We study an energy-constrained random walker on a length-N interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capacity of M on each boundary visit. We establish large N, M distributional asymptotics for the lifetime of the walker, i.e., the first time at which the walker runs out of energy while in the interior. When energy is scarce, there is limit related to a Darling-Mandelbrot law, while when energy is plentiful there is an exponential limit distribution on a suitable scale. We discuss some motivation from random walk models in ecology.

This talk is based on joint work with Michael Grinfeld (Strathclyde).

Consider a random walk whose increments have finite variance, run for a finite number of steps. The convex hull of the trajectory is the (random) minimal convex polytope that encloses all the points visited by the walk. We present some results on laws of the iterated logarithm for intrinsic volumes of the convex hull, in the case where the random walk has a non-zero drift. Analogous results in the case of zero drift (where the scaling is different) were obtained by Khoshnevisan. Our proofs, like Khoshnevisan's, are based on Strassen's functional law of the iterated logarithm. For the special case of the area of the planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the Dido problem.

This talk is based on joint work with Wojciech Cygan, Nikola Sandrić, and Stjepan Šebek.

I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form.

This talk is based on joint work with Nicholas Georgiou (Durham).

Consider a random walk whose increments have finite variance, run for a finite number of steps. The convex hull of the trajectory is the (random) minimal convex polytope that encloses all the points visited by the walk. We present some results on laws of the iterated logarithm for intrinsic volumes of the convex hull, in the case where the random walk has a non-zero drift. Analogous results in the case of zero drift (where the scaling is different) were obtained by Khoshnevisan. Our proofs, like Khoshnevisan's, are based on Strassen's functional law of the iterated logarithm. For the special case of the area of the planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the Dido problem.

This talk is based on joint work with Wojciech Cygan, Nikola Sandrić, and Stjepan Šebek.

I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form.

This talk is based on joint work with Nicholas Georgiou (Durham).

Consider a planar random walk whose increments have finite variance, run for a finite number of steps. The convex hull of the trajectory is the (random) convex polygon of minimal area that encloses all the points visited by the walk. Associated to it are random variables such as its perimeter length, area, and diameter. We present an overview of some results on the large-time distributional limit theory of these random variables. There is different behaviour depending on whether the walk has a non-zero or zero mean increment. In most cases, the limit distribution is non-Gaussian; in one case, it is Gaussian. The methods use some scaling arguments and, in the anomalous case, some martingale ideas.

The results I will present are mostly from joint work with James McRedmond and Chang Xu.

For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. For example, in the case of a planar domain, explosion occurs if and only if the area of the domain is finite.

This talk is based on joint work with Mikhail Menshikov and Aleksandar Mijatović.

In this talk I will give an overview of a class of random graph models constructed on random points in space, with edges added according to a rule based on proximity, including the "on-line" set-up where vertices are added sequentially and the candidate neighbour of a new vertex is chosen from among its predecessors. The examples I will focus on are the so-called minimal directed spanning tree (introduced by Bhatt and Roy) and the on-line nearest neighbour graph (which goes back at least to Steele), in both of which points are connected to nearest neighbours in Euclidean space, with some constraints. The results I will present cover the large-sample asymptotics for the total edge length of these graphs. The main feature of these results is that the limit distribution undergoes a dimension-dependent phase transition between normal (as is common under local dependence) and non-normal limits, with the non-normal component arising due to the presence of unusually long edges.

I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form.

This talk is based on joint work with Nicholas Georgiou (Durham).

I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form.

This talk is based on joint work with Nicholas Georgiou (Durham).

I will talk about an interacting particle model motivated by nanoscale growth of ultra-thin films. Particles are deposited (according to a space-time Poisson process) on an interval substrate and perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We are interested in the induced interval-splitting process. In particular, we show that the long-time evolution converges to a Markovian interval-splitting process, which we describe. The density that appears in this description is derived from an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson. The splitting density has a compact Fourier series expansion but, apparently, no simple closed form.

This talk is based on joint work with Nicholas Georgiou (Durham).

