bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

the fourier spectrum of critical percolation consider the indicator function f of a 2 dimensional percolation crossing event in this paper the fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations various applications of these bounds are derived in particular we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension frac 31 36 almost surely and the corresponding dimension in the half plane is frac 5 9 it is also proved that critical bond percolation on the square grid has exceptional times almost surely also the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

restart could optimize the probability of success in a bernoulli trial the recently noticed ability of restart to reduce the expected completion time of first passage processes allows appealing opportunities for performance improvement in a variety of settings however complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency here we show that restart may have profound consequences on the splitting probabilities of a bernoulli like first passage process i e of a process which can end with one of two outcomes particularly intriguing in this respect is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way we reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first passage under restart

strict inequalities for some critical exponents in two dimensional percolation for 2d percolation we slightly improve a result of chayes and chayes to the effect that the critical exponent for the percolation probability isstrictly less than 1 the same argument is applied to prove that ifl x y x r cos y r sin for some r 0 or and limp p c log p p c 1 log pcr ito is connected to by an occupied path inl then is strictly decreasing in on 0 2 similarly limn logn 1 logp cr ito is connected by an occupied path inl to the exterior of n n n n is strictly decreasing in on 0 2

exceptional planes of percolation consider the standard continuous percolation in 4 and choose the parameters so that the induced percolation on a fixed two dimensional linear subspace is critical although two dimensional critical percolation dies we show that there are exceptional two dimensional linear subspaces in which percolation occurs

universality of local eigenvalue statistics for some sample covariance matrices we consider random complex sample covariance matrices 1 x x where x is a p n random matrix with i i d entries of distribution it has been conjectured that both the distribution of the distance between nearest neighbor eigenva lues in the bulk and that of the smallest eigenvalues become in the limit n p n 1 the same as that identified for a complex gaussian distribution we prove these conjectures for a certain class of probability distributions c 2004 wiley periodicals inc

search in unknown random environments n searchers are sent out by a source in order to locate a fixed object which is at a finite distance d but the search space is infinite and d would be in general unknown each of the searchers has a finite random lifetime and may be subject to destruction or failures and it moves independently of other searchers and at intermediate locations some partial random information may be available about which way to go when a searcher is destroyed or disabled or when it dies naturally after some time the source becomes aware of this and it sends out another searcher which proceeds similarly to the one that it replaces the search ends when one of the searchers finds the object being sought we use n coupled brownian motions to derive a closed form expression for the average search time as a function of d which will depend on the parameters of the problem the number of searchers the average lifetime of searchers the routing uncertainty and the failure or destruction rate of searchers we also examine the cost in terms of the total energy that is expended in the search

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

intermittent search strategies revisited effect of the jump length and biased motion we study the kinetics of a search of a single fixed target by a large number of searchers performing an intermittent biased random walk in a homogeneous medium our searchers carry out their walks in one of two states between which they switch randomly one of these states search phase is a nearest neighbor walk characterized by the probability of stepping in a given direction i e the walks in this state are not necessarily isotropic the other relocation phase is characterized by the length of the jumps i e when in this state a walker does not perform a nearest neighbor walk within such a framework we propose a model to describe the searchers dynamics generalizing results of our previous work we have obtained and numerically evaluated analytic results for the mean number of distinct sites visited up to a maximum evolution time we have studied the dependence of this quantity on both the transition probability between the states and the parameters that characterize each state in addition to our theoretical approach we have implemented monte carlo simulations finding excellent agreement between the theoretical numerical and simulations results

optimal random search for a single hidden target a single target is hidden at a location chosen from a predetermined probability distribution then a searcher must find a second probability distribution from which random search points are sampled such that the target is found in the minimum number of trials here it will be shown that if the searcher must get very close to the target to find it then the best search distribution is proportional to the square root of the target distribution regardless of dimension for a gaussian target distribution the optimum search distribution is approximately a gaussian with a standard deviation that varies inversely with how close the searcher must be to the target to find it for a network where the searcher randomly samples nodes and looks for the fixed target along edges the optimum is either to sample a node with probability proportional to the square root of the out degree plus 1 or not to do so at all

