prym varieties curves with automorphisms and the sato grassmannian the aim of the paper is twofold first some results of shiota and plaza martin on prym varieties of curves with an involution are generalized to the general case of an arbitrary automorphism of prime order second the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of the sato grassmannian are given

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

hydrodynamical methods for analyzing longest increasing subsequences let ln be the length of the longest increasing subsequence of a random permutation of the numbers 1 n for the uniform distribution on the set of permutations we discuss the hydrodynamical approach to the analysis of the limit behavior which probably started with hammersley proceedings of the 6th berkeley symposium on mathematical statistics and probability vol 1 1972 345 394 and was subsequently further developed by several authors we also give two proofs of an exact non asymptotic result announced in rains preprint 2000

large deviations for increasing sequences on the plane we prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among poisson points on the plane the rate function for lower tail deviations is derived from a 1977 result of logan and shepp about young diagrams of random permutations for the upper tail we use a coupling with hammersley s particle process and convex analytic techniques along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric hammersley s process

on the evolution of a reflection coefficient under the korteweg de vries flow we are concerned with the korteweg de vries equation on the full line with real nondecaying initial profiles we find the time evolution of a relative reflection coefficient an inverse spectral formalism is also considered for certain mixed problems on the full line

limiting curves for i i d records we consider the concentration of measure for n i i d two dimensional random variables under the conditioning that they form a record under mild conditions we show that all random variables tend to concentrate as n rightarrow infty around limiting curves which are the solutions of an appropriate variational problem we also show that the same phenomenon occurs without the records conditioning for the longest increasing subsequence in the sample

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

characterizations of jacobians of curves with automorphisms we obtain a characterization of theta functions of jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties p p a v we first give a characterization in terms of finite dimensional orbits for a suitable action in the sato grassmannian secondly the introduction of formal baker akhiezer functions and formal t functions attached to a p p a v for the multipuncture case allows us to characterize in terms of bilinear identities those baker akhiezer functions that are baker akhiezer functions of jacobians of curves with automorphisms further in the case of automorphisms with fixed points we rewrite the previous result as a hierarchy of partial differential equations for the function of a p p a v finally since baker akhiezer and t functions are written in terms of theta functions these results give rise to characterizations of p p a v in terms of their theta functions

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

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

the krichever map vector bundles over algebraic curves and heisenberg algebras we study the grassmanniangr x n consisting of equivalence classes of rankn algebraic vector bundles over a riemann surfacex with an holomorphic trivialization at a fixed pointp commutative subalgebras ofgl n h h being the ring of functions holomorphic on a punctured disc aboutp define flows on the grassmannian giving rise to classes of solutions to multi component kp hierarchies these commutative subalgebras correspond to heisenberg algebras in the kac moody algebra associated togl n h one can obtain by the krichever map points ofgr x n and solutions of mckp from coveringsf y x and other geometric data conversely for every point ofgr x n and for every choice of heisenberg algebra we construct using the cotangent bundle ofgr x n an algebraic curve coveringx and other data thus inverting the krichever map we show the explicit relation between the choice of heisenberg algebra and the geometry of the covering space

structure in the bifurcation diagram of the duffing oscillator we identify four levels of structure in the bifurcation diagram of the two well periodically driven duffing oscillator plotted as a function of increasing control parameter t the period of the driving term the superstructure or bifurcation peninsula repeats periodically as t increases by ensuremath sim 2 ensuremath pi beginning and ending with symmetric period one orbits whose local torsions differ by 2 within each bifurcation peninsula there is a systematic window structure the primary window structure is due to newhouse and newhouse like orbits fine structure is due to a farey sequence of well ordered orbits between the primary windows hyperfine structure consists of very narrow windows associated with non well ordered orbits we construct a template for the duffing oscillator a two dimensional return map and a one dimensional return map which describes the systematics of orbit creation and annihilation all structures are identified by topological indices our predictions are based on and compatible with numerical computations

cohomological structure in soliton equations and jacobian varieties on etudie la structure des orbites du systeme dynamique definie par la hierarchie kp totale on montre que chaque orbite est localement isomorphe a un certain groupe de cohomologie associe a une algebre commutative le systeme dynamique kp restreint a chaque orbite determine un systeme dynamique de mouvements lineaires sur lui par rapport a la structure lineaire du groupe de cohomologie

from hammersley s lines to hammersley s trees we construct a stationary random tree embedded in the upper half plane with prescribed offspring distribution and whose vertices are the atoms of a unit poisson point process this process which we call hammersley s tree process extends the usual hammersley s line process just as hammersley s process is related to the problem of the longest increasing subsequence this model also has a combinatorial interpretation it counts the number of heaps i e increasing trees required to store a random permutation this problem was initially considered by byers et al 2011 and istrate and bonchis 2015 in the case of regular trees we show in particular that the number of heaps grows logarithmically with the size of the permutation

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation

longest increasing subsequences from patience sorting to the baik deift johansson theorem we describe a simple one person card game patience sorting its analysis leads to a broad circle of ideas linking young tableaux with the longest increasing subsequence of a random permutation via the schensted correspondence a recent highlight of this area is the work of baik deift johansson which yields limiting probability laws via hard analysis of toeplitz determinants

hammersley s process with sources and sinks we show that for a stationary version of hammersley s process with poisson sources on the positive x axis and poisson sinks on the positive y axis an isolated second class particle located at the origin at time zero moves asymptotically with probability 1 along the characteristic of a conservation equation for hammersley s process this allows us to show that hammersley s process without sinks or sources as defined by aldous and diaconis probab theory related fields 10 1995 199 213 converges locally in distribution to a poisson process a result first proved in aldous and diaconis 1995 by using the ergodic decomposition theorem and a construction of hammersley s process as a one dimensional point process developing as a function of continuous time on the whole real line as a corollary we get the result that el t t t converges to 2 as t where l t t is the length of a longest north east path from 0 0 to t t the proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem we also prove a version of burke s theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations

the random average process and random walk in a space time random environment in one dimension we study space time fluctuations around a characteristic line for a one dimensional interacting system known as the random average process the state of this system is a real valued function on the integers new values of the function are created by averaging previous values with random weights the fluctuations analyzed occur on the scale n1 4 where n is the ratio of macroscopic and microscopic scales in the system the limits of the fluctuations are described by a family of gaussian processes in cases of known product form invariant distributions this limit is a two parameter process whose time marginals are fractional brownian motions with hurst parameter 1 4 along the way we study the limits of quenched mean processes for a random walk in a space time random environment these limits also happen at scale n1 4 and are described by certain gaussian processes that we identify in particular when we look at a backward quenched mean process the limit process is the solution of a stochastic heat equation

inverse scattering transformation for positons the scattering problem for the one dimensional schrodinger operator with potential equal to the positon solution of the korteweg de vries kdv equation is investigated it is shown that the transition coefficient is equal to zero and different positon potentials can have the same reflection coefficient i e the inverse scattering problem cannot be solved uniquely it is observed that the reflection coefficient calculated for the positon solutions does not change with time in accordance with the inverse scattering method for kdv

exact solvability and asymptotic aspects of generalized xx0 spin chains abstract building on our earlier work saeedian et al 2018 we introduce and study generalized xx0 models we explicitly construct a long range interacting spin chain referred to as the selberg model and study the correlation functions of the selberg and xx0 models using a matrix integral representation of the generalized xx0 model and applying asymptotic analysis in non intersecting brownian motion the phase structure of the selberg model is determined we find that tails of the tracy widom distribution of gaussian unitary ensemble govern a discrete to continuous third order phase transition in selberg model the same method also reproduces the gross witten phase transition of the original xx0 model finally we conjecture universal features for the phase structure of the generalized xx0 model

exact limiting shape for a simplified model of first passage percolation on the plane we derive the limiting shape for the following model of first passage bond percolation on the two dimensional integer lattice the percolation is directed in the sense that admissible paths are nondecreasing in both coordinate directions the passage times of horizontal bonds are bernoulli distributed while the passage times of vertical bonds are all equal to a deterministic constant to analyze the percolation model we couple it with a one dimensional interacting particle system this particle process has nonlocal dynamics in the sense that the movement of any given particle can be influenced by far away particles we prove a law of large numbers for a tagged particle in this process and the shape result for the percolation is obtained as a corollary

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

on the long time behavior of the doubly infinite toda lattice under initial data decaying at infinity we provide rigorous analysis of the long time behavior of the doubly infinite toda lattice under initial data that decay at infinity in the absence of solitions we solve approximately and for large times the riemann hilbert matrix factorization problem equivalent to the related inverse scattering problem with the help of the beals coifman formula by reducing it to a simpler one through a series of contour deformations in the spirit of the deift zhou method

on the solvability of painleve ii and iv we introduce a rigorous methodology for studying the riemann hilbert problems associated with certain integrable nonlinear ordinary differential equations for concreteness we investigate the painleve ii and painleve iv equations we show that the cauchy problems for these equations admit in general global meromorphic in t solutions furthermore for special relations among the monodromy data and for t on stokes lines these solutions are bounded for finite t

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

an unrestricted learning procedure we study learning problems involving arbitrary classes of functions f underlying measures and targets y because proper learning procedures i e procedures that are only allowed to select functions in f tend to perform poorly unless the problem satisfies some additional structural property e g that f is convex we consider unrestricted learning procedures that are free to choose functions outside the given class we present a new unrestricted procedure whose sample complexity is almost the best that one can hope for and holds for almost any problem including heavy tailed situations moreover the sample complexity coincides with what one could expect if f were convex even when f is not and if f is convex then the unrestricted procedure turns out to be proper

asymptotics of the multipositon soliton τ function of the korteweg de vries equation and the supertransparency the asymptotics of the function generating the n positon m soliton solution of the korteweg de vries equation is calculated this allows to prove that solitons do not experience any phase shift in a collision with positons the positons themselves survive mutual collisions unchanged this phenomenon is called the supertransparency or super reflectionless property of the multipositon solutions the linear aspects of this phenomenon are also discussed it is demonstrated that positons acquire two additional but always finite phase shifts in collision with solitons this result admits a natural extension to any number of solitons and positons in interaction

equations of the moduli of pointed curves in the infinite grassmannian the main result of this paper is the explicit computa tion of the equations defining the moduli space of triples c p where c is an integral and complete algebraic curve p a smooth rational point anda certain isomorphism this is achieved by introducing algebraically infinite grassmannians tau and baker ahkiezer functions and by proving an addition formula for tau functions