Consider a random walk in $d$-dimensional space which remembers its most recent $k$ steps, and at each step takes a jump distributed uniformly on a unit ball, excluding the convex hull of the origin and those $k$ most recent sites. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic (i.e., to have a limiting speed and a limiting direction). We establish ballisticity for the finite-memory case, and comment on some open problems. This is joint work with Francis Comets (Paris VII) and Mikhail Menshikov (Durham).

We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2 = a^+ x_1^{\beta^+}$ and $x_2 = -a^- x_1^{\beta^-}$, with $x_1 \geq 0$. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle $\alpha^+$ or $\alpha^-$ to the relevant inwards-pointing normal vector. Here we focus on the case where $\alpha^+$ and $\alpha^-$ are equal but opposite, which includes the case of normal reflection. For $0 \leq \beta^+, \beta^- < 1$, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.

This is joint work with Mikhail Menshikov (Durham) and Aleksandar Mijatović (Warwick).

Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model popular in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson.

This is joint work with Nicholas Georgiou (Durham).

Consider a random walk in $d$-dimensional space which remembers its most recent $k$ steps, and at each step takes a jump distributed uniformly on a unit ball, excluding the convex hull of the origin and those $k$ most recent sites. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic (i.e., to have a limiting speed and a limiting direction). We establish ballisticity for the finite-memory case, and comment on some open problems. This is joint work with Francis Comets (Paris VII) and Mikhail Menshikov (Durham).

Subadditivity is a simple property of the growth of sequences of real numbers which can be used to deduce existence of various limits. We will look at some simple applications of subadditivity in various contexts, and then turn to its role in some probability and counting problems, such as bin packing and self-avoiding walk. Two beautiful applications of subadditivity ideas in probability are the problem of long increasing subsequences of a random permutation (where much progress has been made recently) and the random travelling salesman problem, concerning the length of the shortest path that visits a large number of randomly distributed cities, where some of the main problems still resist mathematical analysis.

Consider a random walk in $d$-dimensional space which remembers its most recent $k$ steps, and at each step takes a jump distributed uniformly on a unit ball, excluding the convex hull of the origin and those $k$ most recent sites. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic (i.e., to have a limiting speed and a limiting direction). We establish ballisticity for the finite-memory case, and comment on some open problems. This is joint work with Francis Comets (Paris VII) and Mikhail Menshikov (Durham).

The Efron–Stein inequality is a relatively simple way of bounding the variance of a function of independent random variables that is flexible enough to cover many applications and sharp enough to give reasonable results in many cases. We take a tour through a number of applications, including kernel density estimation, the random travelling salesman problem, and eigenvalues of random symmetric matrices. We mention the connection to Chernoff's inequality for functions of Gaussian random variables, and give a proof of Steele's version of the Efron–Stein inequality based on martingale differences. The original work of Efron and Stein arose in the context of jackknife estimation, and we touch on that as well.

Consider a random walk in $d$-dimensional space which remembers its most recent $k$ steps, and at each step takes a jump distributed uniformly on a unit ball, excluding the convex hull of the origin and those $k$ most recent sites. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic (i.e., to have a limiting speed and a limiting direction). We establish ballisticity for the finite-memory case, and comment on some open problems. This is joint work with Francis Comets (Paris VII) and Mikhail Menshikov (Durham).

Consider a random walk in $d$-dimensional space which remembers its most recent $k$ steps, and at each step takes a jump distributed uniformly on a unit ball, excluding the convex hull of the origin and those $k$ most recent sites. For $k = \infty$ this is a version of the model introduced by Angel et al., which is conjectured to be ballistic (i.e., to have a limiting speed and a limiting direction). We establish ballisticity for the finite-memory case, and comment on some open problems. This is joint work with Francis Comets (Paris VII) and Mikhail Menshikov (Durham).

Each site of the one-dimensional integer lattice hosts a queue with arrival rate $\lambda$. A single server, starting at the origin, serves its current queue at rate $\mu$ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case $\lambda = \mu$, we show that the server returns to every site infinitely often. We also give an iterated logarithm result for the server's position. In the talk I will try to explain the main ingredients in the analysis: (i) the times between successive queues being emptied exhibit doubly exponential growth, (ii) the probability that the server changes its direction is asymptotically equal to 1/4, and (iii) a martingale construction that facilitates the proofs. This is joint work with James Cruise (Heriot-Watt).

