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Let F be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists.

We consider a spatial (line) model for invasion of a population by a single mutant with a stochastically selectively neutral fitness landscape, independent from the fitness landscape for non-mutants. This model is similar to those considered in Farhang-Sardroodi et al. [PLOS Comput. Biol., 13(11), 2017; J. R. Soc. Interface, 16(157), 2019]. We show that the probability of mutant fixation in a population of size $N$, starting from a single mutant, is greater than $1/N$, which would be the case if there were no variation in fitness whatsoever. In the small variation regime, we recover precise asymptotics for the success probability of the mutant. This demonstrates that the introduction of randomness provides an advantage to minority mutations in this model, and shows that the advantage increases with the system size. We further demonstrate that the mutants have an advantage in this setting only because they are better at exploiting unusually favorable environments when they arise, and not because they are any better at exploiting pockets of favorability in an environment that is selectively neutral overall., Comment: 25 pages, 4 figures

On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which the behavior at infinity may be ignored, and the usual theorems of classical and stratified Morse theory may be applied. This allows for simplified arguments in the field of analytic combinatorics in several variables, and forms the basis for new methods applying to problems beyond the reach of previous techniques., Comment: Updated and simplified presentation and statements of main results

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions d at least 4. This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and Garding. As a consequence, we give a topological explanation for certain asymptotic phenomenon appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions, giving a significant new tool to attack the so-called connection problem for asymptotics of P-recursive sequences. This in turn enables the rigorous determination of integer coefficients in the Morse-Smale complex, which are difficult to determine using direct geometric methods.

Continuous time branching models are used to create random fractals in a Euclidean space, whose Hausdorff dimension is controlled by an input parameter. Finite realizations are applied in modelling the set of sites visited in models of human and animal mobility., Comment: 23 pages, 2 figures

We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout the regime $\ell = \Theta (m)$ and $n = \Theta (m^2)$. Previously, sharp asymptotics were derived by Tak\'acs only in the regime where $|n - \ell m /2| = O(\sqrt{\ell m (\ell + m)})$ using a local central limit theorem. Our approach is to solve a related large deviation problem: we describe the tilted measure that produces configurations whose bounding rectangle has the given aspect ratio and is filled to the given proportion. Our results are sufficiently sharp to yield the first asymptotic estimates on the consecutive differences of these numbers when $n$ is increased by one and $m, \ell$ remain the same, hence significantly refining Sylvester's unimodality theorem., Comment: Updated with additional references to existing literature, slightly different introductory exposition, and increased focus on connection to Kronecker coefficients

We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} , \cdot)$ at criticality in terms of limits of martingales defined from $\mathbf{T}$ (this requires a moment condition depending on $k$); and a proof that the $k\text{th}$ order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton-Watson tree., Comment: 38 pages

We consider asymptotics of power series coefficients of rational functions of the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of coefficients or diagonal asymptotics. We then analyze coefficient asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods. While ACSV sometimes requires considerable overhead and geometric computation, in the case of symmetric multilinear rational functions there are some reductions that streamline the analysis. Our results include diagonal asymptotics across entire classes of functions, for example the general 3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the denominator in terms of elementary symmetric functions is $1 - e_1 + c e_d$ in any number $d$ of variables. The ACSV analysis also explains a discontinuous drop in exponential growth rate for the GRZ class at the parameter value $c = (d-1)^{d-1}$, previously observed for $d=4$ only by separately computing diagonal recurrences for critical and noncritical values of $c$., Comment: To appear in LIPIcs Proceedings of Analysis of Algorithms 2018

The insertion-deletion channel takes as input a bit string ${\bf x}\in\{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called "traces") of the insertion-deletion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of $q\geq 1/2$ or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., $\exp(O( n^{1/3}))$. We also show that our reconstruction algorithm runs in $n^{1+o(1)}$ time. A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of $\bf x$. The alignment is done by viewing the strings as random walks and comparing the increments in the walk associated with the input string and the trace, respectively., Comment: Analysis of running time added and proof simplified. Alex Zhai added as author. 37 pages, 7 figures

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process., Comment: 35 pages, 1 figure

We generalize the results of \cite{SPU, SJPU} by showing how the Gaussian aggregator may be computed in a setting where parameter estimation is not required. We proceed to provide an explicit formula for a "one-shot" aggregation problem with two forecasters., Comment: 21 pages, 5 figures in Statistics and Probability Letters (2016)

Let $(X_1 , \ldots , X_d)$ be random variables taking nonnegative integer values and let $f(z_1, \ldots , z_d)$ be the probability generating function. Suppose that $f$ is real stable; equivalently, suppose that the polarization of this probability distribution is strong Rayleigh. In specific examples, such as occupation counts of disjoint sets by a determinantal point process, it is known~\cite{soshnikov02} that the joint distribution must approach a multivariate Gaussian distribution. We show that this conclusion follows already from stability of $f$.

