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Shannon Starr

Publications and source records attributed to Shannon Starr.

At least 19 recordsLinked to original sources
arXiv2026

About the Cram\'er Large Deviation Property for Bell Polynomials

If $\boldsymbol{w} = (w_1,w_2,\dots)$ is a sequence in $\mathbb{N}=\{1,2,\dots\}$, the partial Bell polynomials based on $\boldsymbol{w}$ are $B_{n,k}$ for $k \in \mathbb{N}$ and $n\in\{k,k+1,\dots\}$. Let $F(z) = \sum_{n=1}^{\infty} (w_n/n!)z^n$ be the exponential generating function for $\boldsymbol{w}$, and assume the radius of convegence is positive $R>0$. Then $F(z)^k = \sum_{n=k}^{\infty} (k!/n!) z^n B_{n,k}$ for $|z|<R$. Alternatively, defining $Q_{k,n} = (k!/n!)B_{n,k}$, we have $Q_{1,n} = w_n/n!$, and $Q_{k+1,n}=\sum_{m=1}^{n-k} Q_{1,m} Q_{k,n-m}$ for $k\geq 1$. Let us say that the Cram\'er-type large deviation property holds if $$ \lim_{\substack{n \to \infty\\ k/n \to \kappa}} \frac{1}{n}\, \ln\left(Q_{k,n}\right)\, =\, \mathcal{G}(\kappa)\, ,$$ for every $\kappa \in (0,1)$, where $\mathcal{G}(\kappa)=\inf_{r \in (0,R)} (\kappa \ln(F(r))-\ln(r))$. The (Hardy-Ramanujan) Erd\"os induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities $\kappa$.

math.CO↗
arXiv2026

Pal's permanent conjecture: proof for block uniform matrices

Consider a symmetric function $\mathcal{C}(x,y)$ on $[0,1]\times[0,1]$ which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let $A^{(n)} = \big(a^{(n)}_{i,j}\, :\, i,j \in [n]\big)$ be the matrix with entries $a^{(n)}_{i,j}\, =\, \exp(-\mathcal{C}(i/n,j/n))$. Soumik Pal conjectured the asymptotics $$\operatorname{perm}\big(A^{(n)}\big)/n!\sim \exp\big(n \Lambda[\mathcal{C}]\big)/ \sqrt{\mathcal{D}[\mathcal{C}]}$$ as $n \to \infty$ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional $\Lambda[\mathcal{C}]$ is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is $\int_{0}^1 \int_0^{1} (\alpha(x)+\beta(y))\, dx\, dy$ where $\alpha(x)+\beta(y)$ is chosen such that $\rho(x,y) := \exp(-\mathcal{C}(x,y)-\alpha(x)-\beta(y))$ has uniform marginals. The algebraic term $\mathcal{D}[c]$ is given by Peter McCullagh's formula for doubly stochastic matrices: $\operatorname{det}_F(I+J-T^*T)$, the Fredholm determinant, where $I$ is the identity on $L^2([0,1])$, $Jf(x) \equiv \int_{0}^1 f(z)\, dz$ (for all $x$) and $Tf(x) = \int_0^1 \rho(x,y) f(y)\, dy$. We prove the conjecture for functions $\mathcal C$ that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.

math.CO↗
arXiv2026

Asymptotics of the d'Arcais Numbers at Small $k$

The d'Arcais numbers are the triangular array $\{A(2,n,k)\, :\, n=0,1,\dots,\, k=0,\dots,n\}$, such that $\sum_{n=0}^{\infty} \sum_{k=0}^{n} A(2,n,k) x^k z^n/n! = ((z;z)_{\infty})^{-x}$. The infinite $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{n=1}^{\infty} (1-q^n)$. Holding $k$ fixed and considering large $n$, we note that the ratio $k! A(2,n,k)/n!$ is asymptotic to $C(k) \sigma_{2k-1}(n)/n^k$ where the divisor sum function is $\sigma_p(n) = \sum_{d|n} d^p$ and $C(k) = (\zeta(2))^k/(\Gamma(k) \zeta(2k))$. This is a slightly generalized version of one of Ramanujan's formulas from his paper, ``On Certain Arithmetical Functions," and it is an immediate consequence of the more recent article of Oliver, Shreshta and Thorne. Heim and Neuhauser made a conjecture, that $A(2,n,k)/A(2,n,k-1)$ is greater than or equal to $A(2,n,k+1)/A(2,n,k)$, for $k=2,3,\dots$ and all $n$. The conjecture is false for $k=2$, and it is true for $k=3,4,\dots$ when $n$ is sufficiently large. We consider the Hardy-Ramanujan circle method as a heuristic step.