conformal invariance of voronoi percolation it is proved that in the voronoi model for percolation in dimension 2 and 3 the crossing probabilities are asymptotically invariant under conformal change of metric

oded schramm s contributions to noise sensitivity we survey in this paper the main contributions of oded schramm related to noise sensitivity we will describe in particular his various works which focused on the spectral analysis of critical percolation and more generally of boolean functions his work on the shape fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

spectral content of fractional brownian motion with stochastic reset we analyse the power spectral density psd s t f with t being the observation time and f the frequency of a fractional brownian motion fbm with an arbitrary hurst index undergoing a stochastic resetting to the origin at a constant rate r the resetting process introduced some time ago as an example of an efficient optimisable search algorithm to this end we first derive an exact expression for the covariance function of an arbitrary not necessarily a fbm process with a reset expressing it through the covariance function of the parental process without a reset which yields the desired result for the fbm in a particular case we then use this result to compute exactly the power spectral density for fbm for all frequency f the asymptotic large frequency f behaviour of the psd turns out to be distinctly different for sub and super diffusive fbms we show that for large f the psd has a power law tail where the exponent for sub diffusive fbm while for all thus somewhat unexpectedly the exponent in the superdiffusive case h 1 2 sticks to its brownian value and does not depend on h

universality of the second mixed moment of the characteristic polynomials of the 1d band matrices real symmetric case we prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the n n 1d gaussian real symmetric band matrices with the width of the band w n1 2 coincides with those for the gaussian orthogonal ensemble here we adapt the approach of shcherbina commun math phys 328 45 82 2014 where the hermitian case was considered

delocalization for a class of random block band matrices we consider n times n hermitian random matrices h consisting of blocks of size m ge n 6 7 the matrix elements are i i d within the blocks close to a gaussian in the four moment matching sense but their distribution varies from block to block to form a block band structure with an essential band width m we show that the entries of the green s function g z h z 1 satisfy the local semicircle law with spectral parameter z e mathbf i eta down to the real axis for any eta gg n 1 using a combination of the supersymmetry method inspired by shcherbina j stat phys 155 3 466 499 2014 and the green s function comparison strategy previous estimates were valid only for eta gg m 1 the new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized

non gaussianity and dynamical trapping in locally activated random walks we propose a minimal model of locally activated diffusion in which the diffusion coefficient of a onedimensional brownian particle is modified in a prescribed way either increased or decreased upon each crossing of the origin such a local mobility decrease arises in the formation of atherosclerotic plaques due to diffusing macrophage cells accumulating lipid particles we show that spatially localized mobility perturbations have remarkable consequences on diffusion at all scales such as the emergence of a non gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state in the context of atherosclerosis this dynamical transition can be viewed as a minimal mechanism that causes macrophages to aggregate in lipid enriched regions and thereby to the formation of atherosclerotic plaques

local eigenvalue density for general manova matrices we consider random n n matrices of the form begin aligned left xx yy right frac 1 2 yy left xx yy right frac 1 2 end aligned where x and y have independent entries with zero mean and variance one these matrices are the natural generalization of the gaussian case which are known as manova matrices and which have joint eigenvalue density given by the third classical ensemble the jacobi ensemble we show that away from the spectral edge the eigenvalue density converges to the limiting density of the jacobi ensemble even on the shortest possible scales of order 1 n up to logn factors this result is the analogue of the local wigner semicircle law and the local marchenko pastur law for general manova matrices

optimal search behavior and classic foraging theory random walk methods and diffusion theory pervaded ecological sciences as methods to analyze and describe animal movement consequently statistical physics was mostly seen as a toolbox rather than as a conceptual framework that could contribute to theory on evolutionary biology and ecology however the existence of mechanistic relationships and feedbacks between behavioral processes and statistical patterns of movement suggests that beyond movement quantification statistical physics may prove to be an adequate framework to understand animal behavior across scales from an ecological and evolutionary perspective recently developed random search theory has served to critically re evaluate classic ecological questions on animal foraging for instance during the last few years there has been a growing debate on whether search behavior can include traits that improve success by optimizing random stochastic searches here we stress the need to bring together the general encounter problem within foraging theory as a mean for making progress in the biological understanding of random searching by sketching the assumptions of optimal foraging theory oft and by summarizing recent results on random search strategies we pinpoint ways to extend classic oft and integrate the study of search strategies and its main results into the more general theory of optimal foraging