The longest increasing subsequence of a random walk with mean zero and finite variance is known to be $n^{1/2 + o(1)}$. We show that this is not universal for symmetric random walks. In particular, the symmetric Ultra-fat tailed random walk has a longest increasing subsequence that is asymptotically at least $n^{0.690}$ and at most $n^{0.815}$. An exponent strictly greater than $1/2$ is also shown for the symmetric stable-$\alpha$ distribution when $\alpha$ is sufficiently small.

We study a coarsening process of one-dimensional cell complexes. We show that if cell boundaries move with velocities proportional to the difference in size of neighboring cells, then the average cell size grows at a prescribed exponential rate and the Poisson distribution is precisely invariant for the distribution of the whole process, rescaled in space by its average growth rate. We present numerical evidence toward the following universality conjecture: starting from any finite mean stationary renewal process, the system when rescaled by $e^{-2t}$ converges to a Poisson point process. For a limited case, this makes precise what has been observed previously in experiments and simulations, and lays the foundation for a theory of universal asymptotic states of dynamical cell complexes.

Prediction polling is an increasingly popular form of crowdsourcing in which multiple participants estimate the probability or magnitude of some future event. These estimates are then aggregated into a single forecast. Historically, randomness in scientific estimation has been generally assumed to arise from unmeasured factors which are viewed as measurement noise. However, when combining subjective estimates, heterogeneity stemming from differences in the participants' information is often more important than measurement noise. This paper formalizes information diversity as an alternative source of such heterogeneity and introduces a novel modeling framework that is particularly well-suited for prediction polls. A practical specification of this framework is proposed and applied to the task of aggregating probability and point estimates from two real-world prediction polls. In both cases our model outperforms standard measurement-error-based aggregators, hence providing evidence in favor of information diversity being the more important source of heterogeneity.

We consider the problem of distributed corruption detection in networks. In this model, each vertex of a directed graph is either truthful or corrupt. Each vertex reports the type (truthful or corrupt) of each of its outneighbors. If it is truthful, it reports the truth, whereas if it is corrupt, it reports adversarially. This model, first considered by Preparata, Metze, and Chien in 1967, motivated by the desire to identify the faulty components of a digital system by having the other components checking them, became known as the PMC model. The main known results for this model characterize networks in which \emph{all} corrupt (that is, faulty) vertices can be identified, when there is a known upper bound on their number. We are interested in networks in which the identity of a \emph{large fraction} of the vertices can be identified. It is known that in the PMC model, in order to identify all corrupt vertices when their number is $t$, all indegrees have to be at least $t$. In contrast, we show that in $d$ regular-graphs with strong expansion properties, a $1-O(1/d)$ fraction of the corrupt vertices, and a $1-O(1/d)$ fraction of the truthful vertices can be identified, whenever there is a majority of truthful vertices. We also observe that if the graph is very far from being a good expander, namely, if the deletion of a small set of vertices splits the graph into small components, then no corruption detection is possible even if most of the vertices are truthful. Finally, we discuss the algorithmic aspects and the computational hardness of the problem.

Randomness in scientific estimation is generally assumed to arise from unmeasured or uncontrolled factors. However, when combining subjective probability estimates, heterogeneity stemming from people's cognitive or information diversity is often more important than measurement noise. This paper presents a novel framework that uses partially overlapping information sources. A specific model is proposed within that framework and applied to the task of aggregating the probabilities given by a group of forecasters who predict whether an event will occur or not. Our model describes the distribution of information across forecasters in terms of easily interpretable parameters and shows how the optimal amount of extremizing of the average probability forecast (shifting it closer to its nearest extreme) varies as a function of the forecasters' information overlap. Our model thus gives a more principled understanding of the historically ad hoc practice of extremizing average forecasts., Comment: This paper has been withdrawn because it was meant to be a revision to arXiv:1406.2148, not an independent submission. This was discovered on 27 May, 2015, when preparing a replacement version to arXiv:1406.2148. This replacement will supersede both that version and the one withdrawn here

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some $\epsilon > 0$ independent of $n$, even when an adversary is allowed to conjugate each of the four by a possibly different element of $\S_n$ (in other words, the cycle types already guarantee generation of $\mathcal{S}_n$). This is closely related to the following random set model. A random set $M \subseteq \mathbb{Z}^+$ is generated by including each $n \geq 1$ independently with probability $1/n$. The sumset $\text{sumset}(M)$ is formed. Then at most four independent copies of $\text{sumset}(M)$ are needed before their mutual intersection is no longer infinite., Comment: 19pages, 1 figure

We consider an analytic function $f$ whose zero set forms a unit intensity Poisson process on the real line. We show that repeated differentiation causes the zero set to converge in distribution to a random translate of the integers.