math.NT↗
arXiv2026

Large Deviations for the d'Arcais Numbers

The d'Arcais polynomials $P_n(z)$ for $n\in\{0,1,\dots\}$ are defined as $\sum_{n=0}^{\infty} P_n(z) q^n = \exp(-z\ln((q;q)_{\infty}))$ where the $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k)$ for $|q|<1$. Denoting the coefficients for $n \in \mathbb{N}$ by the formula $P_n(z) = \sum_{k=1}^{n} A(2,n,k) z^k/n!$, we prove that $k_n! A(2,n,k_n)/n!$ satisfies a Bahadur-Rao type large deviation formula in the limit $n \to \infty$ with $k_n/n \to \kappa \in [0,1)$ as long as $k_n \to \infty$. The large deviation rate function is the Legendre-Fenchel transform $g^*(-\kappa)$ where $g(\kappa) = f^{-1}(\kappa)$ for the function $f : (0,\infty) \to \mathbb{R}$ given by $f(y)= \ln(-\ln((e^{-y};e^{-y})_{\infty}))$. We relate this fact to information about the abundancy index.

math.PR↗
arXiv2025

A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses saddle point analysis, in a context related to the Hardy-Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.

math.PR↗
arXiv2025

Some Observations about the "Generalized Abundancy Index"

Let $\mathcal{A}(\ell,n) \subset S_n^{\ell}$ denote the set of all $\ell$-tuples $(\pi_1,\dots,\pi_{\ell})$, for $\pi_1,\dots,\pi_{\ell} \in S_n$ satisfying: $\forall i<j$ we have $\pi_i\pi_j=\pi_j\pi_i$. Considering the action of $S_n$ on $[n]=\{1,\dots,n\}$, let $\kappa(\pi_1,\dots,\pi_{\ell})$ be equal to the number of orbits of the action of the subgroup $\langle \pi_1,\dots,\pi_{\ell} \rangle \subset S_n$. There has been interest in the study of the combinatorial numbers $A(\ell,n,k)$ equal to the cardinalities $|\{(\pi_1,\dots,\pi_{\ell}) \in \mathcal{A}(\ell,n)\, :\, \kappa(\pi_1,\dots\pi_{\ell})=k\}|$. If one defines $B(\ell,n)=A(\ell,n,1)/(n-1)!$, then it is known that $B(\ell,n) = \sum_{(f_1,\dots,f_{\ell}) \in \mathbb{N}^{\ell}} \mathbf{1}_{\{n\}}(f_1\cdots f_{\ell}) \prod_{r=1}^{\ell-1} f_r^{\ell-r}$. A special case, $\ell=2$, is $B(2,n) = \sum_{d|n} d = \sigma_1(n)$ the sum-of-divisors function. Then $A(2,n,1)/n!=B(2,n)/n$ is called the abundancy index: $\sigma_1(n)/n$. We call $B(\ell,n) n^{-\ell+1}$ the ``generalized abundancy index.'' Building on work of Abdesselam, using the probability model, we prove that $\lim_{N \to \infty} N^{-1} \sum_{n=1}^{N} B(\ell,n) n^{-\ell+1}$ equals $\zeta(2)\cdots \zeta(\ell)$. Motivated by this we state a more precise conjecture for the asymptotics of $-\zeta(2) + N^{-1}\sum_{n=1}^{N} (B(2,n)/n)$.

math.CO↗
arXiv2024

Multifold Convolutions, Generating Functions and 1d Random Walks

We consider multifold convolutions of a combinatorial sequence $(a_n)_{n=0}^{\infty}$: namely, for each $k \in \N$ the $k$-fold convolution is $\mathcal{M}^{(k)}_n(\boldsymbol{a}) = \sum_{j_1+\dots+j_k=n} a_{j_1} \cdots a_{j_k}$. Let $C_n$ be the Catalan numbers, and let $B_n$ be the central binomial coefficients. Then for random Dyck paths or simple random walk bridges, the multifold convolutions give moments of returns to the origin, using the stars-and-bars problem. There are well-known explicit formulas for the multifold convolutions of $C_n$ and $B_n$. But even for combinatorial sequences $B_n^2$ and $B_n^3$, one may determine asymptotics of multifold convolutions for large $n$. We also discuss large deviations: In a second part of the paper we consider an elementary version of the circle method for calculating asymptotics using complex analysis.

