibits spatially separated ordered and disordered regions we show how the boundary between these regions depends on parameters of the model we give some predictions on the behavior of the polarization in the thermodynamic limit and discuss the relation to the arctic circle theorem

nonparametric tests for conditional symmetry abstract we propose omnibus tests for symmetry of the conditional distribution of a time series process about a nonparametric regression function the test statistic is a weighted version of the integrated squared difference between the restricted and unrestricted estimators of the joint characteristic function of nonparametric residuals and explanatory variables whose critical values are estimated with the assistance of a bootstrap technique the test is sensitive to local alternatives converging to the null at the parametric rate t 1 2 with t the sample size we investigate the finite sample performance of the test by means of monte carlo experiments and two empirical applications to test whether losses are more likely than gains in financial markets and whether expansions and contractions are equally likely in business cycles given the relevant information

exact solution of the six vertex model with domain wall boundary conditions critical line between ferroelectric and disordered phases this is a continuation of the papers of bleher and fokin commun math phys 268 223 284 2006 and of bleher and liechty commun math phys 286 777 801 2009 in which the large n asymptotics is obtained for the partition function z n of the six vertex model with domain wall boundary conditions in the disordered and ferroelectric phases respectively in the present paper we obtain the large n asymptotics of z n on the critical line between these two phases

limit theorems for von mises statistics of a measure preserving transformation for a measure preserving transformation t of a probability space x mathcal f mu and some d ge 1 we investigate almost sure and distributional convergence of random variables of the form begin aligned x rightarrow frac 1 c n sum 0 le i 1 ldots i d n f t i 1 x ldots t i d x n 1 2 ldots end aligned where c 1 c 2 ldots are normalizing constants and the kernel f belongs to an appropriate subspace in some l p x d mathcal f otimes d mu d we establish a form of the individual ergodic theorem for such sequences using a filtration compatible with t and the martingale approximation we prove a central limit theorem in the non degenerate case for a class of canonical totally degenerate kernels and d 2 we also show that the convergence holds in distribution towards a quadratic form sum m 1 infty lambda m eta 2 m in independent standard gaussian variables eta 1 eta 2 ldots

six vertex model with domain wall boundary conditions and one matrix model the partition function of the six vertex model on a square lattice with domain wall boundary conditions dwbc is rewritten as a hermitian one matrix model or a discretized version of it similar to sums over young diagrams depending on the phase the expression is exact for finite lattice size which is equal to the size of the corresponding matrix in the thermodynamic limit the matrix integral is computed using traditional matrix model techniques thus providing a complete treatment of the bulk free energy of the six vertex model with dwbc in the different phases in particular in the antiferroelectric phase the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

statistical tests for deterministic effects in broad band time series we derive a normalized version of the indicators of savit and green and prove that these normalized statistics have asymptotically a normal distribution with a mean of zero and standard deviation of one if the time series is random in the sense of being lid independent and identically distributed we verify this result numerically and study the magnitude of the finite size effects we also show that these statistics are very sensitive to the existence of deterministic effects in the series even if the underlying deterministic structure is complex such as those generated by a chaotic system we show that with moderate amounts of data the statistics can easily indicate the presence of an underlying attractor even in the presence of iid noise which is as large as or greater than the signal finally we discuss the generalization of our approach to include 1 other null hypotheses besides lid which express assumptions of specific dependencies and 2 the study of deterministic effects between more than one time series

exact solution of the six vertex model with domain wall boundary conditions antiferroelectric phase we obtain the large n asymptotics of the partition function zn of the six vertex model with domain wall boundary conditions in the antiferroelectric phase region with the weights a sinh t b sinh t c sinh 2 t we prove the conjecture of zinn justin that as n zn c 4 n f 1 o n 1 where and f are given by explicit expressions in and t and 4 z is the jacobi theta function the proof is based on the riemann hilbert approach to the large n asymptotic expansion of the underlying discrete orthogonal polynomials and on the deift zhou nonlinear steepest descent method 2009 wiley periodicals inc

complete positive group presentations abstract a combinatorial property of positive group presentations called completeness is introduced with an effective criterion for recognizing complete presentations and an iterative method for completing an incomplete presentation we show how to directly read several properties of the associated monoid and group from a complete presentation cancellativity or existence of common multiples in the case of the monoid or isoperimetric inequality in the case of the group in particular we obtain a new criterion for recognizing that a monoid embeds in a group of fractions typical presentations eligible for the current approach are the standard presentations of the artin tits groups and the heisenberg group