We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but show that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.
In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix-like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes-Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general $q$-polynomials.
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 provide a variational derivation of the limit shape of minimal difference partitions and discuss the link with exclusion statistics. Also see arXiv:0707.2312 for a related paper.
We study asymptotics of the partition function $Z_N$ of a Laguerre-type random matrix model when the matrix order $N$ tends to infinity. By using the Deift-Zhou steepest descent method for Riemann-Hilbert problems, we obtain an asymptotic expansion of $\log Z_N$ in powers of $N^{-2}$.
We derive the dominant asymptotic form and the order of the correction terms of the finite-perimeter partition function of self-avoiding polygons on the square lattice, which are weighted according to their area A as q^A, in the inflated regime, q>1. The approach q->1^+ of the asymptotic form is examined.
We study random skew 3D partitions weighted by $q^{\textup{vol}}$ and, specifically, the $q\to 1$ asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.
We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [dgg] for the above class of processes. The method used goes back to A.Khintchine.
This survey is a short version of a chapter written by the first two authors in the book [A. Henrot, editor. Shape optimization and spectral theory. Berlin: De Gruyter, 2017] (where more details and references are given) but we have decided here to put more emphasis on the role of the Aharonov-Bohm operators which appear to be a useful tool coming from physics for understanding a problem motivated either by spectral geometry or dynamics of population. Similar questions appear also in Bose-Einstein theory. Finally some open problems which might be of interest are mentioned.
Let $H_h = h^2 L +V$ where $L$ is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and $V$ is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of $H_h$ as $h \to 0$. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive $h$ by the classical partition function.
We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generatin…
In this paper we define the shifted Schur process as a measure on sequences of strict partitions. This process is a generalization of the shifted Schur measure introduced in [TW] and [Mat] and is a shifted version of the Schur process introduced in [OR]. We prove that the shifted Schur process defines a Pfaffian point process. We further apply this fact to compute the bulk scaling limit of the correlation functions for a measure on strict plane partitions which is an analog of the uniform measure on ordinary plane partitions. As a byproduct, we obtain a shifted analog of the famous MacMahon's …
In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for the restricted or coloured partitions p_k^s(n), which is the number of partitions of an integer n into the summand of s^{th} powers of integers such that each power of a given integer may occur utmost k times. While the method is not rigorous, it reproduces the well known asymptotic results for s=1 apar…
We introduce the matrix sums that represent a discrete analog of the matrix models with quartic potential. The probability space is given by the set of all simple n-vertex graphs with the Gibbs weight determined by the graph Laplacian. We study the large-n limit of the free energy per site and show that it is determined by the number of connected acyclic diagrams on the set of two-valent vertices.
We obtain large N asymptotics for the Hermitian random matrix partition function \[Z_N(V)=\int_{\mathbb R^N}\prod_{i<j}(x_i-x_j)^2 \prod_{j=1}^N e^{-N V(x_j)}dx_j,\] in the case where the external potential $V$ is a polynomials such that the random matrix eigenvalues accumulate on two disjoint intervals (the two-cut case). We compute leading and sub-leading terms in the asymptotic expansion for $\log Z_N(V)$, up to terms that are small as $N$ goes to infinity. Our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential $V$.…
We continue to investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence begun in arxiv:0708.0645. We derive the master matrix of the $(2,1)$ minimal and $(3,1)$ minimal matrix model. We use it's characteristic polynomial to understand why the zeros of the FZZT partition function, which is the Airy function, lie on the real axis. We also introduce an iterative procedure that can describe the Riemann $Ξ$ function as a deformed minimal model whose deformation parameters are related to a Konsevich integr…
We present a solution to a problem suggested by Philippe Biane: We prove that a certain Plancherel-type probability distribution on partitions converges, as partitions get large, to a new determinantal random point process on the set {0,1,2,...} of nonnegative integers. This can be viewed as an edge limit ransition. The limit process is determined by a correlation kernel on {0,1,2,...} which is expressed through the Hermite polynomials, we call it the discrete Hermite kernel. The proof is based on a simple argument which derives convergence of correlation kernels from convergence of unbounded …
In this paper, we discuss P(n), the number of ways in which a given integer n may be written as a sum of primes. In particular, an asymptotic form P_as(n) valid for n towards infinity is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This directly gives the asymptotic number of …
The generating function which counts partitions with the Plancherel measure (and its q-deformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the Gromov-Witten invariants of the X_p Calabi-Yau 3-fold, and we prove a conjecture of M. Marino, that the generating functions F_g of Gromov--Witten invariants of X_p, come from a matrix model, and are the symplectic invariants of the mirror spe…
We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic point $ p=0$ is a repulsive fixed point for which the limiting distribution has a Gumbel form. For all positive $p$ the distribution is shown to be Gaussian.
The treatment of the number-theoretical problem of integer partitions within the approach of statistical mechanics is discussed. Historical overview is given and known asymptotic results for linear and plane partitions are reproduced. From numerical analysis of restricted plane partitions an asymptotic formula is conjectured for an intermediate number of parts.
The class of minimal difference partitions MDP($q$) (with gap $q$) is defined by the condition that successive parts in an integer partition differ from one another by at least $q\ge 0$. In a recent series of papers by A. Comtet and collaborators, the MDP($q$) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classic Bose-Einstein ($q=0$) and Fermi-Dirac ($q=1$) cases. This was done by formally allowing values $q \in (0,1)$ using an analytic continuation of the limit shape of the correspondi…
For a finite random graph, we defined a simple model of statistical mechanics. We obtain an annealed asymptotic result for the random partition function for this model on finite random graphs as n; the size of the graph is very large. To obtain this result, we define the empirical bond distribution, which enumerates the number of bonds between a given couple of spins, and empirical spin distribution, which enumerates the number of sites having a given spin on the spinned random graphs. For these empirical distributions we extend the large deviation principle(LDP) to cover random graphs with co…
We consider some aspects of counting BPS operators which are annihilated by two supercharges, in superconformal field theories. For non-zero coupling, the corresponding multi-variable partition functions can be written in terms of generating functions for vector partitions or their weighted generalisations. We derive asymptotics for the density of states for a wide class of such multi-variable partition functions. We also point out a particular factorisation property of the finite N partition functions. Finally, we discuss the concept of a limit curve arising from the large N partition functio…
