Simulation comments

Writing a simulation of a random walk is a fairly trivial task if you have any pro- gramming experience, but writing a program which can simulate millions of steps with thousands of repetitions whilst not making use of a supercomputer becomes far from trivial, especially if you need to calculate the convex hull of these long walks. Whilst

1.4. Examples and simulation comments 35

many of the plots in this thesis are only short simulations (for presentational purposes), the more demanding programs required some tricks to make the simulations feasible. Firstly, we aimed to use a cluster of computer cores to run parts of the code in parallel. By splitting the code up in this way we reduce the time to simulate all the steps but ‘gluing together’ thousands of partial walks is not feasible because the memory required to store all of the individual steps is too great for a normal computer to process, never mind calculating the convex hull of such a walk. However, we can use what we know about convex hulls and convexity to help us.

If we calculate the convex hull of each individual subsection of the walk, and remember the start and endpoints of the walk, we only need to store the vertices of the walk and these two points. In the convex hull of an n step walk there are around log n vertices (more on this later), so the memory required is much more manageable. Moreover, after gluing together all the convex hulls of the subsections, we can calculate the convex hulls of all the vertices which produces the convex hull of the whole walk. All that remains is to balance out the length of the subsections with the number of subsections, in turn balancing out the time increased by calculating each subsection’s convex hull and the time it takes to combine all the individual parts afterwards, with the memory cost of making each individual subsection too long.

As a final comment, we note that we also calculate the diameter by calculating the maximum distance between any two of the vertices in the convex hull. By convexity this will find the two points of the underlying walk which attain the diameter. Of course finding the maximum distance between any two points from a set of size log n is much faster than between points from a set of size n.

Chapter 2

Laws of large numbers and

extensions using classical results

Our first exploration of the convex hull starts by considering the laws of large numbers for the perimeter length and diameter functionals in dimension 2. As discussed in the introduction, there are several results already in the literature, in particular relating to the perimeter length. Most of the results in this section can be heuristically justified by the idea that the walk with drift converges to a line segment under the law of large numbers scaling, and the walk without drift degenerates to a point under the same scaling. For now, we mention this only to explain the intuition behind the results, but this idea is more formally explored in Chapter 3.

The Spitzer and Widom formula (1.1.2) was used by Snyder and Steele to establish the law of large numbers for the perimeter length as described at (1.1.3). Their result requires the condition E(kZk2) < ∞ and was stated for the case µ 6= 0, but their proof works equally well when µ = 0. Our first contribution in this section is to provide a different proof for this result which removes the need for the second moment to be finite. With a few basic observations and an application of Pratt’s lemma, we can extend this to a law of large numbers for the diameter. Despite this extension being relatively simple, and could have been established from Snyder and Steele’s law of large numbers for the perimeter length, albeit with stronger assumptions, it does not seem to have appeared explicitly in the literature. These two laws actually give some justification to our heuristic about the shape of the convex hull by considering the ratio

2.1. Laws of large numbers 37

of the two quantities.

In the case of drift we present some further results. First, we establish the second order term of the asymptotic expansion of E Ln, and then use this result to recast one of the

second order results of Wade and Xu [WX15a] in a stronger form. The expansion of E Ln can be compared with the expansions found by Grebenkov, Lanoiselée and

Majumdar [GLM17], see Section 1.1.3 for details.

Finally, we provide an inequality for the same expansion for the diameter. The exact asymptotic result does not follow from the methods we employ here, and remains, as far as we know, an open problem. The second order results are known, in fact we prove them in Chapter 5; we dedicate a chapter to these results because they require a lengthier proof along the lines of the method Wade and Xu used to establish the perimeter length results.

In this section, we do not use any specific methods to obtain the results, we just make use of some classical probability theory, Cauchy’s formula and some other geometrical facts1_.

2.1

Laws of large numbers

Throughout this chapter, we consider the walk with the notation as described at (W2 µ).

Then our first result is the following law of large numbers for Ln. Theorem 2.1.1. Suppose that E kZk < ∞. Then

lim

n→∞n

−1

Ln = 2kµk, a.s. and in L1.

