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Ross G. Pinsky

Publications and source records attributed to Ross G. Pinsky.

At least 37 records · Page 2Linked to original sources

arXiv2021

A View From The Bridge Spanning Combinatorics and Probability

This paper presents an offering of some of the myriad connections between Combinatorics and Probability, directed in particular toward combinatorialists. The choice of material was dictated by the author's own interests, tastes and familiarity, as well as by a desire to present results with either complete proofs or well developed sketches of proofs, and to ensure that the arguments are rather accessible to combinatorialists. The first several sections collect some concepts and rudimentary results from probability theory that are needed to understand the rest of the paper.

math.HO↗

arXiv2021

Comparing the inversion statistic for distribution-biased and distribution-shifted permutations with the geometric and the GEM distributions

For a distribution $p:=\{p_k\}_{k=1}^\infty$ on the positive integers, there are two natural ways to construct a random permutation in $S_n$ or of $\mathbb{N}$ from IID samples from $p$--the $p$-biased construction and the $p$-shifted construction. First we consider the case that $p$ is the geometric distribution with parameter $1-q\in(0,1)$. In this case, the $p$-shifted random permutation has the Mallows distribution with parameter $q$. Let $P_n^{b;\text{Geo}(1-q)}$ and $P_n^{s;\text{Geo}(1-q)}$denote the biased and the shifted distributions on $S_n$. The expected number of inversions of a permutation under $P_n^{s;\text{Geo}(1-q)}$ is greater than under $P_n^{b;\text{Geo}(1-q)}$, and under either of these, a permutation tends to have many fewer inversions than it would have under the uniform distribution. For fixed $n$, both $P_n^{b;\text{Geo}(1-q)}$ and $P_n^{s;\text{Geo}(1-q)}$ converge weakly as $q\to1$ to the uniform distribution on $S_n$. We compare the biased and the shifted distributions by studying the inversion statistic under $P_n^{b;\text{Geo}(q_n)}$ and $P_n^{s;\text{Geo}(q_n)}$ for various rates of convergence of $q_n$ to 1. Then we consider $p$-biased and $p$-shifted permutations in the case that the distribution $p$ is itself random and distributed as a GEM$(θ)$-distribution. In both the GEM$(θ)$-biased and the GEM$(θ)$-shifted cases, the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-shifted distribution with $θ=\frac q{1-q}$. Thus, one can consider the GEM$(θ)$-shifted case as the random counterpart of the Geo$(q)$-shifted case. We also consider another $p$-biased distribution with random $p$ for which the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-biased case with $θ$ and $q$ as above, and with $θ\to\infty$ and $q\to1$.

math.PR↗

arXiv2021

The Infinite Limit of Separable Permutations

Let $P_n^{\text{sep}}$ denote the uniform probability measure on the set of separable permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions $σ=\{σ_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $σ^{-1}(\mathbb{N})$\rm; that is, if $σ_i=σ_j$, $i\neq j$, then $σ_i=\infty$. Extending permutations $σ\in S_n$ by defining $σ_j=j$, for $j>n$, we have $S_n\subset S(\mathbb{N},\mathbb{N}^*)$. We show that $\{P_n^{\text{sep}}\}_{n=1}^\infty$ converges weakly on $S(\mathbb{N},\mathbb{N}^*)$ to a limiting distribution of regenerative type, which we calculate explicitly.

math.PR↗

arXiv2021

Super-clustering of consecutive numbers in $p$-shifted random permutations

Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$ consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on $\mathbb{N}$ with $p_j>0$. Let $P_n$ denote the probability measure on $S_n$ corresponding to the $p$-shifted random permutation. Our main result, under the additional assumption that $\{p_j\}_{j=1}^\infty$ is non-increasing, is that $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k})=\big(\prod_{j=1}^{k-1}\sum_{i=1}^jp_i\big) \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big), \end{aligned} $$ and that if $\lim_{n\to\infty}\min(k_n,n-k_n)=\infty$, then $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k_n})= \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big)^2. \end{aligned} $$ In particular these limits are positive if and only if $\sum_{j=1}^\infty jp_j<\infty$. We say that super-clustering occurs when the limits are positive. We also give a new characterization of the class of $p$-shifted probability distributions on $S_\infty$.

