dna breathing dynamics analytic results for distribution functions of relevant brownian functionals we investigate dna breathing dynamics by suggesting and examining several brownian functionals associated with bubble lifetime and reactivity bubble dynamics is described as an overdamped random walk in the number of broken base pairs the walk takes place on the poland scheraga free energy landscape we suggest several probability distribution functions that characterize the breathing process and adopt the recently studied backward fokker planck method and the path decomposition method as elegant and flexible tools for deriving these distributions in particular for a bubble of an initial size x 0 we derive analytical expressions for i the distribution p t f x 0 of the first passage time t f characterizing the bubble lifetime ii the distribution p a x 0 of the area a until the first passage time providing information about the effective reactivity of the bubble to processes within the dna iii the distribution p m of the maximum bubble size m attained before the first passage time and iv the joint probability distribution p m t m of the maximum bubble size m and the time t m of its occurrence before the first passage time these distributions are analyzed in the limit of small and large bubble sizes we supplement our analytical predictions with direct numerical simulations of the related langevin equation and obtain a very good agreement in the appropriate limits the nontrivial scaling behavior of the various quantities analyzed here can in principle be explored experimentally

realizing rotation vectors for torus homeomorphisms we consider the rotation set p f for a lift f of a homeomorphism f t2 t2 which is homotopic to the identity our main result is that if a vector v lies in the interior of p f and has both coordinates rational then there is a periodic point x e t2 with the property that

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

statistical mechanics of a single particle in a multiscale random potential parisi landscapes in finite dimensional euclidean spaces we construct an n dimensional gaussian landscape with multiscale translation invariant logarithmic correlations and investigate the statistical mechanics of a single particle in this environment in the limit of high dimension n the free energy of the system and overlap function are calculated exactly using the replica trick and parisi s hierarchical ansatz in the thermodynamic limit we recover the most general version of the derrida s generalized random energy model grem the low temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape if the latter consists of k discrete values the system is characterized by a k step replica symmetry breaking solution we argue that our construction is in fact valid in any finite spatial dimensions n 1 we discuss the implications of our results for the singularity spectrum describing multifractality of the associated boltzmann gibbs measure finally we discuss several generalizations and open problems such as the dynamics in such a landscape and the construction of a generalized multifractal random walk

rotation and entropy for a given map f x x and an observable o x llkd rotation vectors are the limits of ergodic averages of w we study which part of the topological entropy of f is associated to a given rotation vector and which part is associated with many rotation vectors according to this distinction we introduce directional and lost entropies we discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps

extreme value theory for constrained physical systems we investigate extreme value theory for physical systems with a global conservation law which describes renewal processes mass transport models and long range interacting spin models as shown previously a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support we expose exact relationships between constrained extreme value theory and well known quantities of the underlying stochastic dynamics all valid beyond the midpoint in general i e even far from the thermodynamic limit for example for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events in the thermodynamic limit we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory for example for the renewal process we unravel dual scaling of the extreme value distribution pointing out two types of limiting laws a normalizable scaling function for the typical statistics and a non normalized state describing the rare events

extreme value distributions and renormalization group in the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent identically distributed random variables are studied the vast majority of the literature on the subject deals with affine normalization we argue that more general normalizations are natural from a mathematical and physical point of view and work them out the problem is approached using the language of renormalization group transformations in the space of probability densities the limit distributions are fixed points of the transformation and the study of its differential around them allows a local analysis of the domains of attraction and the computation of finite size corrections

generalization errors of the simple perceptron to find an exact form for the generalization error of a learning machine is an open problem even in the simplest case simple perceptron learning we introduce a new approach to tackle the problem the generalization error of the simple perceptron is expressed as a linear combination of extreme values of inputs with the help of extreme value theory in statistics we then obtain an exact form of the generalization error of the simple perceptron in the case of the worst learning generalization errors of the higher order perceptron taking the form of an inverse power law in the number of examples are also considered

extremum statistics in scale free network models we investigate the statistics of the most connected node in scale free networks for a scale free network model with homogeneous nodes we show by means of extensive simulations that the exponential truncation due to the finite size of the network of the degree distribution governs the scaling of the extreme values and that the distribution of maxima follows the gumbel statistics for a scale free network model with heterogeneous nodes we show that scaling no longer holds and that the truncation of the degree distribution no longer controls the maxima distribution

second order asymptotics for matrix models we study several matrix models and show that when the potential is convex and a small perturbation of the gaussian potential the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one in order to do that we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges

fluctuating fronts as correlated extreme value problems an example of gaussian statistics in this paper we view fluctuating fronts made of particles on a one dimensional lattice as an extreme value problem the idea is to denote the configuration for a single front realization at time t by the set of co ordinates k i t identical with k 1 t k 2 t k n t t of the constituent particles where n t is the total number of particles in that realization at time t when k i t are arranged in the ascending order of magnitudes the instantaneous front position can be denoted by the location of the rightmost particle i e by the extremal value k f t max k 1 t k 2 t k n t t due to interparticle interactions k i t at two different times for a single front realization are naturally not independent of each other and thus the probability distribution p k f t based on an ensemble of such front realizations describes extreme value statistics for a set of correlated random variables in view of the fact that exact results for correlated extreme value statistics are rather rare here we show that for a fermionic front model in a reaction diffusion system p k f t is gaussian in a bosonic front model however we observe small deviations from the gaussian

