arXiv subjects
Publications and source records attributed to Pavel Bleher.
We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.
We show that the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlevé equation, constructed from the Airy functions, as well as associated solutions of P$_\mathrm{XXXIV}$ and S$_\mathrm{II}$, can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P$_\mathrm{II}$ that depend only on the first Airy function $ \mathrm{Ai} $ (but not on $ \mathrm{Bi} $) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.
We use the Riemann-Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers $\mathscr{N}_j(g)$ of $4$-valent connected graphs with $j$ vertices on a compact Riemann surface of genus $g$. We explicitly evaluate these numbers for Riemann surfaces of genus $0,1,2,$ and $3$. Also, for a Riemann surface of an arbitrary genus $g$, we also calculate the leading term in the asymptotics of $\mathscr{N}_j(g)$ as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David \cite{DAVID}. These phase transitions are of the following four types: a) one-cut to two-cut through the splitting of the cut at the origin, b) two-cut to three-cut through the birth of a new cut at the origin, c) one-cut to three-cut through the splitting of the cut at two symmetric points, and d) one-cut to three-cut through the birth of two symmetric cuts.
Consider the general complex polynomial external field $$ V(z)=\frac{z^{k}}{k}+\sum_{j=1}^{k-1} \frac{t_j z^j}{j}, \qquad t_j \in \mathbb{C}, \quad k \in \mathbb{N}. $$ Fix an equivalence class $\mathcal{T}$ of admissible contours whose members approach $\infty$ in two different directions and consider the associated max-min energy problem. When $k=2p$, $p \in \mathbb{N}$, and $\mathcal{T}$ contains the real axis, we show that the set of parameters $t_1, \cdots, t_{2p-1}$ which gives rise to a regular $q$-cut max-min (equilibrium) measure, $1 \leq q \leq 2p-1 $, is an open set in $\mathbb{C}^{2p-1}$. We use the implicit function theorem to prove that the endpoint equations are solvable in a small enough neighborhood of a regular $q$-cut point. We also establish the real-analyticity of the real and imaginary parts of the end-points for all $q$-cut regimes, $1 \leq q \leq 2p-1$, with respect to the real and imaginary parts of the complex parameters in the external field. Our choice of even $k$ and the equivalence class $\mathcal{T} \ni \mathbb{R}$ of admissible contours is only for the simplicity of exposition and our proof extends to all possible choices in an analogous way.
We study Dyson's classical $r$-component ferromagnetic hierarchical model with a long range interaction potential $U(i,j)= -l(d(i,j)) d^{-2}(i,j)$, where $d(i,j)$ denotes the hierarchical distance. We prove a conjecture of Dyson, which states that the convergence of the series $l_1+l_2+...$, where $l_n=l(2^n)$, is a necessary and sufficient condition of the existence of phase transition in the model under consideration, and the spontaneous magnetization vanishes at the critical point, i.e. there is no Thouless' effect. We find however that the distribution of the normalized average spin at the critical temperature $T_c$ tends to the uniform distribution on the unit sphere in $\Bbb R^r$ as the volume tends to infinity, a phenomenon which resembles the Thouless effect.
We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.
In a classical work of the 1950's, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee-Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF) zeros. In this paper we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the Migdal-Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical (1,1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal-Kadanoff renormalization transformation of various other hierarchical lattices.
We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights $z_h, z_v$ of the dimer model and arbitrary dimensions of the lattice $m, n$. We assume that $m$ is even and we show that the asymptotic expansion depends on the parity of $n$. We review and extend the results of Ivashkevich, Izmailian, and Hu [6] on the full asymptotic expansion of the partition function of the dimer model, and we give a rigorous estimate of the error term in the asymptotic expansion of the partition function.
We prove the Pfaffian Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifically, we prove that the Pfaffian of the Kasteleyn periodic-periodic matrix is negative, while the Pfaffians of the Kasteleyn periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. As an application, we obtain an asymptotics of the dimer model partition function with an exponentially small error term.
We obtain asymptotic formulas for the partition function of the six-vertex model with domain wall boundary conditions and half-turn symmetry in each of the phase regions. The proof is based on the Izergin--Korepin--Kuperberg determinantal formula for the partition function, its reduction to orthogonal polynomials, and on an asymptotic analysis of the orthogonal polynomials under consideration in the framework of the Riemann--Hilbert approach.
We obtain an asymptotic formula, as $n\to\infty$, for the monomer-monomer correlation function $K_2(x,y)$ in the classical dimer model on a triangular lattice, with the horizontal and vertical weights $w_h=w_v=1$ and the diagonal weight $w_d=t>0$, where $x$ and $y$ are sites $n$ spaces apart in adjacent rows. We find that $t_c=\frac{1}{2}$ is a critical value of $t$. We prove that in the subcritical case, $0<t<\frac{1}{2}$, as $n\to\infty$, $K_2(x,y)=K_2(\infty)\left[1-\frac{e^{-n/ξ}}{n}\,\Big(C_1+C_2(-1)^n+\mathcal O(n^{-1})\Big)\right]$, with explicit formulae for $K_2(\infty)$, $ξ$, $C_1$, and $C_2$. In the supercritical case, $\frac{1}{2} < t < 1$, we prove that as $n\to\infty$, $K_2(x,y)=K_2(\infty)\Bigg[1- \frac{e^{-n/ξ}}{n}\, \Big(C_1\cos(ωn+φ_1)+C_2(-1)^n\cos(ωn+φ_2)+ C_3+C_4(-1)^n$ $+\mathcal O(n^{-1})\Big)\Bigg]$, with explicit formulae for $K_2(\infty)$, $ξ$, $ω$, and $C_1$, $C_2$, $C_3$, $C_4$, $φ_1$, $φ_2$. The proof is based on an extension of the Borodin-Okounkov-Case-Geronimo formula to block Toeplitz determinants and on an asymptotic analysis of the Fredholm determinants in hand.
The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this paper we consider the normal matrix model with cubic plus linear potential. To regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain $Ω$ that we determine explicitly by finding the rational parametrization of its boundary. We also study in details the mother body problem associated to $Ω$. It turns out that the mother body measure $μ_*$ displays a novel phase transition that we call the mother body phase transition: although $\partial Ω$ evolves analytically, the mother body measure undergoes a "one-cut to three-cut" phase transition. To construct the mother body measure, we define a quadratic differential $\varpi$ on the associated spectral curve, and embed $μ_*$ into its critical graph. Using deformation techniques for quadratic differentials, we are able to get precise information on $μ_*$. In particular, this allows us to determine the phase diagram for the mother body phase transition explicitly. Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated $g$-functions significantly more involved, and the critical graph of $\varpi$ becomes the key technical tool in this analysis as well.
We obtain an asymptotic formula for the partition function of the six-vertex model with partial domain wall boundary conditions in the ferroelectric phase region. The proof is based on a formula for the partition function involving the determinant of a matrix of mixed Vandermonde/Hankel type. This determinant can be expressed in terms of a system of discrete orthogonal polynomials, which can then be evaluated asymptotically by comparison with the Meixner polynomials.
In the present paper we calculate explicitly the constant factor $C$ in the large $N$ asymptotics of the partition function $Z_N$ of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and ferroelectric phases. On the critical line the weights $a,b,c$ of the model are parameterized by a parameter $\al>1$, as $a=\frac{\al-1}{2}$, $b=\frac{\al+1}{2}$, $c=1$. The asymptotics of $Z_N$ on the critical line was obtained earlier in the paper \cite{BL2} of Bleher and Liechty: $Z_N=CF^{N^2}G^{\sqrt{N}}N^{1/4}\big(1+O(N^{-1/2})\big)$, where $F$ and $G$ are given by explicit expressions, but the constant factor $C>0$ was not known. To calculate the constant $C$, we find, by using the Riemann-Hilbert approach, an asymptotic behavior of $Z_N$ in the double scaling limit, as $N$ and $\al$ tend simultaneously to $\infty$ in such a way that $\frac{N}{\al}\to t\ge 0$. Then we apply the Toda equation for the tau-function to find a structural form for $C$, as a function of $\al$, and we combine the structural form of $C$ and the double scaling asymptotic behavior of $Z_N$ to calculate $C$.
In the present article we obtain the large $N$ asymptotics of the partition function $Z_N$ of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and antiferroelectric phases. Using the weights $a=1-x,b=1+x,c=2,|x|<1$, we prove that, as $N\rightarrow\infty$, $Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$, where $F$ is given by an explicit expression in $x$ and the $x$-dependency in $C$ is determined. This result reproduces and improves the one given in the physics literature by Bogoliubov, Kitaev and Zvonarev. Furthermore, we prove that the free energy exhibits an infinite order phase transition between the disordered and antiferroelectric phases. Our proofs are based on the large $N$ asymptotics for the underlying orthogonal polynomials which involve a non-analytical weight function, the Deift-Zhou nonlinear steepest descent method to the corresponding Riemann-Hilbert problem, and the Toda equation for the tau-function.
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $n$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $n\times n$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $n$ asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.
In this work we study the minimization problem for the total distance in a cloud computing network on the sphere. We give a solution to this problem in terms of hyperbolic Voronoi diagrams on the sphere. We present results of computer simulations illustrating the solution.
In this paper we return to the classical work by Brézin, Itzykson, Parisi and Zuber, in which, among other things, the authors explicitly calculated the coefficients of the topological expansion in the cubic random matrix model in genus 0. Our main goal will be to rigorously prove the results of Brézin, Itzykson, Parisi and Zuber and to obtain an explicit formula for the coefficients of the topological expansion in genus 1. We will also prove some formulae and asymptotic results for the coefficients of the topological expansion in higher genera.