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area? I will describe recent work on the convex hull of planar random walk, concerned in particular with the large-$n$ asymptotics of its perimeter length and area. We provide variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.

This talk is mostly based on joint work with Chang Xu (Strathclyde); I'll also mention ongoing work with Ostap Hryniv and James McRedmond (Durham).

Each site of the one-dimensional integer lattice hosts a queue with arrival rate $\lambda$. A single server, starting at the origin, serves its current queue at rate $\mu$ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case $\lambda = \mu$, we show that the server returns to every site infinitely often. We also give an iterated logarithm result for the server's position. In the talk I will try to explain the main ingredients in the analysis: (i) the times between successive queues being emptied exhibit doubly exponential growth, (ii) the probability that the server changes its direction is asymptotically equal to 1/4, and (iii) a martingale construction that facilitates the proofs. This is joint work with James Cruise (Heriot-Watt).

These lectures are about the Foster–Lyapunov or semimartingale method for studying the asymptotic behaviour of near-critical stochastic systems. The basic idea of the method is exhibiting a function of the underlying process with a one-dimensional image which satisfies locally a drift condition, which can be used to conclude about e.g. recurrence, transience, or positive-recurrence of the process. If the process is near-critical in the sense of being near some phase boundary in asymptotic behaviour, then the one-dimensional process arising from a suitable Lyapunov function is typically near-critical as well. The prototypical family of near-critical one-dimensional stochastic processes are processes with asymptotically-zero drift, studied in a seminal series of papers by Lamperti. Analysis of these one-dimensional process allows one to study, via the method of Lyapunov functions, asymptotic behaviour of many-dimensional Markov processes. The Lyapunov function method also enables one to study continuous-time Markov chains, and random walks with heavy-tailed increments. These lectures will lead a tour encompassing some of the key aspects of the above topics, and are based on the recently published book "Non-Homogeneous Random Walks" by Menshikov, Popov, and Wade, Cambridge University Press, 2016.

Spatially homogeneous random walks (i.e., partial sums of i.i.d. random vectors) are well understood. The most delicate regime is when the walk has zero drift, where (under mild conditions) the walk is recurrent in dimensions 1 or 2 but transient in dimension 3 or more. If spatial homogeneity is relaxed, very different behaviour can be observed: zero-drift random walks that are recurrent in 3 dimensions, or transient in 2 dimensions, for example. To probe precisely the recurrence-transience phase transition it is natural to study the asymptotically-zero drift regime (in analogy to classical one-dimensional work of Lamperti). I will survey some results on recurrence behaviour and angular asymptotics for this class of spatially non-homogeneous random walks, including joint work with Nicholas Georgiou, Iain MacPhee, Mikhail Menshikov, and Aleksandar Mijatović.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area? I will describe recent work on the convex hull of planar random walk, concerned in particular with the large-$n$ asymptotics of its perimeter length and area. We provide variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.

This talk is mostly based on joint work with Chang Xu (Strathclyde); I'll also mention ongoing work with Ostap Hryniv and James McRedmond (Durham).

We consider a class of spatially non-homogeneous random walks in multidimensional Euclidean space with zero drift, which in any dimension (two or higher) can be recurrent or transient depending on the details of the walk. These walks satisfy an invariance principle, and have as their scaling limits a class of martingale diffusions, with law determined uniquely by an SDE with discontinuous coefficients at the origin. Furthermore, pathwise uniqueness of this SDE may fail. The radial coordinate of the diffusion is a Bessel process of dimension greater than 1. Unique characterization of the law of the diffusion, which must start at the origin, is natural via excursions built around the Bessel process; each excursion has a generalized skew-product-type structure, in which the angular component spins at infinite speed at the start and finish of each excursion. Defining appropriately the Riemannian metric $g$ on the sphere $S$ allows us to give an explicit construction of the angular component (and hence of the entire skew-product decomposition) as a time-changed Brownian motion with drift on the Riemannian manifold $(S,g)$. In particular, this provides a multidimensional generalisation of the Pitman–Yor representation of the excursions of Bessel process with dimension between one and two. Furthermore, the density of the stationary law of the angular component with respect to the volume element of $g$ can be characterised by a linear PDE involving the Laplace–Beltrami operator and the divergence under the metric $g$.