modelling extremal events using gnedenko distributions we address the problem of fitting a gnedenko distribution to a realization of an iid sample thereby deciding upon the heavy light character of the upper tail of the phenomenon under study an illustration based on earthquake magnitude data is supplied

renormalization flow for extreme value statistics of random variables raised to a varying power using a renormalization approach we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power qn that varies monotonically with the sample size n under these conditions a non standard class of max stable limit distributions which mirror the classical ones emerges furthermore a transition mechanism between the classical and the non standard limit distributions is brought to light if qn grows slower than a characteristic function q n the standard limit distributions are recovered while if qn behaves asymptotically as q n non standard limit distributions emerge

renormalization group theory for finite size scaling in extreme statistics we present a renormalization group rg approach to explain universal features of extreme statistics applied here to independent identically distributed variables the outlines of the theory have been described in a previous paper the main result being that finite size shape corrections to the limit distribution can be obtained from a linearization of the rg transformation near a fixed point leading to the computation of stable perturbations as eigenfunctions here we show details of the rg theory which exhibit remarkable similarities to the rg known in statistical physics besides the fixed points explaining universality and the least stable eigendirections accounting for convergence rates and shape corrections the similarities include marginally stable perturbations which turn out to be generic for the fisher tippett gumbel class distribution functions containing unstable perturbations are also considered we find that after a transitory divergence they return to the universal fixed line at the same or at a different point depending on the type of perturbation

universality classes for self similarity of noiseless multidimensional burgers turbulence and interface growth the present work is devoted to the evolution of random solutions of the unforced burgers and kpz equations in d dimensions in the limit of vanishing viscosity we consider a cellular model and as initial condition assign a value for the velocity potential chosen independently within each cell we show that the asymptotic behavior of the turbulence at large times is determined by the tail of the initial potential probability distribution function three classes of initial distribution leading to self similar evolution are identified a distributions with a power law tail b compactly supported potential c stretched exponential tails in class c we find that the mean potential mean height of the surface increases logarithmically with time and the turbulence energy e t mean square gradient of the surface decays as 1 t times a logarithmic correction in classes a and b we find that the changes in the mean potential and energy have a power law time dependence in class c the roughness of the surface measured by its mean square gradient may either decrease or increase with time we discuss also the influence of finite viscosity and long range correlation on the late stage evolution of the burgers turbulence

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

perturbative expansion for the maximum of fractional brownian motion brownian motion is the only random process which is gaussian stationary and markovian dropping the markovian property i e allowing for memory one obtains a class of processes called fractional brownian motion indexed by the hurst exponent h for h 1 2 brownian motion is recovered we develop a perturbative approach to treat the non locality in time in an expansion in varepsilon h 1 2 this allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics the maximum m of the process and the time t text max at which this maximum is reached as well as their joint distribution we test our analytical predictions with extensive numerical simulations for different values of h they show excellent agreement even for h far from 1 2

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

localized growth and branching random walks with time correlations we generalize a model of growth over a disordered environment to a large class of it processes in particular we study how the microscopic properties of the noise influence the macroscopic growth rate the present model can account for growth processes in large dimensions and provides a bed to understand better the tradeoff between exploration and exploitation an additional mapping to the schrodinger equation readily provides a set of disorders for which this model can be solved exactly this mean field approach exhibits interesting features such as a freezing transition and an optimal point of growth which can be studied in detail and gives yet another explanation for the occurrence of the zipf law in complex well connected systems

the topology and dynamics of the hyperspaces of normal fuzzy sets and their inverse limit spaces given a compact and connected metric space continuum x we study topological and dynamical properties of the hyperspace of normal fuzzy sets f1 x equipped with the hausdorff endograph or sendograph metric among the many results we show that it is contractible path connected locally contractible locally path connected locally simply connected and locally connected for the endograph metric the hyperspace f1 x is a continuum and then for a topological graph x we show how using the inverse limit approach of barge and martin the inverse limit of a fuzzy dynamical system on x can be realized as an attractor of a fuzzy dynamical system on a manifold

jamming hysteresis and oscillation in scalar models for shear thickening we investigate shear thickening and jamming within the framework of a family of spatially homogeneous scalar rheological models these are based on the soft glassy rheology model of sollich et al phys rev lett 78 2020 1997 but with an effective temperature x that is a decreasing function of either the global stress ensuremath sigma or the local strain l for appropriate x x ensuremath sigma it is shown that the flow curves include a region of negative slope around which the stress exhibits hysteresis under a cyclically varying imposed strain rate stackrel ifmmode dot else fi ensuremath gamma a subclass of these x ensuremath sigma have flow curves that touch the stackrel ifmmode dot else fi ensuremath gamma 0 axis for a finite range of stresses imposing a stress from this range jams the system in the sense that the strain ensuremath gamma creeps only logarithmically with time t ensuremath gamma t ensuremath sim mathrm ln t these same systems may produce a finite asymptotic yield stress under an imposed strain in a manner that depends on the entire stress history of the sample a phenomenon we refer to as history dependent jamming in contrast when x x l the flow curves are always monotonic but we show that some x l generate an oscillatory strain response for a range of steady imposed stresses similar spontaneous oscillations are observed in a simplified model with fewer degrees of freedom we discuss this result in relation to the temporal instabilities observed in rheological experiments and stick slip behavior found in other contexts and comment on the possible relationship with delay differential equations that are known to produce oscillations and chaos