On the other hand, if E kZk = ∞ then lim supn→∞n−1Ln= ∞, a.s.

Remark 2.1.2. It is a natural question to ask whether, when E kZk = ∞, does it in

fact hold that limn→∞n−1Ln = ∞? We note that the proof employed here does not

directly answer this question, and yet neither have we found a counter example to this statement, so it remains an open problem.

2.1. Laws of large numbers 38

Similarly, we have a law of large numbers for Dn. Theorem 2.1.3. Suppose that E kZk < ∞. Then

lim

n→∞n

−1

Dn= kµk, a.s. and in L1.

On the other hand, if E kZk = ∞ then lim supn→∞n −1_D

n= ∞, a.s.

In the case µ 6= 0, Theorems 2.1.1 and 2.1.3 have the following immediate consequence.

Corollary 2.1.4. Suppose that E kZk < ∞ and that µ 6= 0. Then

lim

n→∞Ln/Dn= 2, a.s.

Before we start on the proofs, we recall that Cauchy’s formula, equation (1.1.1), can be stated in the following form (see e.g. equation (2.1) of [SS93]), for a finite point set {x0, x1, . . . , xn} ⊂ R2, the perimeter length of hull{x0, x1, . . . , xn} is given by

Z 2π

0

max

0≤k≤n(xk· eθ)dθ.

Proof of Theorem 2.1.1. Cauchy’s formula applied to our random walk implies that Ln =

Z 2π

0

max

0≤k≤n(Sk· eθ)dθ. (2.1.1)

First suppose that E kZk < ∞. Then the strong law of large numbers says that for any ε > 0 there exists Nε with P(Nε < ∞) = 1 for which

kSn− nµk < nε, for all n ≥ Nε. (2.1.2)

Since S0 = 0, taking k = 0 and k = n in (2.1.1) and writing x+ := x1{x > 0}, we have

Ln≥

Z 2π

0

(Sn· eθ)+dθ = 2kSnk, (2.1.3)

by Cauchy’s formula for hull{0, Sn}. For n ≥ Nε we have from (2.1.2) that

kSnk ≥ knµk − kSn− nµk ≥ nkµk − nε.

2.1. Laws of large numbers 39

On the other hand, for any ε > 0, we have from (2.1.2) that max 0≤k≤n(Sk· eθ) ≤ max0≤k≤Nε (Sk· eθ) + max Nε≤k≤n (Sk· eθ) ≤ max 0≤k≤Nε kSkk + max 0≤k≤n(k(µ · eθ+ ε)) = max 0≤k≤Nε kSkk + n(µ · eθ+ ε)+. Let Aε:= {θ ∈ [0, 2π] : µ · eθ > −ε}. Then Z 2π 0 (µ · eθ+ ε)+dθ = Z Aε (µ · eθ+ ε)dθ ≤ Z Aε µ · eθdθ + 2πε. But Z Aε µ · eθdθ = Z A0 µ · eθdθ + Z Aε\A0 µ · eθdθ ≤ Z 2π 0 (µ · eθ)+dθ + kµk|Aε\ A0|.

Hence, from (2.1.1) we obtain

Ln ≤ 2π max 0≤k≤Nε kSkk + n Z 2π 0 (µ · eθ)+dθ + 2πnε + nkµk|Aε\ A0|.

Since P(Nε< ∞) = 1, it follows from Cauchy’s formula for hull{0, µ} that, a.s.,

lim sup

n→∞

n−1Ln ≤ 2kµk + 2πε + kµk|Aε\ A0|.

Since ε > 0 was arbitrary, and |Aε\A0| → 0 as ε → 0, we get lim supn→∞n−1Ln≤ 2kµk,

a.s. Thus the almost sure convergence statement is established.

Moreover, from (2.1.1), Ln≤ Z 2π 0 max 0≤k≤nkSkkdθ ≤ 2π max 0≤k≤n k X j=1 kZjk ≤ 2π n X j=1 kZjk.

The strong law shows that, n−1Pn

j=1kZjk a.s.

−→ E kZk < ∞, while E(n−1Pn

j=1kZjk) =