This is joint work with Nicholas Georgiou and Aleksandar Mijatović.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area? I will describe recent work on the convex hull of planar random walk, concerned in particular with the large-$n$ asymptotics of its perimeter length and area. We provide variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.

This talk is mostly based on joint work with Chang Xu (Strathclyde); I'll also mention ongoing work with Ostap Hryniv and James McRedmond (Durham).

I will talk about a Markov chain on a complex of half-lines joined at a common origin, which is partially homogeneous in the sense that on each half-line a given increment distribution is used. Increment distributions are of two types: one-sided (in which the jump always moves towards, and possibly over, the origin) and symmetric. In both cases the tails are polynomial with exponent in $(0,2)$. When the walker jumps over the origin, it is routed to a new half-line according to a stochastic transition matrix. We give a criterion for recurrence or transience. This model generalizes the case of two half-lines, called the `oscillating random walk', studied by Kemperman. In the two half-line case our criterion is linear in the two tail exponents; it is only in the more general case where the non-linear nature of the criterion is revealed.

This is joint work with Dimitri Petritis (Rennes) and Mikhail Menshikov (Durham).

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area? I will describe recent work with Chang Xu (Strathclyde) on the convex hull of planar random walk, concerned in particular with the large-$n$ asymptotics of its perimeter length and area. We assume finite second moments for the steps of the walk. First-order results for the perimeter length include a remarkable expectation formula due to Spitzer and Widom, and a law of large numbers due to Snyder and Steele, who also proved a variance upper bound. We complement these results by variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.

Spatially homogeneous random walks (i.e., partial sums of i.i.d. random vectors) are well understood. The most delicate regime is when the walk has zero drift, where (under mild conditions) the walk is recurrent in dimensions 1 or 2 but transient in dimension 3 or more. If spatial homogeneity is relaxed, very different behaviour can be observed: zero-drift random walks that are recurrent in 3 dimensions, or transient in 2 dimensions, for example. To probe precisely the recurrence-transience phase transition it is natural to study the asymptotically-zero drift regime (in analogy to classical one-dimensional work of Lamperti). I will survey some results on recurrence behaviour and angular asymptotics for this class of spatially non-homogeneous random walks, including joint work with Nicholas Georgiou, Iain MacPhee, Mikhail Menshikov, and Aleksandar Mijatović.

Vertices arrive one at a time at random locations in the unit cube and are joined to existing vertices at random according to a rule that combines preference according to current degree with preference according to spatial proximity. We investigate phase transitions in the structure of the resulting graph as the relative weighting of these two components of the attachment rule is varied. This is joint work with Jonathan Jordan (Sheffield).

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area?

I will describe recent work with Chang Xu (Strathclyde) on the convex hull of planar random walk, concerned in particular with the large-$n$ asymptotics of its perimeter length and area. We assume finite second moments for the steps of the walk.

First-order results for the perimeter length include a remarkable expectation formula due to Spitzer and Widom, and a law of large numbers due to Snyder and Steele, who also proved a variance upper bound.

We complement these results by variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? What is its area?

I will describe recent work with Chang Xu (Strathclyde) on the convex hull of planar random walk, concerned in particular with the large-$n$ asymptotics of its perimeter length and area. We assume finite second moments for the steps of the walk.

First-order results for the perimeter length include a remarkable expectation formula due to Spitzer and Widom, and a law of large numbers due to Snyder and Steele, who also proved a variance upper bound.

We complement these results by variance asymptotics and distributional limit theorems. Of the four combinations of the two quantities (perimeter and area) in the two regimes (zero drift or non-zero drift for the steps of the walk), one limit is Gaussian; three are not.