math.PR↗

arXiv2020

Clustering of consecutive numbers in permutations under Mallows distributions and super-clustering under general $p$-shifted distributions

Let $A^{(n)}_{l;k}\subset S_n$ denote the set of permutations of $[n]$ for which the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions. Under the uniformly probability measure $P_n$ on $S_n$, one has $P_n(A^{(n)}_{l;k})\sim\frac{l!}{n^{l-1}}$ as $n\to\infty$. In one part of this paper we consider the probability of clustering of consecutive numbers under Mallows distributions $P_n^q$, $q>0$. Because of a duality, it suffices to consider $q\in(0,1)$. We show that for $q_n=1-\frac c{n^α}$, with $c>0$ and $α\in(0,1)$, $P_n^q(A^{(n)}_{l;k_n})$ is on the order $\frac1{n^{α(l-1)}}$, uniformly over all sequences $\{k_n\}_{n=1}^\infty$. Thus, letting $N^{(n)}_l=\sum_{k=1}^{n-l+1}1_{A^{(n)}_{l;k}}$ denote the number of sets of $l$ consecutive numbers appearing in sets of consecutive positions, we have \begin{equation*} \lim_{n\to\infty} E_n^{q_n}N^{(n)}_l = \begin{cases}\infty,\ \text{if}\ l<\frac{1+α}α;\\ 0,\ \text{if} \ l>\frac{1+α}α. \end{cases}. \end{equation*} We also consider the cases $α=1$ and $α>1$. In the other part of the paper we consider general $p$-shifted distributions, of which the Mallows distribution is a particular case. We calculate explicitly the quantity $\lim_{l\to\infty} \liminf_{n\to\infty}P_n^q(A^{(n)}_{l;k_n}) = \lim_{l\to\infty}\limsup_{n\to\infty}P_n^q(A^{(n)}_{l;k_n})$ in terms of the $p$-distribution. When this quantity is positive, we say that super-clustering occurs. In particular, super-clustering occurs for the Mallows distribution with parameter $q\neq1$. We also give a new characterization of $p$-shifted distributions.

math.PR↗

arXiv2020

Permutations avoiding a pattern of length three under Mallows distributions

We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern $τ\in S_3-\{321\}$, we obtain rather precise results on the asymptotic probability as $n\to\infty$ that a permutation $σ\in S_n$ under the Mallows distribution with parameter $q\in(0,1)$ avoids the pattern. By a duality between the parameters $q$ and $\frac1q$, we also obtain rather precise results on the above probability for $q>1$ and any pattern $τ\in S_3-\{123\}$.

math.PR↗

arXiv2019

The speed of a general random walk reinforced by its recent history

We consider several variants of a class of random walks whose increment distributions depend on the average value of the process over its most recent $N$ steps. We investigate the speed of the process, and in particular, the limiting speed as the "history window" $N\to\infty$.

math.PR↗

arXiv2019

Kemeny's constant for one-dimensional diffusions

Let $X(\cdot)$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\mathbb{R}$ with invariant probability density $μ(x)$, and let $τ_y=\inf\{t\ge0: X(t)=y\}$ denote the first hitting time of $y$. Let $\mathcal{X}$ be a random variable independent of the diffusion process $X(\cdot)$ and distributed according to the process's invariant probability measure $μ(x)dx$. Denote by $\mathcal{E}^μ$ the expectation with respect to $\mathcal{X}$. Consider the expression $$ \mathcal{E}^μE_xτ_\mathcal{X}=\int_{-\infty}^\infty (E_xτ_y)μ(y)dy, \ x\in\mathbb{R}. $$ In words, this expression is the expected hitting time of the diffusion starting from $x$ of a point chosen randomly according to the diffusion's invariant distribution. We show that this expression is constant in $x$, and that it is finite if and only if $\pm\infty$ are entrance boundaries for the diffusion. This result generalizes to diffusion processes the corresponding result in the setting of finite Markov chains, where the constant value is known as Kemeny's constant.

