local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

pointwise estimates and exponential laws in metastable systems via coupling methods we show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a meta stable set in the example of the random field curie weiss model we show that these ideas can also be used to prove asymptotic exponentiallity of normalized metastable escape times

mixing times for the mean field blume capel model via aggregate path coupling we investigate the relationship between the mixing times of the glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure for this we consider the mean field blume capel model one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure within a two dimensional parameter space there exists a curve at which the model undergoes a second order continuous phase transition a curve where the model undergoes a first order discontinuous phase transition and a tricritical point which separates the two curves we determine the interface between the regions of slow and rapid mixing in order to completely determine the region of rapid mixing we employ a novel extension of the path coupling method successfully proving rapid mixing even in the absence of contraction between neighboring states

thermodynamic versus statistical nonequivalence of ensembles for the mean field blume emery griffiths model abstract we illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean field blume emery griffiths beg model as a test model the novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states for this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy by computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the beg model we show that for values of the mean energy where the microcanonical entropy is nonconcave the microcanonical distributions of states are nowhere realized in the canonical ensemble moreover we show that for values of the mean energy where the microcanonical entropy is strictly concave the equilibrium microcanonical distributions of states can be put in one to one correspondence with equivalent canonical equilibrium distributions of states our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by ellis et al j stat phys 101 2000 999

fisher information and stochastic complexity by taking into account the fisher information and removing an inherent redundancy in earlier two part codes a sharper code length as the stochastic complexity and the associated universal process are derived for a class of parametric processes the main condition required is that the maximum likelihood estimates satisfy the central limit theorem the same code length is also obtained from the so called maximum likelihood code

cutoff for general spin systems with arbitrary boundary conditions the cutoff phenomenon describes a sharp transition in the convergence of a markov chain to equilibrium in recent work the authors established cutoff and its location for the stochastic ising model on the d dimensional torus n d for any d 1 the proof used the symmetric structure of the torus and monotonicity in an essential way here we enhance the framework and extend it to general geometries boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log sobolev constant of the glauber dynamics in particular we show there is cutoff for the stochastic ising model on any sequence of bounded degree graphs with subexponential growth under arbitrary external fields provided the inverse log sobolev constant is bounded for lattices with homogenous boundary such as all plus we identify the cutoff location explicitly in terms of spectral gaps of infinite volume dynamics on half plane intersections analogous results establishing cutoff are obtained for nonmonotone spin systems at high temperatures including the gas hard core model the potts model the antiferromagnetic potts model and the coloring model 2014 wiley periodicals inc

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

cutoff for the ising model on the lattice introduced in 1963 glauber dynamics is one of the most practiced and extensively studied methods for sampling the ising model on lattices it is well known that at high temperatures the time it takes this chain to mix in l 1 on a system of size n is o logn whether in this regime there is cutoff i e a sharp transition in the l 1 convergence to equilibrium is a fundamental open problem if so as conjectured by peres it would imply that mixing occurs abruptly at c o 1 logn for some fixed c 0 thus providing a rigorous stopping rule for this mcmc sampler however obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple markov chains already for the one dimensional ising model showing cutoff is a longstanding open problem