Let $M$ be a random $m$ by $n$ matrix with 0, 1 entries and i.i.d. rows, which follow a specified distribution on their weight (number of ones). We study the number of left null vectors of $M$ with addition mod 2, as $n$ tends to infinity and $m/n$ tends to a given aspect ratio $\alpha$, while the weight distribution converges weakly to a limit distribution. We describe the asymptotics of the expected number of null vectors in terms of analytic properties of the limiting weight distribution. This random matrix model has other interpretations, including a random hypergraph model (where null vectors correspond to hypercycles), randomized XORSAT, and random walk on a generalized hypercube. Most of the existing literature considers the case where the limiting weight distribution is degenerate, i.e., constant. This is joint work with Richard Darling, Mathew Penrose, and Sandy Zabell.

We consider a class of spatially non-homogeneous random walks in multidimensional Euclidean space with zero drift, which in any dimension (two or higher) can be recurrent or transient depending on the details of the walk. These walks satisfy an invariance principle, and have as their scaling limits a class of zero-drift diffusions, with law determined uniquely by an SDE with discontinuous coefficients at the origin. The radial coordinate of the diffusion is a Bessel process of dimension greater than 1 (this component of the invariance principle is related to a theorem of Lamperti). Unique characterization in law of the diffusion, which must start at the origin, is natural via excursions built around the Bessel process; each excursion has a generalized skew-product-type structure, in which the angular component is a diffusion on the sphere time-changed according to a functional of the radial component, and, in general, also driven dependently (in the usual skew-product, the dependence is mediated entirely by the time-change).

This is joint work with Nicholas Georgiou, Mikhail Menshikov, and Aleksandar Mijatović.

Vertices arrive one at a time at random locations in the unit cube and are joined to existing vertices at random according to a rule that combines preference according to current degree with preference according to spatial proximity. We investigate phase transitions in the structure of the resulting graph as the relative weighting of these two components of the attachment rule is varied.

This is joint work with Jonathan Jordan (Sheffield).

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? Denote by $L_n$ the perimeter length of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment. Snyder and Steele showed that $L_n/n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $\text{Var} (L_n) = O(n)$. Further study separates into two cases: (i) zero drift, in which the Brownian scaling limit of the walk entails a scaling limit for the convex hull, once one has set things up correctly; (ii) non-zero drift. I will describe recent work with Chang Xu (Strathclyde) on these problems. Our main result is in the case of non-zero drift, where we show that $n^{-1} \text{Var} (L_n)$ converges, and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $L_n/n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $\text{Var}\ L_n = O(n)$. I will describe recent work with Chang Xu (Strathclyde) in which we show that $n^{-1} \text{Var} L_n$ converges, and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $L_n/n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $\text{Var}\ L_n = O(n)$. I will describe recent work with Chang Xu (Strathclyde) in which we show that $n^{-1} \text{Var} L_n$ converges, and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.

I will describe first some of the general background on non-homogeneous random walks, that is, random walks in d-dimensional space in which the jump distribution can vary according to the spatial position of the walker. Then I will focus on the one-dimensional case. The critical case from the point of view of recurrence/transience is the "Lamperti" case in which the mean drift decays in inverse proportion to the distance from the origin. I will discuss some recent work with Ostap Hryniv and Mikhail Menshikov (Durham) on the path properties of these random walks, including scaling and tail properties of maxima, passage times, and path integrals.

On each of $n$ unsteady steps, a drunken gardener drops a seed. Once the flowers have bloomed, what is the minimum length of fencing required to enclose the garden? Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $L_n/n$ converges almost surely to a deterministic limit, and proved an upper bound on the variance $\text{Var}\ L_n = O(n)$. I will describe recent work with Chang Xu (Strathclyde) in which we show that $n^{-1} \text{Var}\ L_n$ converges, and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for $L_n$ in the non-degenerate case.

Markov processes are stochastic processes for which, informally, "given the present, the future is independent of the past". This Markov property entails the fundamental Chapman-Kolmogorov relation for transition kernels. Viewing transition kernels as operators on an appropriate function space, the Chapman-Kolmogorov relation turns into the semigroup property for operators. In this talk I will review some basic ideas from Markov process theory and give an introduction to aspects of the semigroup approach, including the appropriate conditions under which it is possible to define the infinitesimal generator for the process, and the role played by resolvents.