universality classes for extreme value statistics the equilibrium low temperature physics of disordered systems is governed by the statistics of extremely low energy states it is thus relevant to discuss the possible universality classes for extreme value statistics we compare the usual probabilistic classification to the results of the replica approach we show in detail for several problems including the random energy model and the decaying burgers turbulence that one class of independent variables corresponds exactly to the so called one step replica symmetry breaking solution in the replica language we argue that this universality class holds if the correlations are sufficiently weak and propose a conjecture on the level of correlations which leads to different universality classes

synchronization landscapes in small world connected computer networks motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small world networks embedded in one and two dimensions we find that the synchronization landscape corresponding to the progress of the individual processors exhibits kardar parisi zhang like kinetic roughening on regular networks with short range communication links although the processors on average progress at a nonzero rate their spread the width of the synchronization landscape diverges with the number of nodes desynchronized state hindering efficient data management when random communication links are added on top of the one and two dimensional regular networks resulting in a small world network large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system size limit synchronized state in the resulting synchronization scheme the processors make close to uniform progress with a nonzero rate without global intervention we obtain our results by simulating the simulations based on the exact algorithmic rules supported by coarse grained arguments

generalized extreme value statistics and sum of correlated variables we show that generalized extreme value statistics the statistics of the kth largest value among a large set of random variables can be mapped onto a problem of random sums this allows us to identify classes of non identical and generally correlated random variables with a sum distributed according to one of the three k dependent asymptotic distributions of extreme value statistics namely the gumbel frechet and weibull distributions these classes as well as the limit distributions are naturally extended to real values of k thus providing a clear interpretation to the onset of gumbel distributions with non integer index k in the statistics of global observables this is one of the very few known generalizations of the central limit theorem to non independent random variables finally in the context of a simple physical model we relate the index k to the ratio of the correlation length to the system size which remains finite in strongly correlated systems

extremal driving as a mechanism for generating long term memory it is argued that systems whose elements are renewed according to an extremal criterion can generally be expected to exhibit long term memory this is verified for the minimal extremally driven model which is first defined and then solved for all system sizes n 2 and times t 0 yielding exact expressions for the persistence r t 1 t n 1 1 and the two time correlation function c tw t tw 1 1 n n tw n tw t 1 the existence of long term memory is inferred from the scaling of c tw t tw f t tw denoting aging finally we suggest ways of investigating the robustness of this mechanism when competing processes are present

gumbel central limit theorem for max min and min max the max min and min max of matrices arise prevalently in science and engineering however in many real world situations the computation of the max min and min max is challenging as matrices are large and full information about their entries is lacking here we take a statistical physics approach and establish limit laws akin to the central limit theorem for the max min and min max of large random matrices the limit laws intertwine random matrix theory and extreme value theory couple the matrix dimensions geometrically and assert that gumbel statistics emerge irrespective of the matrix entries distribution due to their generality and universality as well as their practicality these results are expected to have a host of applications in the physical sciences and beyond

rotation entropy and equilibrium states for a dynamical system x t and function f x rd we consider the corresponding generalised rotation set this is the convex subset of rd consisting of all integrals of f with respect to t invariant probability measures we study the entropy h e of rotation vectors e and relate this to the directional entropy lhl l of geller misiurewicz for x t a mixing subshift of finite type and f of summable variation we prove that if the rotation set is strictly convex then the functions h and h are in fact one and the same for those rotation sets which are not strictly convex we prove that 1h e and h e can differ only at non exposed boundary points e

airy distribution function from the area under a brownian excursion to the maximal height of fluctuating interfaces the airy distribution function describes the probability distribution of the area under a brownian excursion over a unit interval surprisingly this function has appeared in a number of seemingly unrelated problems mostly in computer science and graph theory in this paper we show that this distribution function also appears in a rather well studied physical system namely the fluctuating interfaces we present an exact solution for the distribution p h m l of the maximal height h m measured with respect to the average spatial height in the steady state of a fluctuating interface in a one dimensional system of size l with both periodic and free boundary conditions for the periodic case we show that p h m l l 1 2f h m l 1 2 for all l 0 where the function f x is the airy distribution function this result is valid for both the edwards wilkinson ew and the kardar parisi zhang interfaces for the free boundary case the same scaling holds p h m l l 1 2f h m l 1 2 but the scaling function f x is different from that of the periodic case we compute this scaling function explicitly for the ew interface and call it the f airy distribution function numerical simulations are in excellent agreement with our analytical results our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables some of these results were announced in a recent letter s n majumdar and a comtet phys rev lett 92 225501 2004

multifractality and freezing phenomena in random energy landscapes an introduction abstract we start our lectures with introducing and discussing the general notion of multifractality spectrum for random measures on lattices and how it can be probed using moments of that measure then we show that the boltzmann gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet non trivial example of disorder induced multifractal measures the typical values of the multifractality exponents can be extracted from calculating the free energy of the associated statistical mechanics problem to succeed in such a calculation we introduce and discuss in some detail two analytically tractable models for logarithmically correlated potentials the first model uses a special definition of distances between points in space and is based on the idea of multiplicative cascades which originated in theory of turbulent motion it is essentially equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by derrida and spohn 1988 in ref 12 in this way we introduce the notion of the freezing transition which is identified with an abrupt change in the multifractality spectrum second model which allows for explicit analytical evaluation of the free energy is the infinite dimensional version of the problem which can be solved by employing the replica trick in particular the latter version allows one to identify the freezing phenomenon with a mechanism of the replica symmetry breaking rsb and to elucidate its physical meaning the corresponding one step rsb solution turns out to be marginally stable everywhere in the low temperature phase we finish with a short discussion of recent developments and extensions of models with logarithmic correlations in particular in the context of extreme value statistics the first appendix summarizes the standard elementary information about gaussian integrals and related subjects and introduces the notion of the gaussian free field characterized by logarithmic correlations three other appendices provide the detailed exposition of a few technical details underlying the replica analysis of the model discussed in the lectures

probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d 1 2 3 via a monte carlo procedure in the disorder in order to probe with high precision the tails of the ground state energy distribution of disordered spin systems k orner katzgraber and hartmann have recently proposed an importance sampling monte carlo markov chain in the disorder in this paper we combine their monte carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution p d e 0 for the directed polymer in a random medium of dimension d 1 2 3 in d 1 we check the validity of the algorithm by a direct comparison with the exact result namely the tracy widom distribution in dimensions d 2 and d 3 we measure the negative tail up to ten standard deviations which correspond to probabilities of order p d e 0 ensuremath sim 10 ensuremath 22 our results are in agreement with zhang s argument stating that the negative tail exponent ensuremath eta d of the asymptotic behavior mathrm ln phantom rule 0 2em 0ex p d e 0 ensuremath sim ensuremath ensuremath mid e 0 ensuremath mid ensuremath eta d as e 0 ensuremath rightarrow ensuremath ensuremath infty is directly related to the fluctuation exponent ensuremath theta d which governs the fluctuations ensuremath delta e 0 l ensuremath sim l ensuremath theta d of the ground state energy e 0 for polymers of length l via the simple formula ensuremath eta d 1 1 ensuremath ensuremath theta d throughout the paper we comment on the similarities and differences with spin glasses

statistics of extremal intensities for gaussian interfaces the extremal fourier intensities are studied for stationary edwards wilkinson type gaussian interfaces with power law dispersion we calculate the probability distribution of the maximal intensity and find that generically it does not coincide with the distribution of the integrated power spectrum i e roughness of the surface nor does it obey any of the known extreme statistics limit distributions the fisher tippett gumbel limit distribution is however recovered in three cases i in the nondispersive white noise limit ii for high dimensions and iii when only short wavelength modes are kept in the last two cases the limit distribution emerges in nonconventional scenarios

new rotation sets in a family of torus homeomorphisms we construct a family varphi t t in 0 1 of homeomorphisms of the two torus isotopic to the identity for which all of the rotation sets rho varphi t can be described explicitly we analyze the bifurcations and typical behavior of rotation sets in the family providing insight into the general questions of toral rotation set bifurcations and prevalence we show that there is a full measure subset of 0 1 consisting of infinitely many mutually disjoint non trivial closed intervals on each of which the rotation set mode locks to a constant polygon with rational vertices that the generic rotation set in the hausdorff topology has infinitely many extreme points accumulating on a single totally irrational extreme point at which there is a unique supporting line and that although rho varphi t varies continuously with t the set of extreme points of rho varphi t does not the family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set or by any directional ergodic measure

interfaces in a random medium and replica symmetry breaking the authors develop a variational approach for studying interfaces and other manifolds in a disordered quenched medium the method may be applied to problems which range from directed polymers to the interface of an ising model in a random magnetic field they find that replica symmetry is spontaneously broken and the results of the flory approximation are recovered in a simple way corrections to this approximation may be computed in a systematic way

extreme value statistics of hierarchically correlated variables deviation from gumbel statistics and anomalous persistence we study analytically the distribution of the minimum of a set of hierarchically correlated random variables e1 e2 ellipsis e n where e i represents the energy of the ith path of a directed polymer on a cayley tree if the variables were uncorrelated the minimum energy would have an asymptotic gumbel distribution we show that due to the hierarchical correlations the forward tail of the distribution of the minimum energy becomes highly nonuniversal depends explicitly on the distribution of the bond energies epsilon and is generically different from the superexponential forward tail of the gumbel distribution the consequence of these results to the persistence of hierarchically correlated random variables is discussed and the persistence is also shown to be generically anomalous

linearization of conservative toral homeomorphisms we give an equivalent condition for the existence of a semi conjugacy to an irrational rotation for conservative homeomorphisms of the two torus this leads to an analogue of poincare s classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property for minimal toral homeomorphisms the result extends to arbitrary dimensions further we provide a basic classification for the dynamics of toral homeomorphisms with all points non wandering

on a non linear fluctuation theorem for the ageing dynamics of disordered trap models we consider the dynamics of the disordered trap model which is known to be completely out of equilibrium and to present strong localization effects in its ageing phase we are interested in the influence of an external force when it is applied from the very beginning at t 0 or only after a waiting time tw we obtain a non linear fluctuation theorem for the corresponding one time and two time diffusion fronts in any given sample which implies the following consequences i for fixed times there exists a linear response regime where the fluctuation dissipation relation or einstein relation is valid even in the ageing time sector in contrast with other ageing disordered systems ii for a fixed waiting time and fixed external field the validity of the linear response regime is limited in time by a characteristic time depending on the external force iii in the non linear response regime there exists a very simple relation for the asymmetry in the position for the one time and two times disorder averaged diffusion fronts in the presence of an external force in contrast to other models of random walks in random media the present non linear fluctuation theorem is a consequence of a special dynamical symmetry of the trap model