math.PR↗

arXiv2018

Diffusive Search with spatially dependent Resetting

Consider a stochastic search model with resetting for an unknown stationary target $a\in\mathbb{R}$ with known distribution $μ$. The searcher begins at the origin and performs Brownian motion with diffusion constant $D$. The searcher is also armed with an exponential clock with spatially dependent rate $r$, so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it continues its search anew from there. Denote the position of the searcher at time $t$ by $X(t)$. Let $E_0^{(r)}$ denote expectations for the process $X(\cdot)$. The search ends at time $T_a=\inf\{t\ge0:X(t)=a\}$. The expected time of the search is then $\int_{\mathbb{R}}(E_0^{(r)}T_a)\thinspaceμ(da)$. Ideally, one would like to minimize this over all resetting rates $r$. We obtain quantitative growth rates for $E_0^{(r)}T_a$ as a function of $a$ in terms of the asymptotic behavior of the rate function $r$, and also a rather precise dichotomy on the asymptotic behavior of the resetting function $r$ to determine whether $E_0^{(r)}T_a$ is finite or infinite. We show generically that if $r(x)$ is on the order $|x|^{2l}$, with $l>-1$, then $\log E_0^{(r)}T_a$ is on the order $|a|^{l+1}$; in particular, the smaller the asymptotic size of $r$, the smaller the asymptotic growth rate of $E_0^{(r)}T_a$. The asymptotic growth rate of $E_0^{(r)}T_a$ continues to decrease when $r(x)\sim \frac{Dλ}{x^2}$ with $λ>1$; now the growth rate of $E_0^{(r)}T_a$ is more or less on the order $|a|^{\frac{1+\sqrt{1+8λ}}2}$. However, if $λ=1$, then $E_0^{(r)}T_a=\infty$, for $a\neq0$.

math.PR↗

arXiv2018

Probabilistic Proofs of Some Generalized Mertens' Formulas Via Generalized Dickman Distributions

The classical Mertens' formula states that $ \prod_{p\le N}\big(1-\frac1p)^{-1}\sim e^γ\log N, $ where the product is over all primes $p$ less than or equal to $N$, and $γ$ is the Euler-Mascheroni constant. By the Euler product formula, this is equivalent to either of the following statements: $$ \begin{aligned} &i. \lim_{N\to\infty}\frac{\sum_{n:p|n\Rightarrow p\le N}\thinspace\frac1n}{\sum_{n\le N}\frac1n}=e^γ \ &ii. \sum_{n:p|n\Rightarrow p\le N}\thinspace\frac1n\sim e^γ\log N. \end{aligned} $$ Via some random integer constructions and a criterion for weak convergence of distributions to so-called generalized Dickman distributions, we obtain some generalized Mertens' formulas, some of which are new and some of which have been proved using number-theoretic tools. For example, in the spirit of (i), we show that if $A$ is a subset of the primes which has natural density $θ\in(0,1]$ with respect to the set of all primes, then $$ \lim_{N\to\infty}\frac{\sum_{n:p|n\Rightarrow p\le N\thinspace\text{and}\thinspace p\in A}\frac1n} {\sum_{n\le N:p|n\Rightarrow p\in A}\frac1n}=e^{γθ}Γ(θ+1), $$ and also, for any $k\ge2$, $$ \lim_{N\to\infty}\frac{\sum^{'(k)}_{n:p|n\Rightarrow p\le N\thinspace\text{and}\thinspace p\in A}\frac1n} {\sum^{'(k)}_{n\le N:p|n\Rightarrow p\in A}\frac1n}=e^{γθ}Γ(θ+1), $$ where $\sum^{'(k)}$ denotes that the summation is restricted to $k$-free positive integers. In the spirit of (ii), we show for example that $ \sum^{'(k)}_{n:p|n\Rightarrow p\le N}\frac1{n_{\{(k-1)-\text{free}\}}ϕ(n_{\{(k-1)-\text{power}\}})}\sim e^γ\log N, $ where $ϕ$ is the Euler totient function, and $n_{\{(k-1)-\text{free}\}}$ and $n_{\{(k-1)-\text{power}\}}$ are the $(k-1)$-free part and the $(k-1)$-power part of $n$.