We describe an attempt to model a process of deposition on an interval substrate of particles that subsequently perform random walks and interact to form barriers according to an occupation criterion. We set up a continuous-time Markov process for the model. We would like eventually to derive properties of the fragmentation of the interval induced by the barrier formation, under an appropriate scaling regime. But first we would like to understand if a nice scaling limit exists, and if so, what it is. This is very preliminary joint work with Michael Grinfeld.

I will describe recent joint work with Francis Comets, Mikhail Menshikov and Stas Volkov on a self-interacting random walk model. The model is a discrete-time d-dimensional random walk that interacts with its previous trajectory through the centre of mass of the previous points. Specifically, the increment of the walk at time n has an asymptotically small drift away from or towards the centre of mass of the first n locations. The rate at which the drift decays (as a function of the distance from the particle to the current centre of mass) governs the strength of the interaction. The most completely understood case is when the interaction is self-repelling and is strong enough that the walk is transient, with a limiting direction, and with a super-diffusive but sub-ballistic rate of escape. I will describe how analysis of this model leads to a generalization of a classical problem of Lamperti; the techniques are built on martingale ideas. Our methods give an interesting observation for two-dimensional symmetric simple random walk: the walk itself is recurrent (Pólya), the centre of mass is transient (by a result of Grill), but the process of displacements between the two is recurrent.

Percolation theory has proved a challenging and fruitful subject for mathematicians and physicists over the last 50 years or so. Percolation is still fertile ground for interplay between mathematics and physics, including for example phase transitions, renormalization ideas, and conformal field theory. The basic (bond) percolation model declares each edge of an infinite lattice "open" with probability p (otherwise it is "closed"). Fundamental questions involve the existence and properties of any "infinite cluster" formed by open edges. In this talk I will give an overview of some of the central models and results in percolation theory, including classical results of Harris and Kesten and recent work of Lawler, Schramm, Werner and Smirnov on "conformal invariance".

For this talk a random walk is a discrete-time time-homogeneous Markov process on $d$-dimensional Euclidean space. If such a random walk is spatially homogeneous, its position can be expressed as a sum of independent identically distributed random vectors, and these homogeneous random walks are well understood. The most subtle case is when the mean drift (i.e., average increment) of the walk is zero.

The assumption of spatial homogeneity, while simplifying the mathematical analysis, is not always realistic for applications. As soon as the spatial homogeneity assumption is relaxed, the situation becomes much more complicated: a non-homogeneous random walk can be transient in two dimensions, for instance.

I will give an introduction to some results on non-homogeneous random walks with asymptotically zero mean-drift, that is, the magnitude of the drift at a point tends to 0 as the distance of that point from the origin tends to infinity. It turns out that this is the natural regime in which to look for important phase transitions in asymptotic behaviour. This includes work by Lamperti in the 1960s on recurrence/transience behaviour.

I will also discuss recent joint work with Iain MacPhee and Mikhail Menshikov (Durham) concerned with angular asymptotics, i.e., exit-from-cones problems. We show that, in contrast to recurrence/transience behaviour, the angular properties of non-homogeneous random walks are remarkably well-behaved in some sense in the asymptotically zero drift regime.

For this talk a random walk is a discrete-time time-homogeneous Markov process on $d$-dimensional Euclidean space. If such a random walk is spatially homogeneous, its position can be expressed as a sum of independent identically distributed random vectors, and these homogeneous random walks are well understood. The most subtle case is when the mean drift (i.e., average increment) of the walk is zero.

The assumption of spatial homogeneity, while simplifying the mathematical analysis, is not always realistic for applications. As soon as the spatial homogeneity assumption is relaxed, the situation becomes much more complicated: a non-homogeneous random walk can be transient in two dimensions, for instance.

I will give an introduction to some results on non-homogeneous random walks with asymptotically zero mean-drift, that is, the magnitude of the drift at a point tends to 0 as the distance of that point from the origin tends to infinity. It turns out that this is the natural regime in which to look for important phase transitions in asymptotic behaviour. This includes work by Lamperti in the 1960s on recurrence/transience behaviour.