shapes of tree representations of spin glass landscapes much of the information about the multi valley structure of disordered spin systems can be convened in a simple tree structure a barrier tree the leaves and internal nodes of which represent respectively the local minima and the lowest energy saddles connecting those minima here we apply several statistics used in the study of phylogenetic trees to barrier trees that result from the energy landscapes of p spin models these statistics give information about the shape of these barrier trees in particular about balance and symmetry we then ask if they can be used to classify different types of landscapes compare them with results obtained from random trees and investigate the structure of subtrees of the barrier trees we conclude that at least one of the used statistics is capable of distinguishing different types of landscapes that the barrier trees from p spin energy landscapes are quite different from random trees and that subtrees of barrier trees do not reflect the overall tree structure but their structure is correlated with their depth in the tree

convergence of large deviation estimators we study the convergence of statistical estimators used in the estimation of large deviation functions describing the fluctuations of equilibrium nonequilibrium and manmade stochastic systems we give conditions for the convergence of these estimators with sample size based on the boundedness or unboundedness of the quantity sampled and discuss how statistical errors should be defined in different parts of the convergence region our results shed light on previous reports of phase transitions in the statistics of free energy estimators and establish a general framework for reliably estimating large deviation functions from simulation and experimental data and identifying parameter regions where this estimation converges

statistical properties and network structure of the marriage problem abstract we study the statistical properties of the distribution of total energies or costs over the set of n configurations for the marriage problem with n men and n women analytical results and exhaustive numerical calculations for the whole configuration space are presented taking into account several definitions of the individual energies the distribution of the minimal energy is evaluated approximately moreover we point out a natural association between the configuration space of the marriage problem and a network we study the topological and metric properties of this network as well as the distribution of local energy minima on it

density of near extreme events we provide a quantitative analysis of the phenomenon of crowding of near extreme events by computing exactly the density of states dos near the maximum of a set of independent and identically distributed random variables we show that the mean dos converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential faster than pure exponential or as a pure exponential function we argue that some of these results would remain valid even for certain correlated cases and verify it for power law correlated stationary gaussian sequences satisfactory agreement is found between the near maximum crowding in the summer temperature reconstruction data of western siberia and the theoretical prediction

central limit theorem for a class of globally correlated random variables the standard central limit theorem with a gaussian attractor for the sum of independent random variables may lose its validity in the presence of strong correlations between the added random contributions here we study this problem for similar interchangeable globally correlated random variables under these conditions a hierarchical set of equations is derived for the conditional transition probabilities this result allows us to define different classes of memory mechanisms that depend on a symmetric way on all involved variables depending on the correlation mechanisms and statistics of the single variables the corresponding sums are characterized by distinct probability densities for a class of urn models it is also possible to characterize their domain of attraction which as in the standard case is parametrized by the probability density of each random variable symmetric and asymmetric q gaussian attractors q 1 arise in a particular two state case of these urn models

maximum relative height of elastic interfaces in random media the distribution of the maximal relative height mrh of self affine one dimensional elastic interfaces in a random potential is studied we analyze the ground state configuration at zero driving force and the critical configuration exactly at the depinning threshold both for the random manifold and random periodic universality classes these configurations are sampled by exact numerical methods and their mrh distributions are compared with those with the same roughness exponent and boundary conditions but produced by independent fourier modes with normally distributed amplitudes using pickands theorem we derive an exact analytical description for the right tail of the latter after properly rescaling the mrh distributions we find that corrections from the gaussian independent modes approximation are in general small as previously found for the average width distribution of depinning configurations in the large size limit all corrections are finite except for the ground state in the random periodic class whose mrh distribution becomes for periodic boundary conditions indistinguishable from the airy distribution we find that the mrh distributions are in general sensitive to changes of boundary conditions

distribution of the time at which n vicious walkers reach their maximal height we study the extreme statistics of n nonintersecting brownian motions vicious walkers over a unit time interval in one dimension using path integral techniques we compute exactly the joint distribution of the maximum m and of the time m at which this maximum is reached we focus in particular on nonintersecting brownian bridges watermelons without wall and nonintersecting brownian excursions watermelons with a wall we discuss in detail the relationships between such vicious walkers models in watermelon configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium dprm with one free end point on the other hand we also check our results using numerical simulations of dyson s brownian motion and confront them with numerical simulations of the polynuclear growth model png and of a model of dprm on a discrete lattice some of the results presented here were announced in a recent letter j rambeau and g schehr europhys lett 91 60006 2010

on digit frequencies in beta expansions we study the sets df of digit frequencies of expansions of numbers in 0 1 we show that df is a compact convex set with countably many extreme points which varies continuously with that there is a full measure collection of non trivial closed intervals on each of which df mode locks to a constant polytope with rational vertices and that the generic digit frequency set has infinitely many extreme points accumulating on a single non rational extreme point whose components are rationally independent

