study of the iterations of a mapping associated to a spin glass model we study the iterations of the mapping mathcal n f s frac f s 2 f 0 2 s f 0 2 with the constraintsf 1 1 f s ansn an 0 and find that except iff s s n f s approaches either 0 or 1 for s 1 ask

aperiodic interactions on hierarchical lattices an exact criterion for the potts ferromagnet criticality we discuss the critical behaviour of the q state potts model on a diamond like hierarchical lattice with ferromagnetic interactions according to an aperiodic two letter substitutional sequence we show that the geometric deterministic fluctuations become relevant for where is the wandering exponent of the sequence d is the fractal dimension of the lattice and is the critical exponent associated with the specific heat of the uniform model we also point out that the criteria for analysing the relevance of deterministic and random fluctuations are generically different

the phase transition of the directed polymer on disordered hierarchical lattices the directed polymer on disordered hierarchical lattices is studied using an exact renormalization scheme the phase transition is studied and a hyperscaling relation is derived the fixed distributions of the renormalized energies are obtained numerically the specific heat and associated exponents are computed from the fixed distributions

hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

open quantum random walks ergodicity hitting times gambler s ruin and potential theory in this work we study certain aspects of open quantum random walks oqrws a class of quantum channels described by attal et al j stat phys 147 832 852 2012 as a first objective we consider processes which are nonhomogeneous in time i e at each time step a possibly distinct evolution kernel inspired by a spectral technique described by saloff coste and zuniga stoch proc appl 117 961 979 2007 we define a notion of ergodicity for finite nonhomogeneous quantum markov chains and describe a criterion for ergodicity of such objects in terms of singular values as a second objective and based on a quantum trajectory approach we study a notion of hitting time for oqrws and we see that many constructions are variations of well known classical probability results with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical in this way we are able to examine open quantum versions of the gambler s ruin birth and death chain and a basic theorem on potential theory

differential migdal kadanoff renormalization group for disordered systems phase transitions in quenched random systems are studied within the migdal kadanoff approximation exact differential recursion relations for the distribution of coupling are derived the q state potts model with random ferromagnetic couplings is studied in two and three dimensions in three dimensions there is a non perturbative random fixed point which exists when the specific heat exponent is either positive or slightly negative if the specific heat exponent is slightly negative a multicritical point separates strong and weak disorder regimes the analytic results are confirmed by numerical simulations of the recursion relations

invariant measure for quantum trajectories we study a class of markov chains that model the evolution of a quantum system subject to repeated measurements each markov chain in this class is defined by a measure on the space of matrices and is then given by a random product of correlated matrices taken from the support of the defining measure we give natural conditions on this support that imply that the markov chain admits a unique invariant probability measure we moreover prove the geometric convergence towards this invariant measure in the wasserstein metric standard techniques from the theory of products of random matrices cannot be applied under our assumptions and new techniques are developed such as maximum likelihood type estimations

on the digraphs of sign solvable linear systems abstract a linear system of equations ax b is sign solvable if it is solvable and both its solvability and the sign pattern of its solution vector x are uniquely determined by the sign patterns of a and b it was proved in l bassett j maybee j quirk economitrica 36 1968 544 563 r a bruali b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 that the sign solvability of a linear system ax b in standard form and with a square coefficient matrix a can be described in terms of the signed digraph s a of the matrix a with a specified vertex subset depending on the sign pattern of the vector b in this paper we define the notion of w sign solvable signed digraphs which represent precisely the sign solvable linear systems in standard form we give necessary and sufficient conditions for the underlying unsigned digraphs of w sign solvable signed digraphs in terms of forbidden subdigraphs we also give several necessary and sufficient conditions for strongly connected w sign solvable signed digraphs

nonsingularity criteria for matrices involving combinatorial considerations abstract we survey related nonsingularity criteria for matrices which involve combinatorial considerations generally involving combinatorial restrictions on the matrices considered and conditions on various graphs of those matrices these criteria include the existence of nonzero diagonal products for matrices over arbitrary fields sign nonsingularity due to j maybee potential nonsingularity and stable nonsingularity due to w anderson all for real matrices and some recent work for matrices over arbitrary fields involving a closure operator on matrices due to the author and d hershkowitz

the migdal kadanoff approximation optimisation generalisation ising model in external fields and the migdal kadanoff hierarchies the simplest migdal kadanoff approximation is optimised on hypercubic lattices generalisation is made to all regular lattices leading to recursion relations continuous in lattice coordination one variety of these relations shows universality whilst another variety shows duality on planar lattices the ising model in external fields both normal and staggered is investigated phase diagrams and exponents are found under a complete treatment of the ambiguity in the handling of external fields by bond moving finally the underlying exactly solved family of hierarchical lattices is considered it is shown that these lattices exhibit a complete breakdown in the concept of universality