I will also discuss recent joint work with Iain MacPhee and Mikhail Menshikov (Durham) concerned with angular asymptotics, i.e., exit-from-cones problems. We show that, in contrast to recurrence/transience behaviour, the angular properties of non-homogeneous random walks are remarkably well-behaved in some sense in the asymptotically zero drift regime.

Percolation theory has proved a challenging and fruitful subject for mathematicians and physicists over the last 50 years or so. Percolation is still fertile ground for interplay between mathematics and physics, including for example phase transitions, renormalization ideas, and conformal field theory.

The basic (bond) percolation model declares each edge of an infinite lattice "open" with probability p (otherwise it is "closed"). Fundamental questions involve the existence and properties of any "infinite cluster" formed by open edges. In this talk I will give a non-technical overview of some of the central models and results in percolation theory, including classical results of Harris and Kesten and recent work of Lawler, Schramm, Werner and Smirnov on "conformal invariance".

Random walks are often used to model polymer molecules in solution. The classical selfavoiding walk model has some disadvantages. We introduce a new model that is a genuine stochastic process, in which the walk interacts with its previous path. The selfinteraction is mediated by the centre of mass of the previous trajectory. The model can be tuned to model polymers in extended or collapsed phases. In the extended phase, we present rigorous results on the scaling of the model. This is joint work with F. Comets, M.V. Menshikov, and S. Volkov.

For this talk a random walk is a discrete-time time-homogeneous Markov process on $d$-dimensional Euclidean space. If such a random walk is spatially homogeneous, its position can be expressed as a sum of independent identically distributed random vectors. Such homogeneous random walks are classical and the literature devoted to their study extensive, particularly when the state-space is the d-dimensional integer lattice. The most subtle case is when the mean drift (i.e., average increment) of the walk is zero.

The assumption of spatial homogeneity, while simplifying the mathematical analysis, is not always realistic for applications. Thus it is desirable to study non-homogeneous random walks. As soon as the spatial homogeneity assumption is relaxed, the situation becomes much more complicated. Even in the zero-drift case, a non-homogeneous random walk can behave completely differently to a zero-drift homogeneous random walk, and can be transient in two dimensions, for instance. Such potentially wild behaviour means that results for non-homogeneous random walks often have to be stated under rather restrictive conditions, and techniques from the study of homogeneous random walks are difficult to apply.

I will give an introduction to some of the known results on non-homogeneous random walks with asymptotically zero mean-drift, that is, the magnitude of the drift at a point tends to 0 as the distance of that point from the origin tends to infinity. It turns out that this is the natural regime in which to look for important phase transitions in asymptotic behaviour. This includes work by Lamperti in the 1960s on recurrence/transience behaviour.

I will also discuss recent joint work with Iain MacPhee and Mikhail Menshikov (Durham) concerned with angular asymptotics, i.e., exit-from-cones problems. We show that, in contrast to recurrence/transience behaviour, the angular properties of non-homogeneous random walks are remarkably well-behaved in some sense in the asymptotically zero drift regime.

The mathematical modelling of polymers in solution has produced some fascinating but hard problems, most notably the problem of the self-avoiding walk. The sites visited by the walk represent the locations of the monomers; the increments of the walk represent chemical bonds.

Heuristic arguments dating back to Nobel Laureate P.J. Flory in the 1940s predict the scaling behaviour of self-avoiding walk, but very little is known rigorously. The standard formulation of self-avoiding walk cannot be interpreted as a genuine stochastic process in the usual sense. It is of interest to formulate models for polymer molecules that are genuine stochastic processes. To retain the physical motivation, such processes must be self-interacting in some way, i.e., the stochastic evolution must depend upon the entire history of the process. This introduces challenges for analysis.

In this talk I will discuss a model introduced in collaboration with Francis Comets, Mikhail Menshikov, and Stas Volkov, whereby the intraction of a random walk with its previous history is mediated through the barycentre (centre of mass) of its previous trajectory.

Real-world networks often have spatial content and evolve over time by the addition of new nodes. The on-line nearest-neighbour graph is a very simple model of spatial network evolution in which nodes arrive one by one, distributed uniformly in the unit cube, and each new node is joined by an edge to its nearest predecessor. We describe recent results and open problems for the large-sample asymptotic behaviour of the total Euclidean length of the network (more generally, the total power-weighted length, which displays an interesting phase transition). Both Gaussian and non-Gaussian distributions appear as limits. Some of the results presented are joint work with Mathew Penrose (University of Bath).