math.NT↗

arXiv2018

The Infinite Limit of Random Permutations Avoiding Patterns of Length Three

For $τ\in S_3$, let $μ_n^τ$ denote the uniformly random probability measure on the set of $τ$-avoiding permutations in $S_n$. Let $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$ with an appropriate metric and denote by $S(\mathbb{N},\mathbb{N}^*)$ the compact metric space consisting of functions $σ=\{σ_i\}_{ i=1}^\infty$ from $\mathbb{N}$ to $\mathbb{N}^*$ which are injections when restricted to $σ^{-1}(\mathbb{N})$\rm; that is, if $σ_i=σ_j$, $i\neq j$, then $σ_i=\infty$. Extending permutations $σ\in S_n$ by defining $σ_j=j$, for $j>n$, we have $S_n\subset S(\mathbb{N},\mathbb{N}^*)$. For each $τ\in S_3$, we study the limiting behavior of the measures $\{μ_n^τ\}_{n=1}^\infty$ on $S(\mathbb{N},\mathbb{N}^*)$.

math.PR↗

arXiv2018

Optimizing the Drift in a Diffusive Search for a Random Stationary Target

Let $a\in\mathbb{R}$ denote an unknown stationary target with a known distribution $μ\in\mathcal{P(\mathbb{R}})$, the space of probability measures on $\mathbb{R}$. A diffusive searcher $X(\cdot)$ sets out from the origin to locate the target. The time to locate the target is $T_a=\inf\{t\ge0: X(t)=a\}$. The searcher has a given constant diffusion rate $D>0$, but its drift $b$ can be set by the search designer from a natural admissible class $\mathcal{D}_μ$ of drifts. Thus, the diffusive searcher is a Markov process generated by the operator $L=\frac D2\frac{d^2}{dx^2}+b(x)\frac d{dx}$. % equivalently, $X(\cdot)$ satisfies the stochastic differential equation %$X(t)=W(t)+\int_0^tb(X(s))ds$, where $W(\cdot)$ is a standard Brownian motion. For a given drift $b$, the expected time of the search is \begin{equation} \int_{\mathbb{R}} (E^{(b)}_0T_a)\thinspaceμ(da). \end{equation} Our aim is to minimize this expected search time over all admissible drifts $b\in\mathcal{D}_μ$. For measures $μ$ that satisfy a certain balance condition between their restriction to the positive axis and their restriction to the negative axis, a condition satisfied, in particular, by all symmetric measures, we can give a complete answer to the problem. We calculate the above infimum explicitly, we classify the measures for which the infimum is attained, and in the case that it is attained, we calculate the minimizing drift explicitly. For measures that do not satisfy the balance condition, we obtain partial results.

math.PR↗

arXiv2017

A Natural Probabilistic Model on the Integers and its Relation to Dickman-Type Distributions and Buchstab's Function