statistical mechanics in the extended gaussian ensemble the extended gaussian ensemble ege is introduced as a generalization of the canonical ensemble this ensemble is a further extension of the gaussian ensemble introduced by hetherington j low temp phys 66 145 1987 the statistical mechanical formalism is derived both from the analysis of the system attached to a finite reservoir and from the maximum statistical entropy principle the probability of each microstate depends on two parameters b and g which allow one to fix independently the mean energy of the system and the energy fluctuations respectively we establish the legendre transform structure for the generalized ther modynamic potential and propose a stability criterion we also compare the ege probability distribution with the q exponential distribution as an example an application to a system with few independent spins is presented the development of statistical mechanics based on en semble theory is founded on the postulate of equal a priori probabilities which is assumed to apply to all microstates consistent with the given macrostate of an isolated system 1 the corresponding statistical ensemble is the so called microcanonical ensemble a representative system in this en semble has all mechanical variables such as energy e volume v magnetization m etc fixed for convenience in calculations other ensembles are used which invariably sup pose the existence of a subsidiary system or reservoir in con tact with the actual system for instance in the canonical ensemble the walls of the system permit an exchange of en ergy with the reservoir while in the grand canonical en semble both energy and matter can be exchanged in gen eral the different ensembles are constructed by allowing one or more mechanical variables to fluctuate the exchange of each of these variables is controlled by a parameter which is a characteristic of the reservoir for instance in the case of the canonical ensemble this parameter is precisely the tem perature of the reservoir and determines the mean energy of the system actually this is adequate when the reservoir is a very large system that can exchange arbitrary amounts of energy without modification of its intensive properties in practical situations this is not always the case however very few studies have been devoted to analyze the conse quences of possible deviations from these ideal reservoir properties in this paper we develop the statistical mechanics of a system that can exchange energy with a finite reservoir char acterized by two parameters b and g these parameters con trol independently the mean energy of the system and its energy fluctuations respectively the corresponding statisti cal ensemble represents a generalization of the canonical en semble and will be called the extended gaussian ensemble ege a similar ensemble was already developed in a more restricted framework by hetherington 2 the author con sidered that the sample system was in contact with a finite reservoir with size dependent properties the so called gaussian ensemble was introduced so that it is equivalent to the canonical ensemble in the limit of large systems except in the energy range of a first order transition interestingly it enables a smooth interpolation between the microcanonical and the canonical ensembles taking into account these fea tures challa and hetherington 3 4 showed the interest of this ensemble for monte carlo simulation studies of phase transitions they demonstrated a significant reduction in computer time compared to standard simulations in the ca nonical ensemble and its adequacy for distinguishing second order from first order transitions compared to the ege introduced in the present paper the main difference arises from the fact that in the gaussian ensemble the sample and the reservoir are assumed to be statistically independent which implies the additivity of the corresponding entropies this is not assumed in our formalism the consequences are important and will be discussed in depth in this work the present formalism can be considered as an alternative to the statistical mechanics based on nonadditive generalized entropies actually the study of such generalized entropies has generated a lot of interest in the past 15 years the mo tivation for the so called tsallis statistical mechanics has been to extend the standard boltzmann gibbs framework to include nonextensive systems 5 among different interpre tations it has been suggested that tsallis formalism corre sponds to an ensemble describing a system attached to a finite reservoir 6 7 although a large number of papers have been published the physical meaning of many related issues is still open to discussion 8 9 the ege formalism that we propose in this paper provides a clear and consistent frame work for the statistical mechanics with a finite reservoir the paper is organized as follows in secs ii and iii the ege is founded from the analysis of a contact with a finite reservoir and from the maximum statistical entropy prin ciple respectively in sec iv the main thermodynamic rela tions are derived in sec v we highlight the nonadditive nature of the thermodynamic formalism in sec vi a stabil

global well posedness of the dynamic φ4 model in the plane we show global well posedness of the dynamic 4 4 model in the plane the model is a nonlinear stochastic pde that can only be interpreted in a renormalised sense solutions take values in suitable weighted besov spaces of negative regularity

bayesian analysis of contingent claim model error abstract this paper formally incorporates parameter uncertainty and model error into the implementation of contingent claim models we make hypotheses for the distribution of errors to allow the use of likelihood based estimators consistent with parameter uncertainty and model error we then write a bayesian estimator which does not rely on large sample properties but allows exact inference on the relevant functions of the parameters option value hedge ratios and forecasts this is crucial because the common practice of frequently updating the model parameters leads to small samples even for simple error structures and the black scholes model the bayesian estimator does not have an analytical solution markov chain monte carlo estimators help solve this problem we show how they extend to some generalizations of the error structure we apply these estimators to the black scholes given recent work using non parametric function to price options we nest the b s in a polynomial expansion of its inputs despite improved in sample fit the expansions do not yield any out of sample improvement over the b s also the out of sample errors though larger than in sample are of the same magnitude this contrasts with the performance of the popular implied tree methods which produce outstanding in sample but disastrous out of sample fit as dumas fleming and whaley 1997 show this means that the estimation method is as crucial as the model itself

effect of random field disorder on the first order transition in p spin interaction model abstract we study the random field p spin model with ising spins on a fully connected graph using the theory of large deviations in this paper this is a good model to study the effect of quenched random field on systems which have a sharp first order transition in the pure state for p 2 the phase diagram of the model for bimodal distribution of the random field has been well studied and is known to undergo a continuous transition for lower values of the random field h and a first order transition beyond a threshold h t p 0 439 we find the phase diagram of the model for all p 2 with bimodal random field distribution using large deviation techniques we also look at the fluctuations in the system by calculating the magnetic susceptibility for p 2 beyond the tricritical point in the regime of first order transition we find that for h t p h 0 447 magnetic susceptibility increases rapidly even though it never diverges as one approaches the transition from the high temperature side on the other hand for 0 447 h 0 5 the high temperature behaviour is well described by the curie weiss law for all p 2 we find that for larger magnitudes of the random field h h o 1 p the system does not show ferromagnetic order even at zero temperature we find that the magnetic susceptibility for p 3 is discontinuous at the transition point for h h o

chaos butterflies also generate phase transitions we exhibit examples of mixing subshifts of finite type and of continuous potentials such that there are phase transitions but the pressure is always strictly convex more surprisingly we show that the pressure can be analytic on some interval although there exist several equilibrium states

