Recent ucscmath items

Min-max minimal disks with free boundary in riemannian manifolds

Mon, 3 Feb 2025 00:00:00 +0000

We establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for the Plateau problem of minimal disks, which can be used to generalize the famous work by Morse–Tompkins and Shiffman on minimal surfaces in Rn to the Riemannian setting. More precisely, we generalize, to the free boundary setting, the min-max construction of minimal surfaces using harmonic replacement introduced by Colding–Minicozzi. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit 2–disk in R2 into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness...

A characterization of finite étale morphisms in tensor triangular geometry

Thu, 15 Aug 2024 00:00:00 +0000

We provide a characterization of finite \'etale morphisms in tensor triangular geometry. They are precisely those functors which have a conservative right adjoint, satisfy Grothendieck--Neeman duality, and for which the relative dualizing object is trivial (via a canonically-defined map).

The spectrum of derived Mackey functors

Thu, 15 Aug 2024 00:00:00 +0000

We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by “ungluing” closed points. In order to compute the spectrum, we provide a new description of Kaledin’s category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories,...

Stratification and the comparison between homological and tensor triangular support

Thu, 15 Aug 2024 00:00:00 +0000

Abstract: We compare the homological support and tensor triangular support for ‘big’ objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending the work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support.

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Thu, 15 Aug 2024 00:00:00 +0000

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Profinite properties of algebraically clean graphs of free groups

Thu, 15 Aug 2024 00:00:00 +0000

Profinite properties of algebraically clean graphs of free groups

Dropping Bodies

Tue, 2 Jul 2024 00:00:00 +0000

Dropping Bodies

No infinite spin for planar total collision

Tue, 2 Jul 2024 00:00:00 +0000

The infinite spin problem is an old problem concerning the rotational behavior of total collision orbits in the n-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It’s conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with...

No Infinite Spin for Planar Total Collision

Tue, 2 Jul 2024 00:00:00 +0000

The infinite spin problem concerns the rotational behavior of total collision orbits in the $n$-body problem. It has long been known that when a solution tends to total collision then its normalized configuration curve must converge to the set of normalized central configurations. In the planar n-body problem every normalized configuration determines a circle of rotationally equivalent normalized configurations and, in particular, there are circles of normalized central configurations. It's conceivable that by means of an infinite spin, a total collision solution could converge to such a circle instead of to a particular point on it. Here we prove that this is not possible, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Lojasiewicz gradient inequality.

Bicycle paths, elasticae and sub-Riemannian geometry

Tue, 2 Jul 2024 00:00:00 +0000

We relate the sub-Riemannian geometry on the group of rigid motions of the plane to `bicycling mathematics'. We show that this geometry's geodesics correspond to bike paths whose front tracks are either non-inflectional Euler elasticae or straight lines, and that its infinite minimizing geodesics (or `metric lines') correspond to bike paths whose front tracks are either straight lines or `Euler's solitons' (also known as Syntractrix or Convicts' curves).

The Honest Embedding Dimension of a Numerical Semigroup

Tue, 2 Jul 2024 00:00:00 +0000

Attached to a singular analytic curve germ in $d$-space is a numerical semigroup: a subset $S$ of the non-negative integers which is closed under addition and whose complement isfinite. Conversely, associated to any numerical semigroup $S$ is a canonical mononial curve in $e$-space where $e$ is the number of minimal generators of the semigroup. It may happen that $d < e = e(S)$ where $S$ is the semigroup of the curve in $d$-space. Define the minimal (or `honest') embedding of a numerical semigroup to be the smallest $d$ such that $S$ is realized by a curve in $d$-space. Problem: characterize the numerical semigroups having minimal embedding dimension $d$. The answer is known for the case $d=2$ of planar curves and reviewed in an Appendix to this paper. The case $d =3$ of the problem is open. Our main result is a characterization of the multiplicity $4$ numerical semigroups whose minimal embedding dimension is $3$. See figure 1. The motivation for this work came from thinking...

The Kepler Cone, Maclaurin Duality and Jacobi-Maupertuis metrics

Tue, 2 Jul 2024 00:00:00 +0000

The Kepler problem is the special case $\alpha = 1$ of the power law problem: to solve Newton's equations for a central force whose potential is of the form $-\mu/r^{\alpha}$ where $\mu$ is a coupling constant. Associated to such a problem is a two-dimensional cone with cone angle $2 \pi c$ with $c = 1 - \frac{\alpha}{2}$. We construct a transformation taking the geodesics of this cone to the zero energy solutions of the $\alpha$-power law problem. The `Kepler Cone' is the cone associated to the Kepler problem. This zero-energy cone transformation is a special case of a transformation discovered by Maclaurin in the 1740s transforming the $\alpha$- power law problem for any energies to a `Maclaurin dual' $\gamma$-power law problem where $\gamma = \frac{2 \alpha}{2-\alpha}$ and which, in the process, mixes up the energy of one problem with the coupling constant of the other. We derive Maclaurin duality using the Jacobi-Maupertuis metric reformulation of mechanics. We then use the...

Oscillating about coplanarity in the 4 body problem

Tue, 2 Jul 2024 00:00:00 +0000

For the Newtonian 4-body problem in space we prove that any zero angular momentum bounded solution suffers infinitely many coplanar instants, that is, times at which all 4 bodies lie in the same plane. This result generalizes a known result for collinear instants ("syzygies") in the zero angular momentum planar 3-body problem, and extends to the $d+1$ body problem in $d$-space. The proof, for $d=3$, starts by identifying the center-of-mass zero configuration space with real $3 \times 3$ matrices, the coplanar configurations with matrices whose determinant is zero, and the mass metric with the Frobenius (standard Euclidean) norm. Let $S$ denote the signed distance from a matrix to the hypersurface of matrices with determinant zero. The proof hinges on establishing a harmonic oscillator type ODE for $S$ along solutions. Bounds on inter-body distances then yield an explicit lower bound $\omega$ for the frequency of this oscillator, guaranteeing a degeneration within every time interval...

Compactification of the energy surfaces for n bodies

Tue, 2 Jul 2024 00:00:00 +0000

For n bodies moving in Euclidean d-space under the influence of a homogeneous pair interaction we compactify every center-of-mass energy surface, obtaining a 2d(n -1)-1 - dimensional manifold with corners in the sense of Melrose. After a time change, the flow on this manifold is globally defined and non-trivial on the boundary.

Dropping Bodies

Tue, 2 Jul 2024 00:00:00 +0000

The subject is brake orbits for the 3-body problem: orbits where all velocities are zero at some instant. We extract a paradox and a mystery out of a recent database of 30 non-collision periodic brake orbits for the equal mass 3-body problem built up by Li and Liao. We solve the paradox and the mystery and pose an open question.

Evading Anderson localization in a one-dimensional conductor with correlated disorder

Tue, 2 Jul 2024 00:00:00 +0000

Evading Anderson localization in a one-dimensional conductor with correlated disorder

Classical n-body scattering with long-range potentials

Tue, 2 Jul 2024 00:00:00 +0000

Abstract: We consider the scattering of n classical particles interacting via pair potentials which are assumed to be ‘long-range’, i.e. of order O ( r − α ) as r tends to infinity, for some α > 0. We define and focus on the ‘free region’, the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an...

The negative energy N-body problem has finite diameter

Tue, 2 Jul 2024 00:00:00 +0000

The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted $M_E$ where $E$ is the energy of the problem. We show that $M_E$ has finite diameter for $E < 0$. Consequently $M_E$ has no metric rays. Motivation comes from work of Burgos- Maderna and Polimeni-Terracini for the case $E \ge 0$ and from a need to correct an error made in a previous ``proof''. We show that $M_E$ has finite diameter for $E < 0$ by showing that there is a constant $D$ such that all points of the Hill region lie a distance $D$ from the Hill boundary. (When $E \ge 0$ the Hill boundary is empty.) The proof relies on a game of escape which allows us to quantify the escape rate from a closed subset of configuration space, and the reduction of this game to one of escaping the boundary of a polyhedral convex cone into its interior.

Endotrivial complexes

Fri, 21 Jun 2024 00:00:00 +0000

Endotrivial complexes

The homological arrow polynomial for virtual links

Thu, 9 Nov 2023 00:00:00 +0000

The arrow polynomial is an invariant of framed oriented virtual links that generalizes the virtual Kauffman bracket. In this paper, we define the homological arrow polynomial, which generalizes the arrow polynomial to framed oriented virtual links with labeled components. The key observation is that, given a link in a thickened surface, the homology class of the link defines a functional on the surface’s skein module, and by applying it to the image of the link in the skein module this gives a virtual link invariant. We give a graphical calculus for the homological arrow polynomial by taking the usual diagrams for the Kauffman bracket and including labeled “whiskers” that record intersection numbers with each labeled component of the link. We use the homological arrow polynomial to study [Formula: see text]-nullhomologous virtual links and checkerboard colorability, giving a new way to complete Imabeppu’s characterization of checkerboard colorability of virtual links with up to...

Arithmetic of arithmetic Coxeter groups

Thu, 25 May 2023 00:00:00 +0000

In the 1990s, J. H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the "topograph," Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell's equation. It appears that the crux of his method is the coincidence between the arithmetic group [Formula: see text] and the Coxeter group of type [Formula: see text] There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway's topograph and generalizations to other arithmetic Coxeter groups. This includes a study of "arithmetic flags" and variants of binary quadratic forms.

The compactness locus of a geometric functor and the formal construction of the Adams isomorphism

Tue, 19 May 2020 00:00:00 +0000

The compactness locus of a geometric functor and the formal construction of the Adams isomorphism

Chains in CR geometry as geodesics of a Kropina metric

Sat, 27 Jul 2019 00:00:00 +0000

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The correspondence between geodesics of the Kropina metric and chains allows us to use the methods of metric geometry and the calculus of variations to study chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985] that locally any two points of a strictly pseudoconvex CR manifolds can be joined by a chain. Finally, we generalize this result to the global setting by showing that any...

On nonnegatively curved hypersurfaces in Hn+1

Tue, 22 Jan 2019 00:00:00 +0000

In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.

On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds

Tue, 22 Jan 2019 00:00:00 +0000

In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation of stable spheres of constant mean curvature in an asymptotically flat 3-manifold with positive mass.

On $n$-superharmonic functions and some geometric applications

Tue, 22 Jan 2019 00:00:00 +0000

In this paper we study asymptotic behavior of $n$-superharmonic functions at isolated singularity using the Wolff potential and $n$-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro in 2 dimensions. To study $n$-superharmonic functions we use a new notion of $n$-thinness by $n$-capacity motivated by a type of Wiener criterion in Arsove-Huber's paper. To extend Taliaferro's work, we employ the Adams-Moser-Trudinger inequality for the Wolff potential, which is inspired by the one used by Brezis-Merle. For geometric applications, we study the asymptotic end behavior of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. In both geometric applications the strong $n$-capacity lower bound estimate of Gehring in 1961 is brilliantly used. These geometric applications seem to elevate the importance of $n$-Laplace equations and make a closer tie to the classic analysis...

Norm inflation for incompressible magneto-hydrodynamic system ]{Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}_{\infty}^{-1,\infty}$

Tue, 22 Jan 2019 00:00:00 +0000

We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to present a very expository development of the inginious construction of Bourgain and Pavlovi\'{c} for Navier-Stokes equation.

Weakly horospherically convex hypersurfaces in hyperbolic space

Tue, 22 Jan 2019 00:00:00 +0000

In [2], the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $\phi:M^n \to \mathbb{H}^{n+1}$ and a class of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Some of the key aspects of the correspondence and its consequences have dimensional restrictions $n\geq3$ due to the reliance on an analytic proposition from [5] concerning the asymptotic behavior of conformal factors of conformal metrics on domains of $\mathbb{S}^n$. In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of [2] to all dimensions $n\geq2$ in a unified way. In the case of a single point boundary $\partial_{\infty}\phi(M)=\{x\} \subset \mathbb{S}^n$, we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in [2],...

On nonnegatively curved hypersurfaces in hyperbolic space

Tue, 22 Jan 2019 00:00:00 +0000

In this paper we prove the conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.

Compactness of conformally compact Einstein 4-manifolds II

Tue, 22 Jan 2019 00:00:00 +0000

In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee.

Hypersurfaces in Hyperbolic Poincaré Manifolds and Conformally Invariant PDEs

Tue, 22 Jan 2019 00:00:00 +0000

We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincar\'e manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincar\'e manifolds and conformally invariant equations on the conformal infinity. In particular, we generalize an equivalence exhibited in [7] between Christoffel-type problems for hypersurfaces in ${\H}^{n+1}$ and scalar curvature problems on the conformal infinity ${\SS}^n$ to hyperbolic Poincar\'e manifolds.

Hypersurfaces with nonnegative Ricci curvature in hyperbolic space

Tue, 22 Jan 2019 00:00:00 +0000

Based on properties of n-subharmonic functions we show that a complete, noncompact, properly embedded hypersurface with nonnegative Ricci curvature in hyperbolic space has an asymptotic boundary at infinity of at most two points. Moreover, the presence of two points in the asymptotic boundary is a rigidity condition that forces the hypersurface to be an equidistant hypersurface about a geodesic line in hyperbolic space. This gives an affirmative answer to the question raised by Alexander and Currier in 1990.

On the topology of conformally compact Einstein 4-manifolds

Tue, 22 Jan 2019 00:00:00 +0000

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.

Conformal Ricci flow on asymptotically hyperbolic manifolds

Tue, 22 Jan 2019 00:00:00 +0000

In this article we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.

A note on triangulated monads and categories of module spectra

Wed, 24 Oct 2018 00:00:00 +0000

A note on triangulated monads and categories of module spectra

Higher comparison maps for the spectrum of a tensor triangulated category

Wed, 1 Aug 2018 00:00:00 +0000

Higher comparison maps for the spectrum of a tensor triangulated category

The spectrum of the equivariant stable homotopy category of a finite group

Wed, 1 Aug 2018 00:00:00 +0000

The spectrum of the equivariant stable homotopy category of a finite group

Restriction to finite-index subgroups as étale extensions in topology, KK–theory and geometry

Wed, 1 Aug 2018 00:00:00 +0000

For equivariant stable homotopy theory, equivariant KK–theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite étale extensions in algebraic geometry.

Grothendieck–Neeman duality and the Wirthmüller isomorphism

Wed, 1 Aug 2018 00:00:00 +0000

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

On Picard groups of blocks of finite groups

Tue, 19 Jun 2018 00:00:00 +0000

We show that the subgroup of the Picard group of a $p$-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over the a complete discrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraic closure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picard groups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.

Fibered biset functors

Tue, 19 Jun 2018 00:00:00 +0000

The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules. The main result of this paper is the classification of simple fibered biset functors.

The −+ and −+ constructions for biset functors

Tue, 19 Jun 2018 00:00:00 +0000

In this article we define the $-_+$-construction and the $-^+$-construction, that was crucial in the theory of canonical induction formulas (see \cite{Boltje1998b}), in the setting of biset functors, thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors. Additionally, this provides a unified approach to the study of a class of functors including the Burnside ring, the monomial Burnside ring and global representation ring.

Blocks in the asymmetric simple exclusion process

Mon, 14 May 2018 00:00:00 +0000

In earlier work, the authors obtained formulas for the probability in the asymmetric simple exclusion process that the mth particle from the left is at site x at time t. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work, these results are generalized to the case where the mth particle is the left-most one in a contiguous block of L particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general L.

Airy Kernel and Painleve II

Thu, 22 Feb 2018 00:00:00 +0000

We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlev\'e II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the results for the limiting distribution of the largest eigenvalue in the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). This work we did some years ago in a more general setting. These notes, therefore, are not meant for experts in the field.

Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

Wed, 21 Feb 2018 00:00:00 +0000

The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, q_k(t), have the representation q_k(t) = log det(I-lambda K_k) - log det(I-lambda K_{k-1}) where K_k are integral operators. This class includes the n-periodic cylindrical Toda equations. For n=2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n=3 (and with an additional symmetry contraint) they reduce to earlier results for the radial Bullough-Dodd equation.

On ASEP with Step Bernoulli Initial Condition

Wed, 21 Feb 2018 00:00:00 +0000

This paper extends results of earlier work on ASEP to the case of step Bernoulli initial condition. The main results are a representation in terms of a Fredholm determinant for the probability distribution of a fixed particle, and asymptotic results which in particular establish KPZ universality for this probability in one regime. (And, as a corollary, for the current fluctuations.)

Fluctuations in the composite regime of a disordered growth model

Wed, 21 Feb 2018 00:00:00 +0000

We continue to study a model of disordered interface growth in two dimensions. The interface is given by a height function on the sites of the one--dimensional integer lattice and grows in discrete time: (1) the height above the site $x$ adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above $x$ increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at random with a common distribution $F$, and that the initial state is such that the origin is far above the other sites. Provided that the tails of the distribution $F$ at its right edge are sufficiently thin, there exists a nontrivial composite regime in which the fluctuations of this interface are governed by extremal statistics of $p_x$. In the quenched case, the said fluctuations are asymptotically normal, while in the annealed case they satisfy the appropriate...

Asymptotics in ASEP with Step Initial Condition