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

homogeneous open quantum random walks on a lattice we study open quantum random walks oqrws for which the underlying graph is a lattice and the generators of the walk are homogeneous in space using the results recently obtained in carbone and pautrat ann henri poincare 2015 we study the quantum trajectory associated with the oqrw which is described by a position process and a state process we obtain a central limit theorem and a large deviation principle for the position process we study in detail the case of homogeneous oqrws on the lattice mathbb z d with internal space mathfrak h mathbb c 2

locally self similar phase diagram of the disordered potts model on the hierarchical lattice we study the critical behavior of the random q state potts model in the large q limit on the diamond hierarchical lattice with an effective dimensionality d eff 2 by varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points according to our renormalization group study the phase boundary in the vicinity of the multicritical point is self similar it is well represented by a logarithmic spiral we expect an infinite number of reentrances in the thermodynamic limit consequently one cannot define standard thermodynamic phases in this region

regular and singular orthants of tridiagonal matrices with respect to a tridiagonal matrix with variable diagonal vector g an orthant is said to be regular singular if the matrix is nonsingular singular for all g in it we give necessary and sufficient conditions for an orthant to be regular or singular our idea is based on observations of a simple two by two matrix and all the results obtained are original and self contained

marginal relevance of disorder for pinning models the effect of disorder on pinning and wetting models has attracted much attention in theoretical physics in particular it has been predicted on the basis of the harris criterion that disorder is relevant annealed and quenched model have different critical points and critical exponents if the return probability exponent alpha a positive number that characterizes the model is larger than 1 2 weak disorder has been predicted to be irrelevant i e coinciding critical points and exponents if alpha 1 2 recent mathematical work has put these predictions on firm grounds in renormalization group terms the case alpha 1 2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality the question is particularly intriguing also because the case alpha 1 2 includes the classical models of two dimensional wetting of a rough substrate of pinning of directed polymers on a defect line in dimension 3 1 or 1 1 and of pinning of an heteropolymer by a point potential in three dimensional space here we prove disorder relevance both for the general alpha 1 2 pinning model and for the hierarchical version of the model proposed by b derrida v hakim and j vannimenus jsp 1992 in the sense that we prove a shift of the quenched critical point with respect to the annealed one in both cases we work with gaussian disorder and we show that the shift is at least of order exp 1 beta 4 for beta small if beta is the standard deviation of the disorder

a pathwise ergodic theorem for quantum trajectories if the time evolution of an open quantum system approaches equilibrium in the time mean then on any single trajectory of any of its unravellings the time averaged state approaches the same equilibrium state with probability 1 in the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states

an ergodic theorem for quantum counting processes for a quantum mechanical counting process we show ergodicity under the condition that the underlying open quantum system approaches equilibrium in the time mean this implies equality of time average and ensemble average for correlation functions of the detection current to all orders and with probability 1

passage times exit times and dirichlet problems for open quantum walks we consider open quantum walks on a graph and consider the random variables defined as the passage time and number of visits to a given point of the graph we study in particular the probability that the passage time is finite the expectation of that passage time and the expectation of the number of visits and discuss the notion of recurrence for open quantum walks we also study exit times and exit probabilities from a finite domain and use them to solve dirichlet problems and to determine harmonic measures we consider in particular the case of irreducible open quantum walks the results we obtain extend those for classical markov chains

when the sign pattern of a square matrix determines uniquely the sign pattern of its inverse abstract we give a precise characterization in terms of parity digraphs of those square matrices a such that for every matrix b with the same sign pattern as a b 1 exists and has the same sign pattern as a 1

on the recognition of s systems abstract a sequence a1 an of nonempty subsets of r m is an s system if for each choice of a1 a1 an an the nullspace of the matrix a1 an is a line that penetrates the strictly positive orthant of r n for the case in which each aj is a sign cone s systems are basic to the study of the sign solvability of systems of linear equations in that case they can be recognized in time o m2 the present paper provides several characterizations of general s systems and describes a polynomial time algorithm based on linear programming for the recognition of such systems when the aj are arbitrary finitely presented convex cones