a clt for a band matrix model a law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on or above diagonal entries are independent but neither necessarily identically distributed nor necessarily all of the same variance the derivation is based on systematic combinatorial enumeration study of generating functions and concentration inequalities of the poincare type special cases treated with an explicit evaluation of limiting variances are generalized wigner and wishart matrices

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

eigenvector distribution of wigner matrices we consider n n hermitian or symmetric random matrices with independent entries the distribution of the i j th matrix element is given by a probability measure ij whose first two moments coincide with those of the corresponding gaussian ensemble we prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding gaussian ensemble for eigenvectors associated with bulk eigenvalues the same conclusion holds provided the first four moments of the distribution ij coincide with those of the corresponding gaussian ensemble more generally we prove that the joint eigenvector eigenvalue distributions near the spectral edge of two generalized wigner ensembles agree provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate if in addition the first four moments match then this result holds also in the bulk

lotka volterra systems with stochastic resetting we study the dynamics of predator prey systems where prey are confined to a single region of space and where predators move randomly according to a power law l vy dispersal kernel site fidelity an important feature of animal behavior is incorporated in the model through stochastic resetting dynamics of the predators to the prey patch we solve in the long time limit the lotka volterra rate equations that describe the evolution of the two species densities fixing the demographic parameters and the l vy exponent the total population of predators can be maximized for a certain value of the resetting rate this optimal value achieves a compromise between the overexploitation and underutilization of the habitat similarly at a fixed resetting rate there exists a l vy exponent that is optimal regarding predator abundance these findings are supported by 2d stochastic simulations and show that the combined effects of diffusion and resetting can broadly extend the region of species coexistence in ecosystems facing resource scarcity

effects of refractory period on stochastic resetting we consider a stochastic process undergoing resetting after which a random refractory period is imposed in this period the process is quiescent and remains at the resetting position using a first renewal approach we compute exactly the stationary position distribution and analyse the emergence of a delta peak at the resetting position in the case of a power law distribution for the refractory period we find slow relaxation we generalise our results to the case when the resetting period and the refractory period are correlated by computing the laplace transform of the survival probability of the process and the mean first passage time i e the mean time to completion of a task we also compute exactly the joint distribution of the active and absorption time to a fixed target

properties of additive functionals of brownian motion with resetting we study the distribution of additive functionals of reset brownian motion a variation of normal brownian motion in which the path is interrupted at a given rate and placed back to a given reset position our goal is two fold 1 for general functionals we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting 2 for three examples of functionals positive occupation time area and absolute area we investigate the effect of resetting by computing distributions and moments using a formula that links the generating function with resetting to the generating function without resetting

learning monotone decision trees in polynomial time we give an algorithm that learns any monotone boolean function fisafunc to any constant accuracy under the uniform distribution in time polynomial in n and in the decision tree size of f this is the first algorithm that can learn arbitrary monotone boolean functions to high accuracy using random examples only in time polynomial in a reasonable measure of the complexity of f a key ingredient of the result is a new bound showing that the average sensitivity of any monotone function computed by a decision tree of size s must be at most sqrt log s this bound has proved to be of independent utility in the study of decision tree complexity o schramm r o donnell m saks and r servedio every decision tree has an influential variable in proceedings of the 46th annual ieee symposium on foundations of computer science ieee computer society los alamitos ca 2005 pp 31 39 we generalize the basic inequality and learning result described above in various ways specifically to partition size a stronger complexity measure than decision tree size p biased measures over the boolean cube rather than just the uniform distribution and real valued rather than just boolean valued functions