critical weight statistics of the random energy model and of the directed polymer on the cayley tree we consider the critical point of two mean field disordered models i the random energy model rem introduced by derrida as a mean field spin glass model of n spins and ii the directed polymer of length n on a cayley tree dpct with random bond energies both models are known to exhibit a freezing transition between a high temperature phase where the entropy is extensive and a low temperature phase of finite entropy where the weight statistics coincides with the weight statistics of l evy sums with index ensuremath mu t t c l1 in this paper we study the weight statistics at criticality via the entropy s ensuremath ensuremath sum w i phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex w i and the generalized moments y k ensuremath sum w i k where the w i are the boltzmann weights of the 2 n configurations in the rem we find that the critical weight statistics is governed by the finite size exponent ensuremath nu 2 the entropy scales as overline s n t c ensuremath sim n 1 2 the typical values e overline mathrm ln phantom rule 0 2em 0ex y k decay as n ensuremath k 2 and the disorder averaged values overline y k are governed by rare events and decay as n ensuremath 1 2 for any kg1 for the dpct we find that the entropy scales similarly as overline s n t c ensuremath sim n 1 2 whereas another exponent ensuremath nu ensuremath 1 governs the y k statistics the typical values e overline mathrm ln phantom rule 0 2em 0ex y k decay as n ensuremath k and the disorder averaged values overline y k decay as n ensuremath 1 for any kg1 as a consequence the asymptotic probability distribution overline ensuremath pi n ensuremath infty q of the overlap q in addition to the delta function ensuremath delta q which bears the whole normalization contains an isolated point at q 1 as a memory of the delta peak 1 ensuremath t t c ensuremath delta q ensuremath 1 of the low temperature phase tl t c the associated value overline ensuremath pi n ensuremath infty q 1 is finite for the dpct and diverges as overline ensuremath pi n ensuremath infty q 1 ensuremath sim n 1 2 for the rem

aging in two dimensional bouchaud s model let e x be a collection of i i d exponential random variables symmetric bouchaud s model on 2 is a markov chain x t whose transition rates are given by w xy exp e x if x y are neighbours in 2 we study the behaviour of two correlation functions x t w t x t w and x t x t w t t w t w t we prove the sub aging behaviour of these functions when 1

a branching random walk seen from the tip we show that all the time dependent statistical properties of the rightmost points of a branching brownian motion can be extracted from the traveling wave solutions of the fisher kpp equation the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the fisher kpp traveling waves when the initial condition is modified the limiting measure exhibits the surprising property of superposability the statistical properties of the distances between the rightmost points of the union of two realizations of the branching brownian motion shifted by arbitrary amounts are the same as those of a single realization we discuss the extension of our results to more general branching random walks

existence of non contractible periodic orbits for homeomorphisms of the open annulus in this article we consider homeomorphisms of the open annulus mathbb a mathbb r mathbb z times mathbb r which are isotopic to the identity and preserve a borel probability measure of full support focusing on the existence of non contractible periodic orbits assume f is such homeomorphism such that the connected components of the set of fixed points of f are all compact further assume that there exists check f a lift of f to the universal covering of mathbb a such that the set of fixed points of check f is non empty and that this set projects into an open topological disk of mathbb a we prove that in this setting one of the following two conditions must be satisfied 1 f has non contractible periodic points of arbitrarily large periodic or 2 for every compact set k of mathbb a there exists a constant m depending on the compact set such that if check z and check f n check z project on k then their projections on the first coordinate have distance less or equal to m

order statistics of 1 fα signals order statistics of periodic gaussian noise with 1 f power spectrum is investigated using simulations and phenomenological arguments we find three scaling regimes for the average gap d k x k x k 1 between the kth and k 1 st largest values of the signal the result d k k 1 known for independent identically distributed variables remains valid for 0 5 the spectra of average ordered values e k x 1 x k k is also examined the exponent is derived from the gap scaling as well as by relating e k to the density of near extreme states known results for the density of near extreme states combined with scaling suggest that 2 1 2 4 3 2 and 2 are exact values we also show that parallels can be drawn between e k and the quantum mechanical spectra of a particle in power law potentials

a note about proving non γ under a finite non microstates free fisher information assumption we prove that if x1 xn n 1 are self adjoints in a w probability space with finite non microstates free fisher information then the von neumann algebra w x1 xn they generate doesn t have property especially is not amenable this is an analog of a well known result of voiculescu for microstates free entropy we also prove factoriality under finite non microstates entropy

freezing transition of the random bond rna model statistical properties of the pairing weights to characterize the pairing specificity of rna secondary structures as a function of temperature we analyze the statistics of the pairing weights as follows for each base i of the sequence of length n we consider the n 1 pairing weights w i j with the other bases j not equal i of the sequence we numerically compute the probability distributions p1 w of the maximal weight w i max max j w i j the probability distribution pi y 2 of the parameter y2 i summation operator j w i 2 j as well as the average values of the moments y k i summation operator j w i k j we find that there are two important temperatures t c t gap the distribution p1 w vanishes at some value w 0 t 1 and accordingly the moments y k i decay exponentially as w 0 t k in k for t t gap the distributions p1 w and pi y2 present the characteristic derrida flyvbjerg singularities at w 1n and y 2 1 n for n 1 2 in particular there exists a temperature dependent exponent mu t that governs the singularities p1 w approximately 1 w mu t 1 and pi y2 approximately 1 y 2 mu t 1 as well as the power law decay of the moments y k i approximately 1k mu t the exponent mu t grows from the value mu t 0 0 up to mu t gap approximately 2 the study of spatial properties indicates that the critical temperature t c where the large scale roughness exponent changes from the low temperature value zeta approximately 0 67 to the high temperature value zeta approximately 0 5 corresponds to the exponent mu t c 1 for t t c there exists frozen pairs of all sizes whereas for t c t t gap there exists frozen pairs but only up to some characteristic length diverging as xi t approximately 1 t c t nu with nu approximately 2 the similarities and differences with the weight statistics in levy sums and in derrida s random energy model are discussed

microscopic realizations of the trap model monte carlo optimizations of number partitioning and of diophantine approximations are microscopic realizations of trap model dynamics this offers a fresh look at the physics behind this model and points at other situations in which it may apply our results strongly suggest that in any such realization of the trap model the response and correlation functions of smooth observables obey the fluctuation dissipation theorem even in the aging regime our discussion of the number partitioning problem may be relevant for the class of optimization problems whose cost function does not scale linearly with the size and are thus awkward from the statistical mechanic point of view