math.CO↗
arXiv2024

About the Hardy-Ramanujan partition function asymptotics

The Hardy-Ramanujan partition function asymptotics is a famous result in the asymptotics of combinatorial sequences. It was originally derived using complex analysis and number-theoretic ideas by Hardy and Ramanujan. It was later re-derived by Paul Erd\H{o}s using real analytic methods. Later still, D.J.~Newman used just the usual Hayman saddle-point approach, ubiquitous in asymptotic analysis. Fristedt introduced a probabilistic approach, which was further extended by Dan Romik, for restricted partition functions. Our perspective is that the Laplace transform changes the essentially algebraic generating function into an exponential form. Using this, we carry out the exercise of deriving the leading order asymptotics, following the Fristedt-Romik approach. We also give additional examples of the Laplace transform method.

math.CO↗
arXiv2024

Online minimum search for a Brownian bridge

In this short note we consider the computational problem of numerically finding the minimum and arg-min of a Brownian bridge. Using well-known results by Pitman, Tanaka, Vervaat and Williams we are able to show that the bisection method has both a small error and a small probability of failure.

math.PR↗
arXiv2024

About the Moments of the Generalized Ulam Problem

Given $\pi \in S_n$, let $Z_{n,k}(\pi)=\sum_{1\leq i_1<\dots<i_k\leq n} \mathbf{1}(\{ \pi_{i_1}<\dots<\pi_{i_k}\}$ denote the number of increasing subsequences of length $k$. Consider the "generalized Ulam problem," studying the distribution of $Z_{n,k}$ for general $k$ and $n$. For the 2nd moment, Ross Pinsky initiated a combinatorial study by considering a pair of subsequences $i^{(r)}_1<\dots<i^{(r)}_k$ for $r \in \{1,2\}$, and conditioning on the size of the intersection $j = |\{i_1^{(1)},\dots,i^{(1)}_k\} \cap \{i^{(2)}_1,\dots,i^{(2)}_k\}|$. We obtain the exact large deviation rate function for $\mathbf{E}[Z_{n,k} Z_{n,\ell}]$ in the asymptotic regime $k\sim \kappa n^{1/2}$, $\ell \sim \lambda n^{1/2}$ as $n \to \infty$, for $\kappa,\lambda \in (0,\infty)$. This uses multivariate generating function techniques, as found in the textbook of Pemantle and Wilson. The requisite generating function enumerates pairs of up-right paths in $d=2$, which both end at $(k,\ell)$ with a given number of intersections. We also evaluate the analogous generating function for pairs of $(+\boldsymbol{i},+\boldsymbol{j},+\boldsymbol{k})$ paths in $d=3$, which both end at $(k,\ell,m)$, which has some utility in calculating the 3rd moment. Finally, we consider a simpler problem involving partitions instead of permutations, where all moments are calculable and the replica symmetric ansatz can be stated if not proved.

math.CO↗
arXiv2023

Online minimum search for Brownian motion and the Cauchy process: Multiple approaches

The distribution for the minimum of Brownian motion or the Cauchy process is well-known using the reflection principle. Here we consider the problem of finding the sample-by-sample minimum, which we call the online minimum search. We consider the possibility of the golden search method, but we show quantitatively that the bisection method is more efficient. In the bisection method there is a hierarchical parameter, which tunes the depth to which each sub-search is conducted, somewhat similarly to how a depth-first search works to generate a topological ordering on nodes. Finally, we consider the possibility of using harmonic measure, which is a novel idea that has so far been unexplored.

math.PR↗
arXiv2023

Violation of Ferromagnetic Ordering of Energy Levels in Spin Rings for the Singlet

We demonstrate a violation of the ``ferromagnetic ordering of energy levels'' conjecture (FOEL) for even length spin rings. The FOEL conjecture was a guess made by Nachtergaele, Spitzer and an author for the Heisenberg model on certain graphs: a family of inequalities, the first of which is the statement that the spectral gap of the Heisenberg model equals the gap of the random walk. That first guess was originally a conjecture of Aldous which was later proved by Caputo, Liggett and Richthammer. We claim that for spin rings of even length $L>4$, the lowest spin $S=0$ energy is lower than the lowest spin $S=1$ energy. This violates the $(L/2)$-th inequality in the FOEL conjecture. Our methodology is largely numerical: we have applied exact diagonalization up to $L=20$. We also rigorously consider the Hamiltonian of the Heisenberg spin ring for even length $L$ projected to the spin $S=0$ sector. We prove that it has a unique ground state. Then, using the single mode approximation the uniqueness explains the energy turn-around. Important insight comes from reconsideration of previous work by Sutherland, using the Bethe ansatz. Especially important is a work of Dhar and Shastry that goes beyond the Bethe ansatz.

math-ph↗
arXiv2022

Generating Function for Pinsky's Combinatorial Second Moment Formula for the Generalized Ulam Problem