zeta functions of binary hermitian forms and special values of eisenstein series on three dimensional hyperbolic space the present work contains a systematic investigation of zeta functions of binary hermitian forms over the ring o of integers of an imaginary quadratic number field k tl with discriminant d 0 the resulting relations between these zetafunctions have numerous applications to the computation of special values of the eisenstein series for the group psl 2 o acting on the three dimensional hyperbolic space i i in particular we reprove many known formulae for representation numbers for quaternary quadratic forms over z and we obtain many explicit new results for such representation numbers in sect 2 we develop a theory of representations of numbers by binary hermitian forms over o which parallels the classical theory of representations of integers by binary quadratic forms over a key result is a certain bijection described in theorem 2 3 this bijection means that the sum of the numbers of representations modulo sl 2 o units of an integer k 0 by a set of representatives of the sl 2 o classes of binary hermitian forms over o with discriminant a is equal to the number of cosets 2 ko with 2 o 22 a 0 m o d k the number of these cosets is explicitly computed in sect 2 in all cases hence a certain mean value of the representation numbers is known whereas simple formulae for the individual representation numbers do not exist we apply the formulae of sect 2 to the study of zeta functions of definite or indefinite hermitian forms for simplicity we restrict in this introduction to the case of positive definite forms over o although the results of chap i are valid in the indefinite case as well let f be a positive definite binary hermitian form over o with sl 2 o unit group gl f then we define

bulk eigenvalue statistics for random regular graphs ak was partly supported by swiss national science foundation grant 144662 hty was partly supported by nsf grant dms 1307444 and a simons investigator fellowship

a generalized hermite constant for imaginary quadratic fields we introduce the projective hermite constant for positive definite binary hermitian forms associated with an imaginary quadratic number field k it is a lower bound for the classical hermite constant and these two constants coincide when k has class number one using the geometric tools developed by mendoza 6 and vogtmann 15 for their study of the homology of the bianchi groups we compute the projective hermite constants for those k whose absolute discriminants are less than 70 and determine the hermitian forms that attain the projective hermite constants in these cases a comparison of the projective hermitian constant with some other generalizations of the classical hermite constant is also given

optimization in first passage resetting we investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold in an infinite domain this process is nonstationary and its probability distribution exhibits rich features in a finite domain we define a nontrivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point we derive the condition to optimize the net gain in this system namely the reward minus the cost

diffusion with resetting in bounded domains we consider the one dimensional diffusion in a bounded domain with stochastic resetting we start our analysis by presenting a method to derive the master equation for different resetting mechanisms in the next step we compute the non equilibrium steady state for a special case of this differential equation then we consider the existence of an absorbing point in the system and calculate the mean time to absorption of the diffusive particle by the target numerical and analytical calculations of the optimal resetting rate reveal a second order phase transition finally we discuss different special cases of the presented problem

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

optimal stochastic restart renders fluctuations in first passage times universal stochastic restart may drastically reduce the expected run time of a computer algorithm expedite the completion of a complex search process or increase the turnover rate of an enzymatic reaction these diverse first passage time fpt processes seem to have very little in common but it is actually quite the other way around here we show that the relative standard deviation associated with the fpt of an optimally restarted process i e one that is restarted at a constant nonzero rate which brings the mean fpt to a minimum is always unity we interpret further generalize and discuss this finding and the implications arising from it

resetting processes with noninstantaneous return we consider a random two phase process which we call a reset return one the particle starts its motion at the origin the first displacement phase corresponds to a stochastic motion of a particle and is finished at a resetting event the second return phase corresponds to the particle s motion toward the origin from the position it attained at the end of the displacement phase this motion toward the origin takes place according to a given equation of motion the whole process is a renewal one we provide general expressions for the stationary probability density function of the particle s position and for the mean hitting time in one dimension we perform explicit analysis for the brownian motion during the displacement phase and three different types of the return motion return at a constant speed return at a constant acceleration with zero initial speed and return under the action of a harmonic force we assume that the waiting times for resetting events follow an exponential distribution or that resetting takes place after a fixed waiting period for the first two types of return motion and the exponential resetting the stationary probability density function of the particle s position is invariant under return speed acceleration while no such invariance is found for deterministic resetting and for exponential resetting with return under the action of the harmonic force we discuss necessary conditions for such invariance of the stationary pdf of the positions with respect to the properties of the return process and we demonstrate some additional examples when this invariance does or does not take place