Real-world networks often have spatial content and evolve over time by the addition of new nodes. The on-line nearest-neighbour graph is a very simple model of network evolution in which nodes arrive one by one, uniformly distributed in the unit cube, and each new node is joined by an edge to its nearest predecessor. We describe recent results and open problems for the large-sample asymptotic behaviour of the total length of the network (more generally, the total power-weighted length, which displays an interesting phase transition). Some of the work presented will be joint work with Mathew Penrose (University of Bath).

Motivated by ideal gas models in the low density regime, we study a randomly reflecting particle travelling at constant speed in an unbounded domain in the plane with boundary satisfying a polynomial growth condition The growth rate of the domain, together with the reflection distribution, determine the asymptotic behaviour of the process. We give results on recurrence vs. transience, and on almost-sure-bounds for the particle including the rate of escape in the transient case. The proofs exploit a surprising relationship with Lamperti's problem of a process on the half-line with asymptotically zero drift. This is joint work with Mikhail Menshikov and Marina Vachkovskaia.

The on-line nearest-neighbour graph (ONG) joins each point after the first in a sequence of points in $\mathbb{R}^d$ to its nearest predecessor. The ONG is a simple model of network growth that fits into a general scheme of graphs on partially ordered sets. In this talk I will describe some results on the total length of the ONG on random points. This is joint work with Mathew Penrose (Bath).

The minimal directed spanning forest joins each non-minimal vertex of a finite partially ordered set in space to its nearest predecessor. This can be used as a model for spatial drainage networks. We give some results on the total length of the network on random points. This is joint work with Mathew Penrose (Bath).

The on-line nearest-neighbour graph on a sequence of points in Euclidean space joins each point after the first by an edge to its nearest predecessor. This graph is one of the simplest models of spatial network evolution to capture some features of real-world spatial networks. We give some large-sample asymptotic results on the total length of the graph on random points. Some of this talk is based on joint work with Mathew Penrose.

We describe some recent joint work with Mikhail Menshikov (Durham) on random walks in asymptotically homogeneous one-dimensional random environments. Particular examples include random walk in random environment perturbed from Sinai's regime, and simple random walk with asymptotically small random perturbation. We present recurrence/transience properties for these models.

In the one-dimensional random walk in random environment in what is known as Sinai's regime, the random walker typically exhibits "logarithmic speed": after time $t$, he is roughly $(\log t)^2$ from where he started. We show one method of proving this fact, which can be generalized to other types of random environment. We give some new results from work with Mikhail Menshikov on logarithmic speeds for random walks in random environments subject to a vanishing perturbation.

In the one-dimensional random walk in random environment in what is known as Sinai's regime, the random walker typically exhibits "logarithmic" speed: after time $t$, he is roughly $(\log t)^2$ from where he started. We show one method of proving this fact, which can be generalized to other types of random environment. We give some new results from work with Mikhail Menshikov on logarithmic speeds for random walks in random environments subject to a vanishing perturbation.

Consider a random list of jump probabilities, and a random walk defined with those probabilities. This is the random walk in random environment (or RWRE for short). The properties of the RWRE are very different from those of the corresponding simple random walk. A famous result due to Solomon demonstrates the existence of a critical regime (with respect to recurrence, transience and ergodicity) in the case of an i.i.d. random environment, often known as Sinai's regime.

We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on $\mathbb{Z}^+$, with reflection at the origin, in the case where the random environment is not i.i.d., but is subject to a vanishing perturbation from Sinai's regime.

Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique–the method of Lyapunov functions.

This is joint work with Mikhail Menshikov.

In Bhatt and Roy's minimal directed spanning tree construction for $n$ random points in the unit square, all edges must be in a southwesterly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large $n$) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson–Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail. This is joint work with Mathew Penrose.

In Bhatt and Roy's minimal directed spanning tree construction for $n$ random points in the unit square, all edges must be in a southwesterly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large $n$) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed.