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Sep 19, 2013
09/13

by
Pavel Bleher

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We review some results on the critical phenomena in the Dyson hierarchical model and renormalization group.

Source: http://arxiv.org/abs/1010.5855v1

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Sep 22, 2013
09/13

by
Pavel Bleher

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We prove that the error term $\La(R)$ in the Weyl asymptotic formula $$\#\{ E_n\le R^2\}={\Vol M\over 4\pi} R^2+\La(R),$$ for the Laplace operator on a surface of revolution $M$ satisfying a twist hypothesis, has the form $\La(R) =R^{1/2}F(R)$ where $F(R)$ is an almost periodic function of the Besicovitch class $B^2$, and the Fourier series of $F(R)$ in $B^2$ is $\sum_\g A(\g)\cos(|\g|R-\phi)$ where the sum goes over all closed geodesics on $M$, and $A(\g)$ is computed through simple geometric...

Source: http://arxiv.org/abs/alg-geom/9306008v1

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Jun 30, 2018
06/18

by
Pavel Bleher; Karl Liechty

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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.

Topics: Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1407.8483

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Jun 29, 2018
06/18

by
Pavel Bleher; Guilherme Silva

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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 $\Omega$...

Topics: Complex Variables, Classical Analysis and ODEs, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1601.05124

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Sep 21, 2013
09/13

by
Pavel Bleher; Karl Liechty

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This is a continuation of the paper [4] of Bleher and Fokin, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered phase. In the present paper we obtain the large $n$ asymptotics of $Z_n$ in the ferroelectric phase. We prove that for any $\ep>0$, as $n\to\infty$, $Z_n=CG^nF^{n^2}[1+O(e^{-n^{1-\ep}})]$, and we find the exact value of the constants $C,G$ and $F$. The proof is based on the...

Source: http://arxiv.org/abs/0712.4091v2

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Sep 20, 2013
09/13

by
Pavel Bleher; Karl Liechty

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This is a continuation of the papers [4] of Bleher and Fokin and [5] of Bleher and Liechty, in which the large $n$ asymptotics is obtained for the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large $n$ asymptotics of $Z_n$ on the critical line between these two phases.

Source: http://arxiv.org/abs/0802.0690v2

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Sep 19, 2013
09/13

by
Pavel Bleher; Karl Liechty

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We consider the large-$N$ asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh $\frac{1}{N}$, with weight $e^{-NV(x)}$, where $V(x)$ is a real analytic function with sufficient growth at infinity. The proof is based on formulation of an interpolation problem for discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert problem, and steepest descent analysis of this Riemann-Hilbert problem.

Source: http://arxiv.org/abs/0909.1985v3

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Sep 22, 2013
09/13

by
Pavel Bleher; Xiaojun Di

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We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval $(-1,1)$ are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit translation invariant correlations. Then we calculate the correlations between the straightened zeros of the SO(2) random polynomial.

Source: http://arxiv.org/abs/math-ph/0201012v2

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Jun 30, 2018
06/18

by
Pavel Bleher; Karl Liechty

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We obtain asymptotic formulas for the partition function of the six-vertex model with half-turn boundary conditions 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.

Topics: Statistical Mechanics, Mathematical Physics, Condensed Matter, Mathematics

Source: http://arxiv.org/abs/1702.01190

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Sep 21, 2013
09/13

by
Pavel Bleher; Alexander Its

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We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that \epsilon \le (n/N) \le \lambda_{cr} - \epsilon for some \epsilon > 0, where \lambda_{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...

Source: http://arxiv.org/abs/math-ph/9907025v1

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3.0

Jun 29, 2018
06/18

by
Estelle Basor; Pavel Bleher

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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

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1610.08021

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Jul 20, 2013
07/13

by
Pavel Bleher; Caroline Shouraboura

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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.

Source: http://arxiv.org/abs/1203.2951v1

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Sep 22, 2013
09/13

by
Pavel Bleher; Alexander Its

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We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential $V(z)$. Our approach is based on a deformation $\tau_tV(z)$ of $V(z)$ to $z^2$, $0\le t

Source: http://arxiv.org/abs/math-ph/0409082v1

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Sep 23, 2013
09/13

by
Pavel Bleher; Vladimir Fokin

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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...

Source: http://arxiv.org/abs/math-ph/0510033v3

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Sep 22, 2013
09/13

by
Pavel Bleher; Alexander Its

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We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the psi-function for the Hastings-McLeod solution to the Painlev\'e II equation. The proof is based on the Riemann-Hilbert approach.

Source: http://arxiv.org/abs/math-ph/0201003v2

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Sep 23, 2013
09/13

by
Pavel Bleher; Karl Liechty

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We obtain the large $n$ asymptotics of the partition function $Z_n$ of the six-vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights $a=\sinh(\ga-t), b=\sinh(\ga+t), c=\sinh(2\ga), |t|

Source: http://arxiv.org/abs/0904.3088v2

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Sep 23, 2013
09/13

by
Pavel Bleher; Mikhail Lyubich

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We study the analytical continuation in the complex plane of free energy of the Ising model on diamond-like hierarchical lattices. It is known that the singularities of free energy of this model lie on the Julia set of some rational endomorphism $f$ related to the action of the Migdal-Kadanoff renorm-group. We study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin of $f$. We prove that for almost all (with respect to the...

Source: http://arxiv.org/abs/math/9201275v1

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Sep 21, 2013
09/13

by
Pavel Bleher; Alfredo Deaño

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In this paper we return to the classical work by Br\'ezin, 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 Br\'ezin, 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...

Source: http://arxiv.org/abs/1011.6338v3

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57

Sep 22, 2013
09/13

by
Pavel Bleher; Peter Major

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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...

Source: http://arxiv.org/abs/math-ph/0201014v1

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57

Sep 20, 2013
09/13

by
Pavel Bleher; Karl Liechty

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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...

Source: http://arxiv.org/abs/1205.2101v1

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Sep 24, 2013
09/13

by
Pavel Bleher; Denis Ridzal

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We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1,1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1,1) random polynomial of degree $N$ are concentrated in a narrow annulus of the order of $N^{-1}$ around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the...

Source: http://arxiv.org/abs/math-ph/0103037v1

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64

Sep 18, 2013
09/13

by
Pavel Bleher; Robert Mallison; jr

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We derive the large $n$ asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the $n$-th section of the exponential sum into ``genuine zeros'', which approach, as $n\to\infty$, the zeros of the exponential sum, and ``spurious zeros'', which go to infinity as $n\to\infty$. We show that the spurious zeros, after scaling down by the factor of $n$, approach a ``rosette'', a finite collection of curves on the complex plane, resembling the rosette. We derive also...

Source: http://arxiv.org/abs/math-ph/0605066v1

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Jul 20, 2013
07/13

by
Pavel Bleher; Mikhail Lyubich; Roland Roeder

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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...

Source: http://arxiv.org/abs/1107.5764v1

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Sep 23, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations....

Source: http://arxiv.org/abs/math-ph/9904020v1

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Sep 19, 2013
09/13

by
Pavel Bleher; Mikhail Lyubich; Roland Roeder

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In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising models always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function $\RR$ in two variables (the renormalization transformation)....

Source: http://arxiv.org/abs/1009.4691v1

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Sep 22, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the...

Source: http://arxiv.org/abs/math-ph/0011016v2

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Sep 18, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more...

Source: http://arxiv.org/abs/math-ph/9903012v1

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Sep 18, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to\infty$. We fix a point $P$ and focus on the ball...

Source: http://arxiv.org/abs/math-ph/0002039v2

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Sep 22, 2013
09/13

by
Pavel Bleher; Steven Delvaux; Arno B. J. Kuijlaars

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We consider the random matrix model with external source, in case where the potential V(x) is an even polynomial and the external source has two eigenvalues a, -a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures (mu_1,mu_2) that solve a constrained vector equilibrium problem. The proof is based on the steepest descent analysis of the associated Riemann-Hilbert problem for multiple...

Source: http://arxiv.org/abs/1001.1238v1