numerical study of the two replica overlap of the 3d edwards anderson ising spin glass abstract we present results of recent high statistics monte carlo simulations of the edwards anderson ising spin glass model in three dimensions the study is based on a non boltzmann sampling technique the multi self overlap algorithm which is specifically tailored for sampling rare event states we thus concentrate on those properties which are difficult to obtain with standard canonical boltzmann sampling such as the free energy barriers f b q in the probability densities p j q of the parisi overlap parameter q and the behavior of the tails of the disorder averaged density p q p j q av our results for the tails disagree with mean field predictions and support extreme order statistics over many orders of magnitude

law of corresponding states for open collaborations we study the relation between number of contributors and product size in wikipedia and github in contrast to traditional production this is strongly probabilistic but is characterized by two quantitative nonlinear laws a power law bound to product size for increasing number of contributors and the universal collapse of rescaled distributions a variant of the random energy model shows that both laws are due to the heterogeneity of contributors and displays an intriguing finite size scaling property with no equivalent in standard systems the analysis uncovers the right intensive densities enabling the comparison of projects with different numbers of contributors on equal grounds we use this property to expose the detrimental effects of conflicting interactions in wikipedia

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

phase separation in random cluster models i uniform upper bounds on local deviation this is the first in a series of three papers that addresses the behaviour of the droplet that results in the percolating phase from conditioning the planar fortuin kasteleyn random cluster model on the presence of an open dual circuit 0 encircling the origin and enclosing an area of at least or exactly n2 by the fortuin kasteleyn representation the model is a close relative of the droplet formed by conditioning the potts model on an excess of spins of a given type we consider local deviation of the droplet boundary measured in a radial sense by the maximum local roughness mlr 0 this being the maximum distance from a point in the circuit 0 to the boundary of the circuit s convex hull and in a longitudinal sense by what we term maximum facet length mlf 0 namely the length of the longest line segment of which the polygon is formed the principal conclusion of the series of papers is the following uniform control on local deviation that there are constants 0 c c such that the conditional probability that the normalized quantity n 1 3 log n 2 3mlr lies in the interval c c tends to 1 in the high n limit and that the same statement holds for n 2 3 log n 1 3 mlf in this way we confirm the anticipated n1 3 scaling of maximum local roughness and provide a sharp logarithmic power law correction this local deviation behaviour occurs by means of locally gaussian effects constrained globally by curvature and we believe that it arises in many radially defined stochastic interface models including growth models belonging to the kardar parisi zhang universality class

magnetic fluctuations in the classical xy model the origin of an exponential tail in a complex system we study the probability density function for the fluctuations of the magnetic order parameter in the low temperature phase of the mathrm xy model of finite size in two dimensions this system is critical over the whole of the low temperature phase it is shown analytically and without recourse to the scaling hypothesis that in this case the distribution is non gaussian and of universal form independent of both system size and critical exponent ensuremath eta an exact expression for the generating function of the distribution is obtained which is transformed and compared with numerical data from high resolution molecular dynamics and monte carlo simulations the asymptotes of the distribution are calculated and found to be of exponential and double exponential form the calculated distribution is fitted to three standard functions a generalization of gumbel s first asymptote distribution from the theory of extremal statistics a generalized log normal distribution and a ensuremath chi 2 distribution the calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4 despite the fact that critical fluctuations are limited to d 2 these results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

wealth condensation in a simple model of economy abstract we introduce a simple model of economy where the time evolution is described by an equation capturing both exchange between individuals and random speculative trading in such a way that the fundamental symmetry of the economy under an arbitrary change of monetary units is insured we investigate a mean field limit of this equation and show that the distribution of wealth is of the pareto power law type the pareto behaviour of the tails of this distribution appears to be robust for finite range models as shown using both a mapping to the random directed polymer problem as well as numerical simulations in this context a phase transition between an economy dominated by a few individuals and a situation where the wealth is more evenly spread out is found an interesting outcome is that the distribution of wealth tends to be very broadly distributed when exchanges are limited either in amplitude or topologically favouring exchanges and less surprisingly increasing taxes seems to be an efficient way to reduce inequalities

extreme statistics for time series distribution of the maximum relative to the initial value the extreme statistics of time signals is studied when the maximum is measured from the initial value in the case of independent identically distributed iid variables we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn then we turn to correlated periodic gaussian signals with a 1 f ensuremath alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height mathrm mrh mathrm i the exact mathrm mrh mathrm i distribution is derived for ensuremath alpha 0 iid variables ensuremath alpha 2 random walk ensuremath alpha 4 random acceleration and ensuremath alpha ensuremath infty single sinusoidal mode for other intermediate values of ensuremath alpha the distribution is determined from simulations we find that the mathrm mrh mathrm i distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all ensuremath alpha the two main distinguishing features of the mathrm mrh mathrm i distribution are the much larger weight for small relative heights and the divergence at zero height for ensuremath alpha g3 we also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions finally we show that for signals arising from time translationally invariant distributions the density of near extreme states is the same as the mathrm mrh mathrm i distribution this is used in developing a scaling theory for the threshold singularities of the two distributions

