arXiv subjects
Publications and source records attributed to Shannon Starr.
We consider the mixed matrix moments for the complex Ginibre ensemble. These are well-known. We consider the relation to the expected overlap functions of Chalker and Mehlig. This leads to new asymptotic problems for the overlap. We obtain some results, but we also state some remaining open problems.
In this paper we show that d-dimensional Gaussian spin glass models are strongly stochastically stable, fulfill the Ghirlanda-Guerra identities in distribution and the ultrametricity property.
Bernoulli-$p$ thinning has been well-studied for point processes. Here we consider three other cases: (1) sequences $(X_1,X_2,...)$; (2) gaps of such sequences $(X_{n+1}-X_1)_{n\in\mathbb{N}}$; (3) partition structures. For the first case we characterize the distributions which are simultaneously invariant under Bernoulli-$p$ thinning for all $p \in (0,1]$. Based on this, we make conjectures for the latter two cases, and provide a potential approach for proof. We explain the relation to spin glasses, which is complementary to important previous work of Aizenman and Ruzmaikina, Arguin, and Shkolnikov.
We study the antiferromagnetic Potts model on the Poissonian Erdös-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.
The Heisenberg ferromagnet has symmetry group ${\rm SU}(2)$. The property known as ferromagnetic ordering of energy levels (FOEL) states that the minimum energy eigenvalue among eigenvectors with total spin $s$ is monotone decreasing as a function of $s$. While this property holds for certain graphs such as open chains, in this note we demonstrate some counterexamples. We consider the spin 1/2 model on rings of length $2n$ for $n=2,3,...,8$, and show that the minimum energy among all spin singlets is less than or equal to the minimum energy among all spin triplets, which violates FOEL. This also shows some counterexamples to the "Aldous ordering" for the symmetric exclusion process. We also review some of the literature related to these examples.
We consider the interchange process (IP) on the $d$-dimensional, discrete hypercube of side-length $n$. Specifically, we compare the spectral gap of the IP to the spectral gap of the random walk (RW) on the same graph. We prove that the two spectral gaps are asymptotically equivalent, in the limit $n \to \infty$. This result gives further supporting evidence for a conjecture of Aldous, that the spectral gap of the IP equals the spectral gap of the RW on all finite graphs. Our proof is based on an argument invented by Handjani and Jungreis, who proved Aldous's conjecture for all trees. This also has implications for the spectral gap of the quantum Heisenberg ferromagnet.
We review a result obtained with Andrew Ledoan and Marco Merkli. Consider a random analytic function $f(z) = \sum_{n=0}^{\infty} a_n X_n z^n$, where the $X_n$'s are i.i.d., complex valued random variables with mean zero and unit variance, and the coefficients $a_n$ are non-random and chosen so that the variance transforms covariantly under conformal transformations of the domain. If the $X_n$'s are Gaussian, this is called a Gaussian analytic function (GAF). We prove that, even if the coefficients are not Gaussian, the zero set converges in distribution to that of a GAF near the boundary of the domain.
We consider the class of quantum spin chains with arbitrary $U_q(\mathfrak{sl}_2)$-invariant nearest neighbor interactions, sometimes called $\textrm{SU}_q(2)$ for the quantum deformation of $\textrm{SU}(2)$, for $q>0$. We derive sufficient conditions for the Hamiltonian to satisfy the property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron-Frobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of "cascade operators," a special new basis of $U_q(\mathfrak{sl}_2)$ intertwiners we define. We also state applications to interacting particle processes.
The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of inversions of $π$ is equal to the number of pairs $i π_j$. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that $n$ tends to infinity and $q$ tends to 1 in such a way that $n(1-q)$ has a limit in $\R$.
We consider random analytic functions defined on the unit disk of the complex plane as power series such that the coefficients are i.i.d., complex valued random variables, with mean zero and unit variance. For the case of complex Gaussian coefficients, Peres and Virág showed that the zero set forms a determinantal point process with the Bergman kernel. We show that for general choices of random coefficients, the zero set is asymptotically given by the same distribution near the boundary of the disk, which expresses a universality property. The proof is elementary and general.
We construct a W^*-dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved Lieb-Robinson bounds for such systems on finite lattices.
The Mallows model on $S_n$ is a probability distribution on permutations, $q^{d(π,e)}/P_n(q)$, where $d(π,e)$ is the distance between $π$ and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs $(i,j)$ where $1\leq i π_j$. Analyzing the normalization $P_n(q)$, Diaconis and Ram calculated the mean and variance of $d(π,e)$ in the Mallows model, which suggests the appropriate $n \to \infty$ limit has $q_n$ scaling as $1-β/n$. We calculate the distribution of the empirical measure in this limit, $u(x,y) dx dy = \lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^{n} δ_{(i,π_i)}$. Treating it as a mean-field problem, analogous to the Curie-Weiss model, the self-consistent mean-field equations are $\frac{\partial^2}{\partial x \partial y} \ln u(x,y) = 2 βu(x,y)$, which is an integrable PDE, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process, and the ground state of the $\mathcal{U}_q(\mathfrak{sl}_2)$-symmetric XXZ ferromagnet.
We develop a resonance theory to describe the evolution of open systems with time-dependent dynamics. Our approach is based on piecewise constant Hamiltonians: we represent the evolution on each constant bit using a recently developed dynamical resonance theory, and we piece them together to obtain the total evolution. The initial state corresponding to one time-interval with constant Hamiltonian is the final state of the system corresponding to the interval before. This results in a non-markovian dynamics. We find a representation of the dynamics in terms of resonance energies and resonance states associated to the Hamiltonians, valid for all times $t\geq 0$ and for small (but fixed) interaction strengths. The representation has the form of a path integral over resonances. We present applications to a spin-fermion system, where the energy levels of the spin may undergo rather arbitrary crossings in the course of time. In particular, we find the probability for transition between ground- and excited state at all times.
Using a correlation inequality of Contucci and Lebowitz for spin glasses, we demonstrate existence of the thermodynamic limit for short-ranged spin glasses, under weaker hypotheses than previously available, namely without the assumption of the annealed bound.
We obtain bounds to show that the pressure of a two-body, mean-field spin glass is a Lipschitz function of the underlying distribution of the random coupling constants, with respect to a particular semi-norm. This allows us to re-derive a result of Carmona and Hu, on the universality of the SK model, by a different proof, and to generalize this result to the Viana-Bray model. We also prove another bound, suitable when the coupling constants are not independent, which is what is necessary if one wants to consider ``canonical'' instead of ``grand canonical'' versions of the SK and Viana-Bray models. Finally, we review Viana-Bray type models, using the language of Lévy processes, which is natural in this context.
We investigate the low-lying excited states of the spin J ferromagnetic XXZ chain with Ising anisotropy Delta and kink boundary conditions. Since the third component of the total magnetization, M, is conserved, it is meaningful to study the spectrum for each fixed value of M. We prove that for J>= 3/2 the lowest excited eigenvalues are separated by a gap from the rest of the spectrum, uniformly in the length of the chain. In the thermodynamic limit, this means that there are a positive number of excitations above the ground state and below the essential spectrum.
We compute the pressure of the random energy model (REM) and generalized random energy model(GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra's ``broken replica symmetry bounds",and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand's concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.
We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature $β$ and the magnitude of the quantum spins $\CalS$ satisfy $β\ll\sqrt\CalS$. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with $\CalS\gg1$. The most notable examples are the quantum orbital-compass model on $\Z^2$ and the quantum 120-degree model on $\Z^3$ which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.