unraveling the large deviation statistics of markovian open quantum systems we analyze dynamical large deviations of quantum trajectories in markovian open quantum systems in their full generality we derive a quantum level 2 5 large deviation principle for these systems which describes the joint fluctuations of time averaged quantum jump rates and of the time averaged quantum state for long times like its level 2 5 counterpart for classical continuous time markov chains which it contains as a special case this description is both explicit and complete as the statistics of arbitrary time extensive dynamical observables can be obtained by contraction from the explicit level 2 5 rate functional we derive our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states for quantum reset processes we show that the unraveled dynamics is semi markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations we finish by discussing how our level 2 5 approach can be used to study large deviations of nonlinear functions of the state such as measures of entanglement

directed polymers in a random medium abstract the problem of a polymer in a random medium can be formulated as follows consider a regular lattice with on each bond ij a random energy ij the energy e w of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk then the problem is to compute the partition function z w exp e w t several recent results will be presented 1 the mean field version of this problem can be solved exactly and one observes a low temperature phase similar to the spin glass phase 2 for finite dimensional lattices one can derive bounds on the transition temperature in high enough dimension 3 for hierarchical lattices the problem can be reduced to finding stable laws when one combines random variables in a non linear way for example z z 1 z 2 z 3 z 4

critical behavior of the random bond ashkin teller model a monte carlo study the critical behaviour of a bond disordered ashkin teller model on a square lattice is investigated by intensive monte carlo simulations a duality transformation is used to locate a critical plane of the disordered model this critical plane corresponds to the line of critical points of the pure model along which critical exponents vary continuously along this line the scaling exponent corresponding to randomness varies continuously and is positive so that randomness is relevant and different critical behaviour is expected for the disordered model we use a cluster algorithm for the monte carlo simulations based on the wolff embedding idea and perform a finite size scaling study of several critical models extrapolating between the critical bond disordered ising and bond disordered four state potts models the critical behaviour of the disordered model is compared with the critical behaviour of an anisotropic ashkin teller model which is used as a refference pure model we find no essential change in the order parameters critical exponents with respect to those of the pure model the divergence of the specific heat c is changed dramatically our results favor a logarithmic type divergence at tc c logl for the random bond ashkin teller and four state potts models and c log logl for the random bond ising model

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

remarks on disorder and aperiodicity in a model for interacting polymers abstract we present a comparative study of the effects of random and aperiodically distributed interactions on the critical behavior of a model for two interacting polymers on a diamond hierarchical lattice the problem is formulated in terms of exact renormalization group rg recursion relations in the disordered case it is possible to develop a perturbative treatment in order to obtain the fixed points of the moments associated with the random distribution of interactions fully uncorrelated disorder may become relevant driving the system away from a homogeneous fixed point layered disorder may lead to a breakdown of the perturbative treatment in the case of aperiodic interactions we also show some examples of relevance and irrelevance of geometric fluctuations and further investigate the models by resorting to an independent transfer matrix tm analysis which fully corroborates the scaling results

on the hardness of efficiently approximating maximal non l submatrices abstract the sign pattern of a real matrix a is the matrix obtained by replacing each entry of a by its sign a real matrix a is an l matrix if every real matrix with the same sign pattern as a has linearly independent columns l matrices arise naturally in and are essential to the study of sign solvability and related notions in special cases the l matrix property has connections to the even dicycle problem pfaffian orientations and polya s permanent problem unfortunately the problem of recognizing l matrices is known to be co np complete in general we elaborate in this vein by showing a polynomial time inapproximability result for max not l rows a particular optimization version of l matrix recognition by means of an approximation preserving reduction from the previously studied problem max not equal 2 sat

quantum dynamical entropy chaotic unitaries and complex hadamard matrices we introduce two information theoretical invariants for the projective unitary group acting on a finite dimensional complex hilbert space pvm and povm dynamical quantum entropies which are analogues of the classical kolmogorov sinai entropy rate they quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator we study the class of chaotic unitaries i e the ones of maximal entropy or equivalently such that they can be represented by suitably rescaled complex hadamard matrices in some orthonormal bases we provide necessary conditions for a unitary operator to be chaotic which become also sufficient for qubits and qutrits these conditions are expressed in terms of the relation between the trace and the determinant of the operator we also compute the volume of the set of chaotic unitaries in dimensions two and three and the average pvm dynamical entropy over the unitary group in dimension two we prove that this mean value behaves as the logarithm of the dimension of the hilbert space which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity

qualitative comparative statics and audits of model performance abstract it is often difficult to determine if large computer based models actually conform to the mathematical descriptions given in their documentation one way to study this problem is through a sensitivity or comparative statics analysis of the model at issue is the degree to which characteristics of the jacobian matrix associated with a solution to the model can be shown to imply characteristics of the inverse jacobian matrix any necessary characteristics of the inverse jacobian can be tested for via comparative statics sign nonsingularity of the jacobian provides an opportunity to audit a model s mathematical documentation through a comparative statics analysis prototype software was developed that tested for sign nonsingularity and applied to the jacobian matrix corresponding to the u s department of energy s oil market simulation model it was found that the signs of all of the elements of the inverse jacobian matrix could be found this

decrease of fourier coefficients of stationary measures let mu be a borel probability measure on mathrm sl 2 mathbb r with a finite exponential moment and assume that the subgroup vargamma mu generated by the support of mu is zariski dense let nu be the unique mu stationary measure on mathbb p 1 we prove that the fourier coefficients hat nu k of nu converge to 0 as k tends to infinity our proof relies on a generalized renewal theorem for the cartan projection

high temperature scaling limit for directed polymers on a hierarchical lattice with bond disorder diamond lattices are sequences of recursively defined graphs that provide a network of directed pathways between two fixed root nodes a and b the construction recipe for diamond graphs depends on a branching number b in mathbb n and a segmenting number s in mathbb n for which a larger value of the ratio s b intuitively corresponds to more opportunities for intersections between two randomly chosen paths by attaching i i d random variables to the bonds of the graphs we construct a random gibbs measure on the set of directed paths by assigning each path an energy through summing the random variables along the path for the case b s we propose a scaling regime in which the temperature grows along with the number of hierarchical layers of the graphs and the partition function the normalization factor of the gibbs measure appears to converge in law we prove that all of the positive integer moments of the partition function converge in this limiting regime the motivation of this work is to prove a functional limit theorem that is analogous to a previous result obtained in the b s case

rates of convergence in invariance principles for random walks on linear groups via martingale methods in this paper we give explicit rates in the central limit theorem and in the almost sure invariance principle for general r d valued cocycles that appear in the study of the left random walk on linear groups our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying markov structure concerning the martingale part the available results in the literature are not accurate enough to give almost optimal rates whether in the central limit theorem for the wasserstein distance or in the strong approximation a part of this paper is devoted to circumvent this issue we then exhibit near optimal rates both in the central limit theorem in terms of wasserstein distance and in the almost sure invariance principle for r d valued martingales with stationary increments having moments of order p 2 3 the case of sequences of reversed martingale differences is also considered note also that as an application of our results for general r d valued cocycles a special attention is paid to the iwasawa cocycle and the cartan projection for reductive lie groups

almost sure functional central limit theorem for the linear random walk on the torus let rho be a probability measure on vargamma mathrm sl d mathbb z consider the random walk defined by rho on the torus mathbb t d mathbb r d mathbb z d for any x in mathbb t d the walk starting at x is defined by begin aligned left begin array l x 0 x x n 1 g n 1 x n end array right end aligned where g n in vargamma mathbb n is chosen with the law rho otimes mathbb n bourgain furmann lindenstrauss and mozes proved that under an assumption on the group generated by the support of rho the random walk starting at any irrational point equidistributes in the torus in this article we study the functional central limit theorem and the almost sure functional central limit theorem for this walk starting at some points having good diophantine properties

on the harris criterion for hierarchical lattices the authors discuss the condition necessary for the harris criteria to be valid on hierarchical lattices they prove that disorder is always relevant when the specific heat exponent alpha p is positive and show how to construct lattices for which disorder becomes relevant while alpha p is still negative

effect of random impurities on fluctuation driven first order transitions we analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of n coupled two dimensional ising models for n 2 the pure model exhibits a fluctuation driven first order transition characterized by runaway renormalization group behaviour we show that the addition of weak randomness acts to stabilize these flows in such a way that the trajectories ultimately flow back towards the pure decoupled ising fixed point with the usual critical exponents apart from logarithmic corrections we also show by examples that in higher dimensions such transitions may either become continuous or remain first order in the presence of randomness