local parametric analysis of hedging in discrete time when continuous time portfolio weights are applied to a discrete time hedging problem errors are likely to occur this paper evaluates the overall importance of the discretization induced tracking error it does so by comparing the performance of black scholes hedge ratios against those obtained from a novel estimation procedure namely local parametric estimation in the latter the weights of the duplicating portfolio are estimated by fitting parametric models in this paper black scholes in the neighborhood of the derivative s moneyness and maturity local parametric estimation directly incorporates the error from hedging in discrete time results are shown where the root mean square tracking error is reduced up to 41 for short maturity options the performance can still be improved by combining locally estimated hedge portfolio weights with standard analysis based on historically estimated parameters the root mean square tracking error is thereby reduced by about 18 for long maturity options plots of the locally estimated volatility parameter against moneyness and maturity reveal the biases of the black scholes model when hedging in discrete time in particular there is a sharp smile effect in the relation between estimated volatility and moneyness for short maturity options as well as a significant wave effect in the relation with maturity for deep out of the money options 1997 elsevier science s a

rigorous results on the bipartite mean field model we consider a bipartite mean field model in which interaction coefficients and magnetic fields depend only on the groups the particles belong to we rigorously compute the value of the limiting pressure per particle using tail estimation techniques we study the phase space of the model in the symmetric regime without an external field and when the interaction coefficients within the two groups are identical magnetic field perturbations are considered

the aggregate path coupling method for the potts model on bipartite graph in this paper we derive the large deviation principle for the potts model on the complete bipartite graph kn n as n increases to infinity next for the potts model on kn n we provide an extension of the method of aggregate path coupling that was originally developed in the work of kovchegov otto and titus j stat phys 144 5 1009 1027 2011 for the mean field blume capel model and in kovchegov and otto j stat phys 161 3 553 576 2015 for a general mean field setting that included the generalized curie weiss potts model analyzed in the work of jahnel et al markov process relat fields 20 601 632 2014 we use the aggregate path coupling method to identify and determine the threshold value s separating the rapid and slow mixing regimes for the glauber dynamics of the potts model on kn n

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

on first order phase transitions in microcanonical and canonical non extensive systems abstract two examples of microcanonical potts models two dimensional nearest neighbor and mean field are considered via exact enumeration of states and analytical asymptotic methods in the interval of energies corresponding to a first order phase transition both of these models exhibit a convex dip in the entropy vs energy plot and a region with negative specific heat within the dip it is observed that in the nearest neighbor model the dip flattens and disappears as the lattice size grows while in the mean field model the dip persists even in the limit of an infinite system if formal transitions from microcanonical to canonical ensembles and back are performed for an infinite but non extensive system the convex dip in the microcanonical entropy plot disappears

exact solution for a class of random walk on the hypercube a class of families of markov chains defined on the vertices of the n dimensional hypercube n 0 1 n is studied the single step transition probabilities pn ij with i j n are given by p n ij frac 1 alpha d ij 2 alpha n where 0 1 and dij is the hamming distance between i and j this corresponds to flip independently each component of the vertex with probability frac 1 alpha 2 alpha the m step transition matrix p n ij m is explicitly computed in a close form the class is proved to exhibit cutoff a model independent result about the vanishing of the first m terms of the expansion in of p n ij m is also proved

mixing time of vertex weighted exponential random graphs abstract exponential random graph models have become increasingly important in the study of modern networks ranging from social networks economic networks to biological networks they seek to capture a wide variety of common network tendencies such as connectivity and reciprocity through local graph properties sampling from these exponential distributions is crucial for parameter estimation hypothesis testing as well as understanding the features of the network in question we inspect the efficiency of a popular sampling technique the glauber dynamics for vertex weighted exponential random graphs letting n be the number of vertices in the graph we identify a region in the parameter space where the mixing time for the glauber dynamics is n log n the high temperature phase and a complementary region where the mixing time is exponentially slow of order e n the low temperature phase lastly we give evidence that along a critical curve in the parameter space the mixing time is o n 2 3

asymptotics of the mean field heisenberg model we consider the mean field classical heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity in particular we obtain cramer and sanov type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second order phase transition in the gibbs measures we also study the asymptotics of the total spin throughout the phase transition using stein s method proving central limit theorems in the sub and supercritical phases and a nonnormal limit theorem at the critical temperature

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

limit theorems for the bipartite potts model we consider the bipartite potts model in which interaction strengths depend only on the groups the particles belong to we first rigorously compute the exact value of the free energy then put forward two scaling limit theorems for the joint distribution of two empirical vectors which serve as order parameters it turns out that the limit distribution can be gaussian or the exponential distribution of higher order under different conditions

the ising model on diluted graphs and strong amenability say that a graph has persistent transition if the ising model on the graph can exhibit a phase transition nonuniqueness of gibbs measures in the presence of a nonzero external field we show that for nonamenable graphs for bernoulli percolation with p close to 1 all the infinite clusters have persistent transition on the other hand we show that for transitive amenable graphs the infinite clusters for any stationary percolation do not have persistent transition this extends a result of georgii for the cubic lattice a geometric consequence of this latter fact is that the infinite clusters are strongly amenable i e their anchored cheeger constant is 0 finally we show that the critical temperature for the ising model with no external field on the infinite clusters of bernoulli percolation with parameter p on an arbitrary bounded degree graph is a continuous function of p

subspace based multivariable system identification from frequency response data two noniterative subspace based algorithms which identify linear time invariant mimo multi input multioutput systems from frequency response data are presented the algorithms are related to the recent time domain subspace identification techniques the first algorithm uses equidistantly in frequency spaced data and is strongly consistent under weak noise assumptions the second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure

mean field driven first order phase transitions in systems with long range interactions we consider a class of spin systems on d with vector valued spins sx that interact via the pair potentials jx ysx sy the interactions are generally spread out in the sense that the jx y s exhibit either exponential or power law fall off under the technical condition of reflection positivity and for sufficiently spread out interactions we prove that the model exhibits a first order phase transition whenever the associated mean field theory signals such a transition as a consequence e g in dimensions d 3 we can finally provide examples of the 3 state potts model with spread out exponentially decaying interactions which undergoes a first order phase transition as the temperature varies similar transitions are established in dimensions d 1 2 for power law decaying interactions and in high dimensions for next nearest neighbor couplings in addition we also investigate the limit of infinitely spread out interactions specifically we show that once the mean field theory is in a unique state then in any sequence of translation invariant gibbs states various observables converge to their mean field values and the states themselves converge to a product measure

non equivalence of dynamical ensembles and emergent non ergodicity dynamical ensembles have been introduced to study constrained stochastic processes in the microcanonical ensemble the value of a dynamical observable is constrained to a given value in the canonical ensemble a bias is introduced in the process to move the mean value of this observable the equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result in this paper we study the physical conditions associated with ensemble equivalence and the consequences of non equivalence for continuous time markov jump processes we show that ergodicity guarantees ensemble equivalence for non ergodic systems or systems with emergent ergodicity breaking we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition we illustrate our results on the infinite range ising model by characterizing the fluctuations of magnetization and activity we discuss the emergence of non ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins

some diagnostics for binary logistic regression via smoothing summary the paper discusses some diagnostic tools for binary logistic regression which use smoothing techniques smoothed binary data are employed to devise a battery of diagnostic plots from simple procedures which indicate only the adequacy or inadequacy of a maximum likelihood logistic fit to partial residual plots and delta local deviance plots which reveal the type of model inadequacy suggesting perhaps the addition of nonlinear terms or interactions properties of the methods are studied by simulation the diagnostic tools are illustrated on a set of data in which the probability of having telephone service is related to the demographic variables income and education

glauber dynamics of 2d kac blume capel model and their stochastic pde limits abstract we study the glauber dynamics of a two dimensional blume capel model or dilute ising model with kac potential parametrized by the inverse temperature and the chemical potential we prove that the locally averaged spin field rescales to the solution of the dynamical 4 equation near a curve in the plane and to the solution of the dynamical 6 equation near one point on this curve our proof relies on a discrete implementation of da prato debussche method 13 as in 33 but an additional coupling argument is needed to show convergence of the linearized dynamics

metastability and low lying spectra in reversible markov chains we study a large class of reversible markov chains with discrete state space and transition matrix pn we define the notion of a set of metastable points as a subset of the state space n such that i this set is reached from any point x n without return to x with probability at least bn while ii for any two points x y in the metastable set the probability t 1x y to reach y from x without return to x is smaller than an 1 bn under some additional non degeneracy assumption we show that in such a situation

universality of the mean field for the potts model we consider the potts model with q colors on a sequence of weighted graphs with adjacency matrices a n allowing for both positive and negative weights under a mild regularity condition on a n we show that the mean field prediction for the log partition function is asymptotically correct whenever mathrm tr a n 2 o n in particular our results are applicable for the ising and the potts models on any sequence of graphs with average degree going to infty using this we establish the universality of the limiting log partition function of the ferromagnetic potts model for a sequence of asymptotically regular graphs and that of the ising model for bi regular bipartite graphs in both ferromagnetic and anti ferromagnetic domain we also derive a large deviation principle for the empirical measure of the colors for the potts model on asymptotically regular graphs