holder continuity of cumulative distribution functions for noncommutative polynomials under finite free fisher information abstract this paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory more precisely we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non microstates free fisher information highlighting the special case of lipschitz conjugate variables for the first time in this generality it is shown that the analytic distributions of those evaluations have holder continuous cumulative distribution functions with an explicit holder exponent that depends only on the degree of the considered polynomial for linear polynomials we reach in the case of finite non microstates free fisher information the optimal holder exponent 2 3 and get lipschitz continuity in the case of lipschitz conjugate variables in particular our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite non microstates free entropy which partially settles a conjecture of charlesworth and shlyakhtenko 8 we further provide a very general criterion that gives for weak approximations of measures having holder continuous cumulative distribution functions explicit rates of convergence in terms of the kolmogorov distance finally we combine these results to study the asymptotic eigenvalue distributions of polynomials in gues or matrices with more general gibbs laws for gibbs laws this extends the corresponding result obtained in 21 from convergence in distribution to convergence in kolmogorov distance in the gue case we even provide explicit rates which quantify results of 23 24 in terms of the kolmogorov distance

maximal and minimal height distributions of fluctuating interfaces maximal and minimal height distributions mahd mihd of two dimensional interfaces grown with the nonlinear equations of kardar parisi zhang kpz second order and of villain lai das sarma vlds fourth order are shown to be different two universal curves may be mahd or mihd of each class depending on the sign of the relevant nonlinear term which is confirmed by results of several lattice models in the kpz and vlds classes the difference between mahd and midh is connected with the asymmetry of the local height distribution a simple exactly solvable deposition erosion model is introduced to illustrate this feature the average extremal heights scale with the same exponent of the average roughness in contrast to other correlated systems generalized gumbel distributions do not fit those mahd and mihd nor those of edwards wilkinson growth

a random matrix approach to the lack of projections in cred f2 abstract in 1982 pimsner and voiculescu computed the k 0 and k 1 groups of the reduced group c algebra c red f k of the free group f k on k generators and settled thereby a long standing conjecture c red f k has no projections except for the trivial projections 0 and 1 later simpler proofs of this conjecture were found by methods from k theory or from non commutative differential geometry in this paper we provide a new proof of the fact that c red f k is projectionless the new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup ext c red f k is not a group for k 2 by the same type of methods we also obtain that two phenomena proved by bai and silverstein for certain classes of random matrices no eigenvalues outside a small neighbourhood of the support of the limiting distribution and exact separation of eigenvalues by gaps in the limiting distribution also hold for arbitrary non commutative selfadjoint polynomials of independent gue goe or gse random matrices with matrix coefficients

global fluctuations and gumbel statistics we explain how the statistics of global observables in correlated systems can be related to extreme value problems and to gumbel statistics this relationship then naturally leads to the emergence of the generalized gumbel distribution ga x with a real index a in the study of global fluctuations to illustrate these findings we introduce an exactly solvable nonequilibrium model describing an energy flux on a lattice with local dissipation in which the fluctuations of the global energy are precisely described by the generalized gumbel distribution

forcing theory for transverse trajectories of surface homeomorphisms this paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and brouwer theory the main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations this allows us to derive new proofs for some known results as well as some new applications among which we note the following we extend franks and handel s classification of zero entropy maps of mathbb s 2 for non wandering homeomorphisms we show that if f is a hamiltonian homeomorphism of the annulus then the rotation set of f is either a singleton or it contains zero in the interior proving a conjecture posed by boyland we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms proving a first case of the franks misiurewicz conjecture we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms

single big jump principle in physical modeling the big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution it states that the tail of the distribution of the sum is the same as the distribution of the largest summand in practice it means that when in a stochastic process the relevant quantity is a sum of variables the mechanism leading to rare events is peculiar instead of being caused by a set of many small deviations all in the same direction one jump the biggest of the lot provides the main contribution to the rare large fluctuation we reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations finite cutoffs continuous paths memory and quenched disorder doing so we are able to predict rare events using the extended big jump principle in l evy walks in a model of laser cooling in a scattering process on a heterogeneous structure and in a class of l evy walks with memory we argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions ranging from contamination spreading to active transport in the cell

the behaviour of aging functions in one dimensional bouchaud s trap model let i be a collection of i i d positive random variables with distribution in the domain of attraction of an stable law with 1 the symmetric bouchaud s trap model on is a markov chain x t whose transition rates are given by w xy 2 x 1 if x y are neighbours in we study the behaviour of two correlation functions x t w t x t w and open image in new window it is well known that for any of these correlation functions a time scale t f t w such that aging occurs can be found we study these correlation functions on time scales different from f t w and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of bouchaud s trap model

anomalous diffusion localization aging and subaging effects in trap models at very low temperature we study in detail the dynamics of the one dimensional symmetric trap model via a real space renormalization procedure which becomes exact in the limit of zero temperature in this limit the diffusion front in each sample consists of two delta peaks which are completely out of equilibrium with each other the statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit t 0 we first compute disorder averages of one time observables such as the diffusion front the thermal width the localization parameters the two particle correlation function and the generating function of thermal cumulants of the position we then study aging and subaging effects our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two time correlations we also extend the real space renormalization group method to include systematic corrections to the previous zero temperature procedure via a series expansion in t we then consider the generalized trap model with parameter alpha in 0 1 and obtain that the large scale effective model at low temperature does not depend on alpha in any dimension so that the only observables sensitive to alpha are those that measure the local persistence such as the probability to remain exactly in the same trap during a time interval finally we extend our approach at a scaling level for the trap model in d 2 and obtain the two relevant time scales for aging properties