hierarchical lattices some examples with a comparison of intrinsic dimension and connectivity and ising model exponents the recently recognised class of hierarchical lattices is examined through a number of examples definitions of length intrinsic dimension and connectivity are made and used to discuss variations of exponents calculated on the ising model generalisations of regular lattice results are found for exponents at discontinuity fixed points and in 1 epsilon dimensions

a real space renormalisation group approach to electrical and noise properties of percolation clusters the migdal kadanoff renormalisation scheme which becomes exact on hierarchical lattices is used to investigate electrical properties of 2d and 3d percolation clusters the author studies in detail the frequency dependence of the conductivity and the loss angle and the amplification of resistance flicker noise this type of noise exhibits critical amplification with its own exponents x and y which are related to the exponent b recently defined by rammal et al 1985

graph theoretical approach to qualitative solvability of linear systems the structure of sign solvable and strongly sign solvable systems is studied here by a refinement of the graph theoretic approach first suggested by maybee both sign solvability and strong sign solvability are characterized in terms of an associated digraph the problem of recognizing sign solvability is reduced to two subproblems recognizing strong sign solvability and recognizing sign nonsingularity under fairly general conditions on the sign patterns of a it is possible to determine all sign patterns for b which render the system ax b sign solvable

critical behavior of spin and polymer models with aperiodic interactions we review and extend some recent investigations of the effects of aperiodic interactions on the critical behavior of ferromagnetic q state potts models by considering suitable diamond or necklace hierarchical lattices and assuming a distribution of interactions according to a class of two letter substitution rules the problem can be formulated in terms of recursion relations in parameter space the analysis of stability of the fixed points leads to an exact criterion to gauge the relevance of geometric fluctuations for irrelevant fluctuations the critical behavior remains unchanged with respect to the uniform systems for relevant fluctuations there appears a two cycle of saddle point character in parameter space a scaling analysis supported by direct numerical thermodynamic calculations shows the existence of novel critical universality classes associated with relevant geometric fluctuations also we show that similar qualitative results are displayed by a simple model of two directed polymers on a diamond hierarchical structure with aperiodic bond interactions

sign nonsingular matrices and even cycles in directed graphs a sign nonsingular matrix or l mahix a is a real m x n matrix such that the columns of any real m x n matrix with the same sign pattern as a are linearly independent the problem of recognizing square l matrices is equivalent to that of finding an even cycle in a directed graph in this paper we use graph theoretic methods to investigate l matrices in particular we determine the maximum number of nonzero elements in square l matrices and we characterize completely the semicomplete l matrices i e the square l matrices a j such that at least one of a and ajl is nonzero for any i j and those square l matrices which are combinatorially symmetric i e the main diagonal has only nonzero entries and a j 0 iff a jr 0 we also show that for any n x n l matrix there is an i such that the total number of nonzero entries in the i th row and i th column is less than n unless the matrix has a completely specified structure finally we discuss the algorithmic aspects

hierarchical pinning model with site disorder disorder is marginally relevant we study a hierarchical disordered pinning model with site disorder for which like in the bond disordered case derrida et al in j stat phys 66 1189 1213 1992 and giacomin et al in probab theor rel fields 2009 arxiv 0711 4649 math pr there exists a value of a parameter b which enters in the definition of the hierarchical lattice that separates an irrelevant disorder regime and a relevant disorder regime we show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model the proof goes beyond the technique used in giacomin et al probab theor rel fields 2009 arxiv 0711 4649 math pr and it takes explicitly advantage of the inhomogeneous character of the green function of the model

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

recursive structure of s matrices and an o m2 algorithm for recognizing strong sign solvability abstract an s matrix is an m m 1 real matrix a such that for each matrix a of the same sign pattern as a a s columns are the vertices of an m simplex in r m that has the origin in its interior an s matrix is one that can be made into an s matrix by replacing some columns with their negatives such matrices are of interest because of their fundamental role in the study of sign solvability new results on the recursive structure of these classes of matrices are presented here and are used as the basis of algorithms of time complexity o m 2 for recognizing members of the classes and for testing the strong sign solvability of linear systems