Given a uniform random permutation $\pi \in S_n$, let $Z_{n,k}$ be equal to the number of increasing subsequences of length $k$: so $Z_{n,k}=|\{(i_1,\dots,i_k) \in \mathbb{Z}^k\, :\, 1\leq i_1<\dots<i_k\leq n\, ,\ \pi_{i_1}<\dots<\pi_{i_k}\}|$. In an important paper, Ross Pinsky proved $\mathbf{E}\big[Z_{n,k}^2\big]$ is equal to $\sum_{i} A(k-i,i)B(n,2k-i)$, where for any nonnegative integers $N$ and $j$, we have $B(N,j) = \binom{N}{j}/j!$ and $A(N,j)$ is a particular nonnegative integer, which Pinsky characterized in two different ways. One characterization of $A(N,j)$ involves the occupation time of the $x$-axis prior to a first return to the origin. Using this, he proved a law of large numbers for the sequence $Z_{n,k_n}$ when $k_n=o(n^{2/5})$ as $n \to \infty$. In a follow-up paper, he also proved the sequence $Z_{n,k_n}$ fails to obey a law of large numbers when $1/k_n = o(1/n^{4/9})$ as $n \to \infty$. Here, we return to his combinatorial formula for the the second moment of $Z_{n,k}$, and we obtain a generating function for the $A(N,j)$ triangular array. We are motivated by the hope of applying spin glass techniques to the well-known Ulam's problem to see if this gives a new perspective.

math.CO↗
arXiv2019

Two Examples of COM Bounds using Spectral Gaps: Length of the LIS in a Random Permutation and Lipschitz Functions of 1d Markov Chains

We consider two examples for a well-known method for obtaining concentration of measure (COM) bounds for a given observable in a given measure. The method is to consider an auxiliary Markov chain for which the invariant distribution is the measure of interest. Then one obtains COM bounds involving two quantities. The first is the spectral gap of the Markov transition matrix. The second is an appropriate Lipschitz constant for the observable of interest with respect to 1 step of the Markov chain. We consider two examples of the basic method. The first is to obtain rough COM bounds for the length of the longest increasing subsequence (LIS) in a uniform random permutation. The bounds are similar to well-known bounds of Talagrand using his isoperimetric inequality. The second example is to consider a 1d Markov chain: $X_0,X_1,\dots,X_n$. We assume the invariant measure for the chain $μ$ is reversible, and let the initial distribution of $X_0$ be $μ$. Then the observable of interest is any function $f(X_0,X_1,\dots,X_n)$, which is Lipschitz with respect to replacement of single variables. One case of this is `target frequency analysis,' which is of interest in biostatistics. The auxiliary Markov chain is Glauber dynamics which is gapped in 1d.

math.PR↗
arXiv2018

Rough Bounds for Emptiness Formation Probability in the 2d Dimer model using Reflection Positivity

We summarize how to obtain rough bounds for one version of the emptiness formation probability in the 2d dimer model. The methods we use are the same as have been developed to obtain EFP bounds in the 1d XXZ model in a paper with Crawford, Ng and one of the authors. A main tool is reflection positivity for the basic dimer model, as proved by Heilmann and Lieb. We also state the corollary for a 1d quantum spin system which Suzuki showed is related to the 2d Ising model.

math-ph↗
arXiv2018

Phase Uniqueness for the Mallows Measure on Permutations

For a positive number $q$ the Mallows measure on the symmetric group is the probability measure on $S_n$ such that $P_{n,q}(π)$ is proportional to $q$-to-the-power-$\mathrm{inv}(π)$ where $\mathrm{inv}(π)$ equals the number of inversions: $\mathrm{inv}(π)$ equals the number of pairs $i π_j$. One may consider this as a mean-field model from statistical mechanics. The weak large deviation principle may replace the Gibbs variational principle for characterizing equilibrium measures. In this sense, we prove absence of phase transition, i.e., phase uniqueness.

math.PR↗
arXiv2016

Emptiness Formation Probability

We present rigorous upper and lower bounds on the emptiness formation probability for the ground state of a spin-$1/2$ Heisenberg XXZ quantum spin system. For a $d$-dimensional system we find a rate of decay of the order $\exp(-c L^{d+1})$ where $L$ is the sidelength of the box in which we ask for the emptiness formation event to occur. In the $d=1$ case this confirms previous predictions made in the integrable systems community, though our bounds do not achieve the precision predicted by Bethe ansatz calculations. On the other hand, our bounds in the case $d \geq 2$ are new. The main tools we use are reflection positivity and a rigorous path integral expansion which is a variation on those previously introduced by Toth, Aizenman-Nachtergaele and Ueltschi.

math-ph↗