Plinths and Pedestals
In this paper we explain what are the plinths and the pedestals of the skyscrapers (=plane partitions), and how one can use them in order to count the skyscrapers.
math.CO↗
arXiv subjects
Senya Shlosman
Publications and source records attributed to Senya Shlosman.
The critical temperature $T_{cr}$(Ising) is DS-computable
We show that the Dobrushin-Shlosman conditions CV for the uniqueness of the Gibbs state provide the exact value for the critical temperature of the d-dimensional Ising model.
math-ph↗
The major index (maj) and its Sch\"utzenberger dual
We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $\lambda/\mu.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $\lambda/\mu.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function. To do this we define for every SsYT $T$ its plinth, $\mathsf{p}\left( T\right) ,$ which is a SsYT of the same shape $\lambda/\mu.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\left( \mathsf{p}\left( T\right) ,Y\left( T-\mathsf{p}\left( T\right) \right) \right) ,$ where $Y\left( T-\mathsf{p}\left( T\right) \right) $ is the reading Young diagram of the SsYT $\left( T-\mathsf{p}\left( T\right) \right) $. \newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\mathsf{p}\left( P\right) $. The two statistics of SYT-s -- the volume $\left\vert \mathsf{p}\left( P\right) \right\vert $ and $\mathsf{maj}\left( P\right) $ -- are related via the Sch\"{u}tzenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \]
math.CO↗
The miracle of integer eigenvalues
For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.
math.CO↗
The Airy$_2$ process and the 3D Ising model
The Ferrari-Spohn diffusion process arises as limit process for the 2D Ising model as well as random walks with area penalty. Motivated by the 3D Ising model, we consider $M$ such diffusions conditioned not to intersect. We show that the top process converges to the Airy$_2$ process as $M\to\infty$. We then explain the relation with the 3D Ising model and present some conjectures about it.
math.PR↗
The Schonmann projection as a g-measure-, how Gibbsian is it?
We study the one-dimensional projection of the extremal Gibbs measures of the two-dimensional Ising model, the "Schonmann projection". These measures are known to be non-Gibbsian at low temperatures, since their conditional probabilities as a function of the two-sided boundary conditions are not continuous. We prove that they are g-measures, which means that their conditional probabilities have a continuous dependence on one-sided boundary condition.
math-ph↗
Art of Unlocking
We describe our recent results concerning the rigidity/unlockability properties of clusters of rigid bodies sliding over the unit sphere.
math.MG↗
Critical prewetting in the 2d Ising model
In this paper we develop a detailed analysis of critical prewetting in the context of the two-dimensional Ising model. Namely, we consider a two-dimensional nearest-neighbor Ising model in a $2N\times N$ rectangular box with a boundary condition inducing the coexistence of the $+$ phase in the bulk and a layer of $-$ phase along the bottom wall. The presence of an external magnetic field of intensity $h=λ/N$ (for some fixed $λ>0$) makes the layer of $-$ phase unstable. For any $β>β_{\rm c}$, we prove that, under a diffusing scaling by $N^{-2/3}$ horizontally and $N^{-1/3}$ vertically, the interface separating the layer of unstable phase from the bulk phase weakly converges to an explicit Ferrari-Spohn diffusion.
math.PR↗
Brownian flights over a circle
The stationary radial distribution, $P(ρ)$, of the random walk with the diffusion coefficient $D$, which winds with the tangential velocity $V$ around the impenetrable disc of radius $R$ for $R\gg 1$ converges to the distribution involving the Airy function. Typical trajectories are localized in the circular strip $[R, R+ δR^{1/3}]$, where $δ$ is the constant which depends on the parameters $D$ and $V$ and is independent on $R$.
math.PR↗
Critical configurations of solid bodies and the Morse theory of MIN functions
We study the manifold of clusters of nonintersecting congruent solid bodies, all touching the central ball $B\subset\mathbb{R}^{3}$ of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. We introduce the notion of \textit{critical cluster} and we study several critical clusters of balls and of cylinders. For the case of cylinders some of our critical clusters are new. We also establish the criticality properties of clusters, introduced earlier by W. Kuperberg.
math.MG↗
The Six Cylinders Problem: $\mathbb{D}_{3}$-symmetry Approach
Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of 6 non-intersecting infinite cylinders of radius $r,$ all touching the unit ball in $\mathbb{R}^{3}.$ We find a configuration with \[ r=\frac{1}{8}\left( 3+\sqrt{33}\right) \approx1.093070331\ .\] We believe that this value is the maximal possible.
math.MG↗
Glassy states: the free Ising model on a tree
We consider the ferromagnetic Ising model on the Cayley tree and we investigate the decomposition of the free state into extremal states below the spin glass temperature. We show that this decomposition has uncountably many components. The tail observable showing that the free state is not extremal is related to the Edwards-Anderson parameter, measuring the variance of the (random) magnetization obtained from drawing boundary conditions from the free state.
cond-mat.stat-mech↗
Platonic Compounds of Cylinders
Generalizing the octahedral configuration of six congruent cylinders touching the unit sphere, we exhibit configurations of congruent cylinders associated to a pair of dual Platonic bodies.
math.MG↗
Extremal Cylinder Configurations II: Configuration $O_6$
We study the octahedral configurations $O_6$ of six equal cylinders touching the unit sphere. We show that the configuration $O_6$ is a local sharp maximum of the distance function. Thus it is not unlockable and, moreover, rigid.
math.MG↗
Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime
The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span $\left< d(t) \right> \sim t^γ$ of the trajectory of $t$ steps above the top of the semicircle or the triangle. We show that $γ= \frac{1}{3}$, i.e. $\left< d(t) \right>$ shares the KPZ statistics for the semicircle, while $γ=0$ for the triangle. We propose heuristic derivations of scaling exponents $γ$ for different geometries, justify them by explicit analytic computations and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.
cond-mat.stat-mech↗
Extremal Cylinder Configurations I: Configuration $C_{\mathfrak{m}}$
We study the path $Γ=\{ C_{6,x}\ \vert\ x\in [0,1]\}$ in the moduli space of configurations of 6 equal cylinders touching the unit sphere. Among the configurations $C_{6,x}$ is the record configuration $C_{\mathfrak{m}}$ of \cite{OS}. We show that $C_{\mathfrak{m}}$ is a local sharp maximum of the distance function, so in particular the configuration $C_{\mathfrak{m}}$ is not only unlockable but rigid. We show that if $\frac{(1 + x) (1 + 3 x)}{3}$ is a rational number but not a square of a rational number, the configuration $C_{6,x}$ has some hidden symmetries, part of which we explain.
math.MG↗
Topological Tverberg Theorem: the proofs and the counterexamples
I describe the history of Topological Tverberg Theorem. I present some important constructions and discuss their properties. In particular, I describe in details the cell structure of the classifying space $K\left( S_{r},1\right),$ where $S_{r}$ is the permutation group. I also clarify some bibliographical issues.
math.CO↗
Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem
The problem of twelve spheres is to understand, as a function of $r \in (0,r_{max}(12)]$, the configuration space of $12$ non-overlapping equal spheres of radius $r$ touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of $N$ spheres of radius $r$ touching a central unit sphere, with emphasis on $3 \le N \le 14$. The problem of determining the maximal radius $r_{max}(N)$ is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.
math.MG↗