We show that the longest k-alternating substring of a random permutation has length asymptotic to 2 (n-k) / 3.

Randomness in scientific estimation is generally assumed to arise from unmeasured or uncontrolled factors. However, when combining subjective probability estimates, heterogeneity stemming from people's cognitive or information diversity is often more important than measurement noise. This paper presents a novel framework that models the heterogeneity arising from experts that use partially overlapping information sources, and applies that model to the task of aggregating the probabilities given by a group of experts who forecast whether an event will occur or not. Our model describes the distribution of information across experts in terms of easily interpretable parameters and shows how the optimal amount of extremizing of the average probability forecast (shifting it closer to its nearest extreme) varies as a function of the experts' information overlap. Our model thus gives a more principled understanding of the historically ad hoc practice of extremizing average forecasts.

The algebraic relations between the principal minors of an $n\times n$ matrix are somewhat mysterious, see e.g. [lin-sturmfels]. We show, however, that by adding in certain \emph{almost} principal minors, the relations are generated by a single relation, the so-called hexahedron relation, which is a composition of six cluster mutations. We give in particular a Laurent-polynomial parameterization of the space of $n\times n$ matrices, whose parameters consist of certain principal and almost principal minors. The parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex $2n$-gon. A matrix is associated to an equivalence class of tilings, all related to each other by Yang-Baxter-like transformations. By specializing the initial data we can similarly parametrize the space of Hermitian symmetric matrices over $\mathbb R, \mathbb C$ or $\mathbb H$ the quaternions. Moreover by further specialization we can parametrize the space of \emph{positive definite} matrices over these rings.

We define and study a recurrence relation in ${\mathbb Z}^3$, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition sums for edge configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations. The Kashaev difference equation arising in the Ising model star-triangle relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.

This was the basis of two lectures in the Current Developments in Mathematics conference in 2011. These lectures survey the theory of hyperbolic and stable polynomials, from their origins in the theory of linear PDE's to their present uses in combinatorics and probability theory.

In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on the complex numbers. We conjecture that the zero set of f' always converges in distribution to mu as n goes to infinity. We prove this for measures with finite one-dimensional energy. When mu is uniform on the unit circle this condition fails. In this special case the zero set of f' converges in distribution to that the IID Gaussian random power series, a well known determinantal point process.

We consider a bivariate rational generating function F(x,y) = P(x,y) / Q(x,y) = sum_{r, s} a_{r,s} x^r y^s under the assumption that the complex algebraic curve $\sing$ on which $Q$ vanishes is smooth. Formulae for the asymptotics of the coefficients a_{rs} were derived by Pemantle and Wilson (2002). These formulae are in terms of algebraic and topological invariants of the pole variety, but up to now these invariants could be computed only under a minimality hypothesis, namely that the dominant saddle lies on the boundary of the domain of convergence. In the present paper, we give an effective method for computing the topological invariants, and hence the asymptotics of the values a_{r,s}, without the minimality assumption. This leads to a theoretically rigorous algorithm, whose implementation is in progress at http://www.mathemagix.org .

Let X_1 ,..., X_n be a collection of binary valued random variables and let f : {0,1}^n -> R be a Lipschitz function. Under a negative dependence hypothesis known as the {\em strong Rayleigh} condition, we show that f - E f satisfies a concentration inequality generalizing the classical Gaussian concentration inequality for sums of independent Bernoullis: P (S_n - E S_n > a) < exp (-2 a^2 / n). The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar examples from these classes are generalized negative binomials and spanning tree measures. For instance, the number of vertices of odd degree in a uniform random spanning tree of a graph satisfies a Gaussian concentration inequality with n replaced by |V|, the number of vertices. We also prove a continuous version for concentration of Lipschitz functionals of a determinantal point process.

Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for $f(n)$ include an $n \times n$ determinant whose entries are binomial coefficients (Kreweras, 1965) and, in the special case of $b_j = rj+s$, a short explicit formula (Proctor, 1988, p.320). A relatively easy bivariate recursion, decomposing all sequences according to $n$ and $a_n$, leads to a bivariate generating function, then a univariate generating function, then a linear recursion for $\{f(n) \}$. Moreover, the coefficients of the bivariate generating function have a probabilistic interpretation, leading to an analytic inequality which is an identity for certain values of its argument.

We consider saddle point integrals in d variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The method is via contour shifting in complex d-space. This work is motivated by applications to asymptotic enumeration.

We apply results from Baryshnikov, Brady, Bressler and Pemantle (2008) to compute limiting probability profiles for various quantum random walks in one and two dimensions. Using analytic machinery we show some features of the limit distribution that are not evident in an empirical intensity plot of the time 10,000 distribution. Some conjectures are stated and computational techniques are discussed as well.

In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to rapidly determine a subsequence whose product is a square (which we call a square product). In his lecture at the 1994 International Congress of Mathematicians, Pomerance observed that the following problem encapsulates all of the key issues: Select integers a_1, a_2, >... at random from the interval [1,x], until some (non-empty) subsequence has product equal to a square. Find good estimate for the expected stopping time of this process. A good solution to this problem should help one to determine the optimal choice of parameters for one's factoring algorithm, and therefore this is a central question. Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability 1-o(1) the first subsequence whose product equals a square occurs after at least J_0^{1-o(1)} integers have been selected, but no more than J_0, for an appropriate (explicitly determined) J_0=J_0(x). Herein we determine this expected stopping time up to a constant factor, tightening Pomerance's interval to $$[ (\pi/4)(e^{-\gamma} - o(1))J_0, (e^{-\gamma} + o(1)) J_0],$$ where $\gamma = 0.577...$ is the Euler-Mascheroni constant. We will also confirm the well established belief that, typically, none of the integers in the square product have large prime factors. We believe the upper of the two bounds to be asymptotically sharp.

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of Pemantle and Wilson (2002).

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations., Comment: substantial corrections

Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean., Comment: 37 pages; to appear in Annales de l'institut Henri Poincare (B)

Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean $p\leq1/2$. How many of these Bernoullis one must look at in order to find a path of length $n$ from the root which maximizes, up to a factor of $1-\epsilon$, the sum of the Bernoullis along the path? In the case $p=1/2$ (the critical value for nontriviality), it is shown to take $\Theta(\epsilon^{-1}n)$ steps. In the case $p<1/2$, the number of steps is shown to be at least $n\cdot\exp(\operatorname {const}\epsilon^{-1/2})$. This last result matches the known upper bound from [Algorithmica 22 (1998) 388--412] in a certain family of subcases., Comment: Published in at http://dx.doi.org/10.1214/08-AAP585 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas., Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $F$. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have the property that generating functions derived from them are amenable to our asymptotic analyses, and we describe further machinery that facilitates computations for these classes of examples., Comment: 87 pages, many figures. To appear in SIAM Review

An infinite sequence of 0's and 1's evolves by flipping each~1 to a~0 exponentially at rate one. When a~1 flips, all bits to its right also flip. Starting from any configuration with finitely many 1's to the left of the origin, we show that the leftmost~1 moves right with linear speed. Upper and lower bounds are given on the speed., Comment: 21pp

We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an $L^3$ capacity arises. In particular, on a spherically symmetric tree that has $n^{\alpha}b^n$ vertices at level $n$ (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if $\alpha\le1/2$. Our proofs are based on a new link between nonlinear recursions on trees and $L^p$ capacities., Comment: Published in at http://dx.doi.org/10.1214/09-AOP482 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a multivariate generating function F(z), whose coefficients are indexed by d-tuples of nonnegative integers: F(z) = sum_r a_r z^r where z^r denotes the product of z_j^{r_j} over j = 1, ..., d. Suppose that F(z) is meromorphic in some neighborhood of the origin in complex d-space. Let V be the set where the denominator of F vanishes. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of V. In the first article in this series, we treated the case of smooth points of V. In this article we deal with multiple points of V. Our results show that the central limit (Ornstein-Zernike) behavior typical of the smooth case does not hold in the multiple point case. For example, when V has a multiple point singularity at the point (1, ..., 1), rather than a_r decaying on the order of |r|^{-1/2} as |r| goes to infinity, a_r is a polynomial plus a rapidly decaying term.

Let $n = b_1 + ... + b_k = b_1' + \cdot + b_k'$ be a pair of compositions of $n$ into $k$ positive parts. We say this pair is {\em irreducible} if there is no positive $j < k$ for which $b_1 + ... b_j = b_1' + ... b_j'$. The probability that a random pair of compositions of $n$ is irreducible is shown to be asymptotic to $8/n$. This problem leads to a problem in probability theory. Two players move along a game board by rolling a die, and we ask when the two players will first coincide. A natural extension is to show that the probability of a first return to the origin at time $n$ for any mean-zero variance $V$ random walk is asymptotic to $\sqrt{V/(2 \pi)} n^{-3/2}$. We prove this via two methods, one analytic and one probabilistic.