first passage under stochastic resetting in an interval we consider a brownian particle diffusing in a one dimensional interval with absorbing end points we study the ramifications when such motion is interrupted and restarted from the same initial configuration we provide a comprehensive study of the first passage properties of this trapping phenomena we compute the mean first passage time and derive the criterion on which restart always expedites the underlying completion we show how this set up is a manifestation of a success failure problem we obtain the success and failure rates and relate them with the splitting probabilities namely the probability that the particle will eventually be trapped on either of the boundaries without hitting the other one numerical studies are presented to support our analytic results

spectral form factor in a random matrix theory in the theory of disordered systems the spectral form factor s ensuremath tau the fourier transform of the two level correlation function with respect to the difference of energies is linear for ensuremath tau mathrm ensuremath tau mathrm c and constant for ensuremath tau g mathrm ensuremath tau mathrm c near zero and near mathrm ensuremath tau mathrm c it exhibits oscillations which have been discussed in several recent papers in problems of mesoscopic fluctuations and quantum chaos a comparison is often made with a random matrix theory it turns out that even in the simplest gaussian unitary ensemble these oscillations have not yet been studied there for random matrices the two level correlation function ensuremath rho ensuremath lambda 1 ensuremath lambda 2 exhibits several well known universal properties in the large n limit its fourier transform is linear as a consequence of the short distance universality of ensuremath rho ensuremath lambda 1 ensuremath lambda 2 however the crossover near zero and mathrm ensuremath tau mathrm c requires one to study these correlations for finite n for this purpose we use an exact contour integral representation of the two level correlation function which allows us to characterize these crossover oscillatory properties this representation is then extended to the case in which the hamiltonian is the sum of a deterministic part mathrm h 0 and of a gaussian random potential v finally we consider the extension to the time dependent case

a continuous time ehrenfest model with catastrophes and its jump diffusion approximation we consider a continuous time ehrenfest model defined over the integers from n to n and subject to catastrophes occurring at constant rate the effect of each catastrophe istantaneously resets the process to state 0 we investigate both the transient and steady state probabilities of the above model further the first passage time through state 0 is discussed we perform a jump diffusion approximation of the above model which leads to the ornstein uhlenbeck process with catastrophes the underlying jump diffusion process is finally studied with special attention to the symmetric case arising when the ehrenfest model has equal upward and downward transition rates

path integral formalism for stochastic resetting exactly solved examples and shortcuts to confinement we study the dynamics of overdamped brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space dependent rate of resetting we present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as i the propagator prior to first reset ii the distribution of the first reset time and iii the spatial distribution of the particle at long times we apply our approach to several representative and hitherto unexplored examples of resetting dynamics a particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle we find that using energy dependent resetting processes is more effective in achieving spatial confinement of brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

analysis of reflected diffusions via an exponential time based transformation let x t be a time homogeneous diffusion process with state space 0 infty where 0 is a reflecting or entrance endpoint and let z denote a random variable that describes the process x t evaluated at an exponentially distributed random time we propose a method to obtain closed form expressions for the conditional density and the mean of a new diffusion process y t with the same state space and with the same infinitesimal variance whose drift depends on the infinitesimal moments of x t and on the hazard rate function of z this method also allows us to obtain the laplace transform of the first passage time density of y t through a lower constant boundary we then discuss the relation between y t and the process x t subject to catastrophes as well as the interpretation of y t as a diffusion in a decreasing potential we study in detail some special cases concerning diffusion processes obtained when x t is the wiener ornstein uhlenbeck bessel and rayleigh process