abrupt convergence and escape behavior for birth and death chains we link two phenomena concerning the asymptotical behavior of stochastic processes i abrupt convergence or cut off phenomenon and ii the escape behavior usually associated to exit from metastability the former is characterized by convergence at asymptotically deterministic times while the convergence times for the latter are exponentially distributed we compare and study both phenomena for discrete time birth and death chains on with drift towards zero in particular this includes energy driven evolutions with energy functions in the form of a single well under suitable drift hypotheses we show that there is both an abrupt convergence towards zero and escape behavior in the other direction furthermore as the evolutions are reversible the law of the final escape trajectory coincides with the time reverse of the law of cut off paths thus for evolutions defined by one dimensional energy wells with sufficiently steep walls cut off and escape behavior are related by time inversion

the cutoff phenomenon in finite markov chains abstract natural mixing processes modeled by markov chains often show a sharp cutoff in their convergence to long time behavior this paper presents problems where the cutoff can be proved card shuffling the ehrenfests urn it shows that chains with polynomial growth drunkard s walk do not show cutoffs the best general understanding of such cutoffs high multiplicity of second eigenvalues due to symmetry is explored examples are given where the symmetry is broken but the cutoff phenomenon persists

fluctuation results for general block spin ising models we study a block spin mean field ising model i e a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices and where pair interactions are given according to their blocks for the vector of block magnetizations we prove large deviation principles and central limit theorems under general assumptions for the block interaction matrix using the exchangeable pair approach of stein s method we establish a rate of convergence in the central limit theorem for the block magnetization vector in the high temperature regime

the kernel estimate of a regression function in likelihood based models abstract smoothing splines have a penalized likelihood motivation good and gaskins 1971 allowing direct application to nonparametric regression in likelihood based models the notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed generalizing the local likelihood theory of tibshirani and hastie 1987 let the data be of the form xi yi i 1 n where xi 0 1 d are lattice points and the yi are independent random variables from a family of distributions with parameter i g xi with g having continuous partial derivatives of order k 2 the goal is to arrive at a nonparametric estimate o of o g xo for a fixed point xo 0 1 d we consider the estimator o that maximizes the weighted likelihood function w n i 1 w xo xi b log f yi with f the density of yi w a symmetric kernel with compact support and b the bandwidth that controls the degree of smoothing sufficient conditions for consistency and asympt

variable bandwidth and local linear regression smoothers in this paper we introduce an appealing nonparametric method for estimating the mean regression function the proposed method combines the ideas of local linear smoothers and variable bandwidth hence it also inherits the advantages of both approaches we give expressions for the conditional mse and mise of the estimator minimization of the mise leads to an explicit formula for an optimal choice of the variable bandwidth moreover the merits of considering a variable bandwidth are discussed in addition we show that the estimator does not have boundary effects and hence does not require modifications at the boundary the performance of a corresponding plug in estimator is investigated simulations illustrate the proposed estimation method

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

ensemble inequivalence in random graphs we present a complete analytical solution of a system of potts spins on a random k regular graph in both the canonical and microcanonical ensembles using the large deviation cavity method ldcm the solution is shown to be composed of three different branches resulting in a non concave entropy function the analytical solution is confirmed with numerical metropolis and creutz simulations and our results clearly demonstrate the presence of a region with negative specific heat and consequently ensemble inequivalence between the canonical and microcanonical ensembles

multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point we derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean field version of the blume emery griffiths model phys rev a4 1971 1071 1077 these probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles mdps for the total spin the model under study is defined by a probability distribution that depends on the parameters n and k which represent respectively the number of spins the inverse temperature and the interaction strength the intricate structure of the phase transitions is revealed by the existence of 18 scaling limits and 18 mdps for the total spin these limit results are obtained as k converges along appropriate sequences n kn to points belonging to various subsets of the phase diagram which include a curve of second order points and a tricritical point the forms of the limiting densities in the scaling limits and of the rate functions in the mdps reflect the influence of one or more sets that lie in neighborhoods of the critical points and the tricritical point of all the scaling limits the structure of those near the tricritical point is by far the most complex exhibiting new types of critical behavior when observed in a limit theorem phase diagram in the space of the two parameters that parametrize the scaling limits

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

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

negative temperature states and large scale long lived vortices in two dimensional turbulence we study onsager s theory of large coherent vortices in turbulent flows in the approximation of the point vortex model for two dimensional euler hydrodynamics in the limit of a large number of point vortices with the energy perpair of vortices held fixed we prove that the entropy defined from the microcanonical distribution as a function of the pair specific energy has its maximum at a finite value and thereafter decreases yielding the negative temperature states predicted by onsager we furthermore show that the equilibrium vorticity distribution maximizes an appropriate entropy functional subject to the constraint of fixed energy and under regularity assumptions obeys the joyce montgomery mean field equation we also prove that under appropriate conditions the vorticity distribution is the same as that for the canonical distribution a form of equivalence of ensembles we establish a large fluctuation theory for the microcanonical distributions which is based on a level 3 large deviations theory for exchangeable distributions we discuss some implications of that property for the ergodicity requirements to justify onsager s theory and also the theoretical foundations of a recent extension to continuous vorticity fields by r robert and j miller although the theory of two dimensional vortices is of primary interest our proofs actually apply to a very general class of mean field models with long range interactions in arbitrary dimensions

reconstruction of the interior sound pressure of a room using the probabilistic approach in this paper a probabilistic approach is introduced and used to find the optimal values assigned to the uncertain parameters of a room acoustic model which is used for the reconstruction of the interior sound pressure distribution the acoustic model selected here to examine the capacity of the probabilistic approach is a rectangular room with two air leakages that is subject to an external uniform noise the model is set up using the modal analysis method the values of the uncertain parameters of the model are identified using the time domain sound pressure responses at selected measurement points in the simulation results the comparisons between the proposed approach and the typical least error square method show that the former clearly assigns optimal values to the uncertain model parameters for the best prediction while the latter generates a set of values for which the prediction errors are very similar and close to the minimum in other words it is difficult to identify the optimal values using the least error square method moreover the optimal values for the uncertain parameters can be found individually unlike those from the least error square method it is also found that the measurement points should not be located at the nodal points of the dominant acoustic modes otherwise the identification process becomes unidentifiable for cases in which large modal truncation errors or modeling errors exist the identification process also becomes unidentifiable

regression by local fitting methods properties and computational algorithms abstract local regression is a procedure for estimating regression surfaces by the local fitting of linear or quadratic functions of the independent variables in a moving fashion that is analogous to how a moving average is computed for a time series the advantage of the methodology over the global fitting of parametric functions of the independent variables by least squares the current paradigm in regression studies is that a much wider class of regression functions can be estimated without distortion in this paper we discuss the methods their statistical properties and computational algorithms

reflection couplings and contraction rates for diffusions we consider contractivity for diffusion semigroups w r t kantorovich l 1 wasserstein distances based on appropriately chosen concave functions these distances are inbetween total variation and usual wasserstein distances it is shown that by appropriate explicit choices of the underlying distance contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w r t standard wasserstein distances fails applications include overdamped langevin diffusions with locally non convex potentials products of these processes and systems of weakly interacting diffusions both of mean field and nearest neighbour type

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

weakly asymmetric non simple exclusion process and the kardar parisi zhang equation we analyze a class of non simple exclusion processes and the corresponding growth models by generalizing the discrete cole hopf transformation of gartner stoch process appl 27 233 260 1987 we identify the main non linearity and eliminate it by imposing a gradient type condition for hopping range at most 3 using the generalized transformation we prove the convergence of the exclusion process toward the kardar parisi zhang kpz equation this is the first universality result under the weak asymmetry concerning interacting particle systems while this class of exclusion processes are not explicitly solvable under the weak asymmetry we obtain the exact one point limiting distribution for the step initial condition by using the previous result of amir et al commun pure appl math 64 4 466 537 2011 and our convergence result

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

system identification of an enclosure with leakages using a probabilistic approach this paper presents a model based method for the system identification of a rectangular enclosure with an unknown number of air leakages subjected to uniform external noise according to the probabilistic approach the method aims to identify the number and corresponding locations and sizes of air leakages utilizing a set of measured interior sound pressure data in the frequency domain system identification of an enclosure with an unknown number of air leakages is not trivial different classes of acoustic models are required to simulate an enclosure with different numbers of leakages by following the traditional system of identification techniques the optimal class of models is selected by minimizing the discrepancy between the measured and modeled interior sound pressure by doing this the most complicated model class that is the one with the highest number of uncertain parameters will always be selected therefore the traditional system identification techniques found in the literature to date cannot be employed to solve this problem our proposed system identification methodology relies on the bayesian information criterion bic to identify accurately the number of leakages in an enclosure unlike all deterministic system identification approaches the proposed methodology aims to calculate the posterior updated probability density function pdf of leakage locations and sizes therefore the uncertainties introduced by measurement noise and modeling error can be explicitly addressed the coefficient of variable cov of uncertain parameters which can be easily calculated from the pdf provides valuable information about the reliability of the identification results

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

entropy driven cutoff phenomena in this paper we present in the context of diaconis paradigm a general method to detect the cutoff phenomenon we use this method to prove cutoff in a variety of models some already known and others not yet appeared in literature including a non reversible random walk on a cylindrical lattice all the given examples clearly indicate that a drift towards the opportune quantiles of the stationary measure could be held responsible for this phenomenon in the case of birth and death chains this mechanism is fairly well understood our work is an effort to generalize this picture to more general systems such as systems having stationary measure spread over the whole state space or systems in which the study of the cutoff may not be reduced to a one dimensional problem in those situations the drift may be looked for by means of a suitable partitioning of the state space into classes using a statistical mechanics language it is then possible to set up a kind of energy entropy competition between the weight and the size of the classes under the lens of this partitioning one can focus the mentioned drift and prove cutoff with relative ease

scaling limits for multi species statistical mechanics mean field models we study the limiting thermodynamic behavior of the normalized sums of spins in multi species curie weiss models we find sufficient conditions for the limiting random vector to be gaussian or to have an exponential distribution of higher order and compute the covariance matrix in terms of model parameters

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

metastability within the generalized canonical ensemble we discuss a property of our recently introduced generalized canonical ensemble m costeniuc r s ellis h touchette b turkington the generalized canonical ensemble and its universal equivalence with the microcanonical ensemble j stat phys 119 2005 1283 we show that this ensemble can be used to transform metastable or unstable nonequilibrium states of the standard canonical ensemble into stable equilibrium states within the generalized canonical ensemble equilibrium calculations within the generalized canonical ensemble can thus be used to obtain information about nonequilibrium states in the canonical ensemble r 2006 elsevier b v all rights reserved

mixing and concentration by ricci curvature abstract we generalise the coarse ricci curvature method of ollivier by considering the coarse ricci curvature of multiple steps in the markov chain this implies new spectral bounds and concentration inequalities we also extend this approach to the bounds for mcmc empirical averages obtained by joulin and ollivier we prove a recursive lower bound on the coarse ricci curvature of multiple steps in the markov chain making our method broadly applicable applications include the split merge random walk on partitions glauber dynamics with random scan and systemic scan for statistical physical spin models and random walk on a binary cube with a forbidden region

total variation cutoff in birth and death chains the cutoff phenomenon describes a case where a markov chain exhibits a sharp transition in its convergence to stationarity diaconis proc natl acad sci usa 93 4 1659 1664 1996 surveyed this phenomenon and asked how one could recognize its occurrence in families of finite ergodic markov chains peres american institute of mathematics aim research workshop palo alto http www aimath org wwn mixingtimes 2004 noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing time and spectral gap tends to infinity and conjectured that in many settings this condition should also be sufficient diaconis and saloff coste ann appl probab 16 4 2098 2122 2006 verified this conjecture for continuous time birth and death chains started at an endpoint with convergence measured in separation it is natural to ask whether the conjecture holds for these chains in the more widely used total variation distance in this work we confirm the above conjecture for all continuous time or lazy discrete time birth and death chains with convergence measured via total variation distance namely if the product of the mixing time and spectral gap tends to infinity the chains exhibit cutoff at the maximal hitting time of the stationary distribution median with a window of at most the geometric mean between the relaxation time and mixing time in addition we show that for any lazy or continuous time birth and death chain with stationary distribution the separation 1 p t x y y is maximized when x y are the endpoints together with the above results this implies that total variation cutoff is equivalent to separation cutoff in any family of such chains

information percolation and cutoff for the stochastic ising model we introduce a new framework for analyzing glauber dynamics for the ising model the traditional approach for obtaining sharp mixing results has been to appeal to estimates on spatial properties of the stationary measure from within a multi scale analysis of the dynamics here we propose to study these simultaneously by examining information percolation clusters in the space time slab using this framework we obtain new results for the ising model on z nz throughout the high temperature regime total variation mixing exhibits cutoff with an o 1 window around the time at which the magnetization is the square root of the volume previously cutoff in the full high temperature regime was only known in dimensions d 2 and only with an o log log n window furthermore the new framework opens the door to understanding the effect of the initial state on the mixing time we demonstrate this on the 1d ising model showing that starting from the uniform disordered initial distribution asymptotically halves the mixing time whereas almost every deterministic starting state is asymptotically as bad as starting from the ordered all plus state

rates of convergence in the blume emery griffiths model we derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean field version of the blume emery griffith model the theorems consist of scaling limits for the total spin the model depends on the inverse temperature beta and the interaction strength k the rates of convergence results are obtained as beta k converges along appropriate sequences beta n k n to points belonging to various subsets of the phase diagram which include a curve of second order points and a tricritical point we apply stein s method for normal and non normal approximation avoiding the use of transforms and supplying bounds such as those of berry esseen quality on approximation error

on the computational complexity of high dimensional bayesian variable selection we study the computational complexity of markov chain monte carlo mcmc methods for high dimensional bayesian linear regression under sparsity constraints we rst show that a bayesian approach can achieve variable selection consistency under relatively mild conditions on the design matrix we then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the mcmc algorithm by introducing a truncated sparsity prior for variable selection we provide a set of conditions that guarantee both variable selection consistency and rapid mixing of a particular metropolis hastings algorithm the mixing time is linear in the number of covariates up to a logarithmic factor our proof controls the spectral gap of the markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection

time dependent critical fluctuations of a one dimensional local mean field model one dimensional stochastic ising systems with a local mean field interaction kac potential are investigated it is shown that near the critical temperature of the equilibrium gibbs distribution the time dependent process admits a scaling limit given by a nonlinear stochastic pde the initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way earlier results of bertini presutti rudiger saada are improved the proof is based on an energy inequality obtained by coupling the glauber dynamics to its voter type linear approximation

kpz equation limit of higher spin exclusion processes we prove that under a particular weak scaling the 4 parameter interacting particle system introduced by corwin and petrov comm math phys 343 2016 651 700 converges to the kardar parisi zhang kpz equation this expands the relatively small number of systems for which weak universality of the kpz equation has been demonstrated

mean field theory of the potts gas we consider a gas of classical particles in r d having q distinct colours interacting via a mean field potts potential and subject to an external field a colour independent molecular interaction of mean field type is also admitted in contrast to the usual lattice potts model the potts gas exhibits the specific volume v as a parameter in addition to colour ordering the v dependence of the potts gas is studied in detail and the complete phase diagram is derived it turns out that for q 3 the transition to colour ordering implies a jump of density for q 2 this transition is continuous but may become discontinuous under the influence of a suitable molecular interaction

censored glauber dynamics for the mean field ising model we study glauber dynamics for the ising model on the complete graph on n vertices known as the curie weiss model it is well known that at high temperature 1 it is exp n recently levin luczak and peres considered a censored version of this dynamics which is restricted to non negative magnetization they proved that for fixed 1 the mixing time of this model is nlog n analogous to the high temperature regime of the original dynamics furthermore they showed cutoff for the original dynamics for fixed 1 the question whether the censored dynamics also exhibits cutoff remained unsettled

nonlinear fluctuations of weakly asymmetric interacting particle systems we introduce what we call the second order boltzmann gibbs principle which allows one to replace local functionals of a conservative one dimensional stochastic process by a possibly nonlinear function of the conserved quantity this replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states as an application of this second order boltzmann gibbs principle we introduce the notion of energy solutions of the kpz and stochastic burgers equations under minimal assumptions we prove that the density fluctuations of one dimensional stationary weakly asymmetric conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic burgers equation we also show that the fluctuations of the height function associated to these models are given by energy solutions of the kpz equation in this sense unfortunately we lack a uniqueness result for these energy solutions we conjecture these solutions to be unique and we show some regularity results for energy solutions of the kpz burgers equation supporting this conjecture

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

estimation of local microcanonical averages in two lattice mean field models using coupling techniques we consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states for instance between certain microcanonical canonical and grand canonical ensemble expectations a particular goal is to obtain good bounds for how such errors will decay with increasing system size as explicit examples we focus on two well studied mean field models the discrete model of a paramagnet and the mean field spherical model of a continuum field both of which are related to the curie weiss model the proof is based on a construction of suitable probabilistic couplings between the relevant states using wasserstein fluctuation distance to control the difference between the expectations in the thermodynamic limit

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model

fast initial conditions for glauber dynamics in the study of markov chain mixing times analysis has centered on the performance from a worst case starting state here in the context of glauber dynamics for the one dimensional ising model we show how new ideas from information percolation can be used to establish mixing times from other starting states at high temperatures we show that the alternating initial condition is asymptotically the fastest one and surprisingly its mixing time is faster than at infinite temperature accelerating as the inverse temperature beta ranges from 0 to beta 0 frac 1 2 mathrm arctanh frac 1 3 moreover the dominant test function depends on the temperature at beta beta 0 it is autocorrelation whereas at beta beta 0 it is the hamiltonian

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

microcanonical solution of the mean field φ 4 model comparison with time averages at finite size abstract we study the mean field 4 model in an external magnetic field in the microcanonical ensemble using two different methods the first one is based on rugh s microcanonical formalism and leads to express macroscopic observables such as temperature specific heat magnetization and susceptibility as time averages of convenient functions of the phase space the approach is applicable for any finite number of particles n the second method uses large deviation techniques and allows us to derive explicit expressions for microcanonical entropy and for macroscopic observables in the n limit assuming ergodicity we evaluate time averages in molecular dynamics simulations and using rugh s approach we determine the value of macroscopic observables at finite n these averages are affected by a slow time evolution often observed in systems with long range interactions we then show how the finite n time averages of macroscopic observables converge to their corresponding n values as n is increased as expected finite size effects scale as n 1

sampling from a gibbs measure with pair interaction by means of pca we consider the problem of approximate sampling from the finite volume gibbs measure with a general pair interaction we exhibit a parallel dynamics probabilistic cellular automaton which efficiently implements the sampling in this dynamics the product measure that gives the new configuration in each site contains a term that tends to favour the original value of each spin this is the main ingredient that allows one to prove that the stationary distribution of the pca is close in total variation to the gibbs measure the presence of the parameter that drives the inertial term mentioned above gives the possibility to control the degree of parallelism of the numerical implementation of the dynamics