Let $\{p_j\}_{j=1}^\infty$ denote the set of prime numbers in increasing order, let $Ω_N\subset \mathbb{N}$ denote the set of positive integers with no prime factor larger than $p_N$ and let $P_N$ denote the probability measure on $Ω_N$ which gives to each $n\inΩ_N$ a probability proportional to $\frac1n$. This measure is in fact the distribution of the random integer $I_N\inΩ_N$ defined by $I_N=\prod_{j=1}^Np_j^{X_{p_j}}$, where $\{X_{p_j}\}_{j=1}^\infty$ are independent random variables and $X_{p_j}$ is distributed as Geom$(1-\frac1{p_j})$. We show that $\frac{\log n}{\log N}$ under $P_N$ converges weakly to the Dickman distribution. Let $D_{\text{nat}}(A)$ denote the natural density of $A\subset\mathbb{N}$, if it exists, and let $D_{\text{log-indep}}(A)=\lim_{N\to\infty}P_N(A\capΩ_N)$ denote the density of $A$ arising from $\{P_N\}_{N=1}^\infty$, if it exists. We show that the two densities coincide on a natural algebra of subsets of $\mathbb{N}$. We also show that they do not agree on the sets of $n^\frac1s$-\it smooth numbers \rm\ $\{n\in\mathbb{N}: p^+(n)\le n^\frac1s\}$, $s>1$, where $p^+(n)$ is the largest prime divisor of $n$. This last consideration concerns distributions involving the Dickman function. We also consider the sets of $n^\frac1s$-\it rough numbers \rm\ $\{n\in\mathbb{N}:p^-(n)\ge n^{\frac1s}\}$, $s>1$, where $p^-(n)$ is the smallest prime divisor of $n$. We show that the probabilities of these sets, under the uniform distribution on $[N]=\{1,\ldots, N\}$ and under the $P_N$-distribution on $Ω_N$, have the same asymptotic decay profile as functions of $s$, although their rates are necessarily different. This profile involves the Buchstab function. We also prove a new representation for the Buchstab function.

math.PR↗

arXiv2017

Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets

We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.

math.PR↗

arXiv2017

On the strange domain of attraction to generalized Dickman distributions for sums of independent random variables

Let $\{B_k\}_{k=1}^\infty, \{X_k\}_{k=1}^\infty$ all be independent random variables. Assume that $\{B_k\}_{k=1}^\infty$ are $\{0,1\}$-valued Bernoulli random variables satisfying $B_k\stackrel{\text{dist}}{=}\text{Ber}(p_k)$, with $\sum_{k=1}^\infty p_k=\infty$, and assume that $\{X_k\}_{k=1}^\infty$ satisfy: $X_k>0,\ \ \ μ_k\equiv EX_k<\infty, \ \ \ \lim_{k\to\infty}\frac{X_k}{μ_k}\stackrel{\text{dist}}{=}1$. Let $M_n=\sum_{k=1}^np_kμ_k$, assume that $M_n\to\infty$ and define the normalized sum of independent random variables $W_n=\frac1{M_n}\sum_{k=1}^nB_kX_k$. We give a general condition under which $W_n\stackrel{\text{dist}}{\to}c$, for some $c\in[0,1]$, and a general condition under which $W_n$ converges in distribution to a generalized Dickman distribution GD$(θ)$. In particular, we obtain the following concrete results, which reveal a strange domain of attraction to generalized Dickman distributions. Let $J_μ,J_p$ be nonnegative integers, let $c_μ,c_p>0$ and let $$ \begin{aligned} &μ_n\sim c_μn^{a_0}\prod_{j=1}^{J_μ}(\log^{(j)}n)^{a_j}, &p_n\sim c_p\big({n^{b_0}\prod_{j=1}^{J_p}(\log^{(j)}n)^{b_j}}\big)^{-1}, \ b_{J_p}\neq0. \end{aligned} $$ If $$ \begin{aligned} &i.\ J_p\le J_μ; &ii.\ b_j=1, \ 0\le j\le J_p; &iii.\ a_j=0, \ 0\le j\le J_p-1,\ \text{and}\ \ a_{J_p}>0, \end{aligned} $$ then $ \lim_{n\to\infty}W_n\stackrel{\text{dist}}{=}\frac1θ\text{GD}(θ),\ \text{where}\ θ=\frac{c_p}{a_{J_p}}. $ Otherwise, $\lim_{n\to\infty}W_n\stackrel{\text{dist}}{=}c$, for some $c\in[0,1]$. We also give an application to the statistics of the number of inversions in certain shuffling schemes.

math.PR↗

arXiv2016

Transience/Recurrence and Growth Rates for Diffusion Processes in Time-Dependent Domains

Let $\mathcal{K}\subset R^d$, $d\ge2$, be a smooth, bounded domain satisfying $0\in\mathcal{K}$, and let $f(t),\ t\ge0$, be a smooth, continuous, nondecreasing function satisfying $f(0)>1$. Define $D_t=f(t)\mathcal{K}\subset R^d$. Consider a diffusion process corresponding to the generator $\frac12Δ+b(x)\nabla$ in the time-dependent domain $D_t$ with normal reflection at the time-dependent boundary. Consider also the one-dimensional diffusion process corresponding to the generator $\frac12\frac{d^2}{dx^2}+B(x)\frac d{dx}$ on the time-dependent domain $(1,f(t))$ with reflection at the boundary. We give precise conditions for transience/recurrence of the one-dimensional process in terms of the growth rates of $B(x)$ and $f(t)$. In the recurrent case, we also investigate positive recurrence, and in the transient case, we also consider the asymptotic growth rate of the process. Using the one-dimensional results, we give conditions for transience/recurrence of the multi-dimensional process in terms of the growth rates of $B^+(r)$, $B^-(r)$ and $f(t)$, where $B^+(r)=\max_{|x|=r}b(x)\cdot\frac x{|x|}$ and $B^-(r)=\min_{|x|=r}b(x)\cdot\frac x{|x|}$.

math.PR↗

arXiv2015

Transience, Recurrence and the Speed of a Random Walk in a Site-Based Feedback Environment

We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required to switch it to the $p$-mode, and $L$ consecutive left jumps from a site in the $p$-mode are required to switch it to the $q$-mode. From a site in the $p$-mode the walk jumps right with probability $p$ and left with probability $1-p$, while from a site in the $q$-mode these probabilities are $q$ and $1-q$. We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $α= α(p,q,R,L)$. For $α> 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $α< 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $α= 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. Nevertheless, we are able to give a characterization of transience/recurrence in many instances, including when either $R=1$ or $L=1$ and when $R=L=2$. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed.

math.PR↗

arXiv2014

The Behavior of the Free Boundary for Reaction-Diffusion Equations with Convection in an Exterior Domain with Neumann or Dirichlet Boundary Condition

Let \begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} \end{equation*} be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, \begin{equation*} \begin{aligned} &Lu=Λu^p\ \text{in}\ \mathbb{R}^d-D;\\ &\nabla u\cdot \bar n=-h\ \text{on}\ \partial D;\\ &u\ge0 \ \text{is minimal}, \end{aligned} \end{equation*} where $p\in(0,1)$, $d\ge2$, $h$ and $Λ$ are continuous positive functions, $D\subset R^d$ is bounded, and $\bar n$ is the unit inward normal to the domain $\mathbb{R}^d-\bar D$. Consider also the same equations with the Neumann boundary condition replaced by the Dirichlet boundary condition; namely, $u=h$ on $\partial D$. The solutions to the above equations may possess a free boundary. When $D=\{|x| R:u(r)=0\}$. We normalize the diffusion coefficient to be on unit order, consider the convection vector field to be on order $r^m$, $m\in R$, pointing either inward $(-)$ or outward $(+)$, and consider the reaction coefficient $Λ$ to be on order $r^{-j}$, $j\in R$. For both the Neumann boundary case and the Dirichlet boundary case, we show for which choices of $m$, $(\pm)$ and $j$ a free boundary exists, and when it exists, we obtain its growth rate in $h$ as a function of $m$, $(\pm)$ and $j$. These results are then used to study the free boundary in the non-radially symmetric case.

math.AP↗