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

diffusion with optimal resetting we consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r we consider several generalisations of the model of m r evans and s n majumdar 2011 diffusion with stochastic resetting phys rev lett 106 160601 i a space dependent resetting rate r x ii resetting to a random position z drawn from a resetting distribution p z iii a spatial distribution for the absorbing target pt x as an example of i we show that the introduction of a non resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target we address the problem of optimal resetting that is minimising the mean time to absorption for a given target distribution for an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows

on the scaling limits of planar percolation we prove tsirelson s conjecture that any scaling limit of the critical planar percolation is a black noise our theorems apply to a number of percolation models including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid we also suggest a natural construction for the scaling limit of planar percolation and more generally of any discrete planar model describing connectivity properties

stability of the matrix dyson equation and random matrices with correlations we consider real symmetric or complex hermitian random matrices with correlated entries we prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum the correlations have fast decay but are otherwise of general form the key novelty is the detailed stability analysis of the corresponding matrix valued dyson equation whose solution is the deterministic limit of the resolvent

first passage percolation has sublinear distance variance let 0 1 the distance dist 0 from the origin to a vertex 2 has variance bounded by c log where c c a b d is a constant which may only depend on a b and d some related variants are also discussed

first passage statistics under stochastic resetting in bounded domains we investigate the first passage problem where a diffusive searcher stochastically resets to a fixed position at a constant rate in a bounded domain we put forward an analytical framework for this

random covariance matrices universality of local statistics of eigenvalues we study the eigenvalues of the covariance matrix 1 n m m of a large rectangular matrix m mn p ij 1 i p 1 j n whose entries are i i d random variables of mean zero variance one and having finite c0th moment for some sufficiently large constant c0 the main result of this paper is a four moment theorem for i i d covariance matrices analogous to the four moment theorem for wigner matrices established by the authors in acta math 2011 random matrices universality of local eigenvalue statistics see also comm math phys 298 2010 549 572 we can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions as a byproduct of our arguments we also extend our previous results on random hermitian matrices to the case in which the entries have finite c0th moment rather than exponential decay

michaelis menten reaction scheme as a unified approach towards the optimal restart problem we study the effect of restart and retry on the mean completion time of a generic process the need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of michaelis and menten stopping a process in its midst only to start it all over again may prolong leave unchanged or even shorten the time taken for its completion here we are interested in the optimal restart problem i e in finding a restart rate which brings the mean completion time of a process to a minimum we derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest we then continue to discover regimes at which solutions to the problem take on universal details independent forms which further give rise to optimal scaling laws the formalism we develop and the results obtained can be utilized when optimizing stochastic search processes and randomized computer algorithms an immediate connection with kinetic proofreading is also noted and discussed

dynamical transition in the temporal relaxation of stochastic processes under resetting a stochastic process when subject to resetting to its initial condition at a constant rate generically reaches a nonequilibrium steady state we study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems we show that as time progresses an inner core region around the resetting point reaches the steady state while the region outside the core is still transient the boundaries of the core region grow with time as power laws at late times with new exponents alternatively at a fixed spatial point the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale t x we calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core these singularities act as separatrices between typical and atypical trajectories our results are verified in the numerical simulations of several models such as simple diffusion and fluctuating one dimensional interfaces

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

large deviations for markov processes with resetting markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches foraging and population dynamics here we study the large deviations of time additive functions or observables of markov processes with resetting by deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables characterizing the likelihood of their fluctuations in the long time limit we consider as an illustration the large deviations of the area of the ornstein uhlenbeck process with resetting other applications involving diffusions random walks and jump processes with resetting or catastrophes are discussed

bulk universality and related properties of hermitian matrix models we give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models assuming that the potential that determines the model is globally c 2 and locally c 3 function see theorem 3 1 the proof as our previous proof in pastur and shcherbina in j stat phys 86 109 147 1997 is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials rather we obtain the sin kernel as a unique solution of a certain non linear integro differential equation that follows from the determinant formulas for the correlation functions of the model we also give a simplified and strengthened version of paper boutet de monvel et al in j stat phys 79 585 611 1995 on the existence and properties of the limiting normalized counting measure of eigenvalues we use these results in the proof of universality and we believe that they are of independent interest

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices