Large Deviations of the Range of the Planar Random Walk on the Scale of the Mean

Abstract

We prove an upper large deviation bound on the scale of the mean for a symmetric random walk in the plane satisfying certain moment conditions. This result complements the study by Phetpradap for the random walk range, which is restricted to dimension three and higher, and of van den Berg, Bolthausen and den Hollander, for the volume of the Wiener sausage.

Appendices

Appendix

1.1 Properties of the Rate Function

Various properties of the rate function $I^{g_a}$ in (1.2) were established in [30]. The results are applicable to our rate function I in (2.2) due to the fact that the function I agrees with $I^{g_a}$ up to a multiplicative constant. We only give the statement and refer the reader for the complete proof to the original paper.

Theorem 5

[30, Theorem 3, 4]

• 1. For every $b>0$, $I(b)=(4\pi )^{-1}\chi \left( b/(2\pi )\right)$, where $\chi :(0,\infty )\rightarrow [0,\infty )$ and

  $$\begin{aligned} \chi (u)=\inf \Big \{{\Vert \nabla \psi \Vert }_2^2:\psi \in H^1({\mathbb {R}}^2),\, {\Vert \psi \Vert }_2=1,\, \int _{{\mathbb {R}}^2}\left( 1-\mathrm{e}^{-\psi ^2}\right) \le u\Big \}. \end{aligned}$$

  (4.1)

• 2. $\chi$ is continuous on $(0,\infty )$ and strictly decreasing on (0, 1). Furthermore, $\chi (u)=0$ for $u\ge 1$.

• 3. $u\mapsto u\chi (u)$ is strictly decreasing on (0, 1) and

  $$\begin{aligned} \lim _{u\downarrow 0}u\chi (u)=-\lambda _2, \end{aligned}$$

  where $\lambda _2$ is the smallest Dirichlet eigenvalue of $-\varDelta$ on a ball of unit volume.

• 4. $u\mapsto (1-u)^{-1}\chi (u)$ is strictly decreasing in (0, 1) and

  $$\begin{aligned} \lim _{u\uparrow 1}(1-u)^{-1}\chi (u)=2\mu _2, \end{aligned}$$

  with

  $$\begin{aligned} \mu _2=\inf \{{\Vert \nabla \psi \Vert }_2^2:\psi \in H^1({\mathbb {R}}^2):{\Vert \psi \Vert }_2=1,\,{\Vert \psi \Vert }_4=1\}, \end{aligned}$$

  satisfying $0<\mu _2<\infty$.

• 5. For all $u\in (0,1)$, the minimization problem (4.1) has at least one minimizer. It is strictly positive, radially symmetric and strictly decreasing in the radial component. All other minimizers are of the same type.

1.2 Technical Supplement

We collect some technical results we have used in previous sections.

Lemma 3

Fix $N>0$. There exists some $\delta >0$ such that we have for $a,b\in \Lambda _N$ and $\varepsilon >0$

$$\begin{aligned} p^{\langle N \rangle }_{\varepsilon {T_\tau }}(a,b) =\frac{1}{{T_\tau }}\mathfrak {p}^{(N)}_\varepsilon (a,b)+{{\mathcal {O}} }\left( \frac{1}{{T_\tau }^{1+\delta }}\right) . \end{aligned}$$

(4.2)

Proof

For any $a,b\in \Lambda _N$, let us set $m={T_\tau }^{1/4+\delta }$ for some $\delta >0$ (on which we will put some constraints later), we decompose the probability

$$\begin{aligned} \begin{aligned} p^{\langle N \rangle }_{\varepsilon {T_\tau }}(a,b) =&\sum _{ \begin{array}{c} z\in {\mathbb {Z}}^2 \\ |z|\le m \end{array}} p_{\varepsilon {T_\tau }}\left( {a}^+,{b}^++ {z^+}N \right) + \sum _{ \begin{array}{c} z\in {\mathbb {Z}}^2 \\ |z|>m \end{array}} p_{\varepsilon {T_\tau }}\left( {a}^+,{b}^++ {z^+}N \right) \\ =&\sum _{ \begin{array}{c} z\in {\mathbb {Z}}^2 \\ |z|\le m \end{array}} P_{{a}^+}(S_{2\varepsilon {T_\tau }}=b^++z^+N) + {{\mathcal {E}} }, \end{aligned} \end{aligned}$$

with ${{\mathcal {E}} }\le P_0\left( \max _{0\le j\le \varepsilon {T_\tau }}|S_j|\ge mN{T_\tau }^{1/2}\right)$. For each summand in the first term, the transition probability of going from the point ${a}^+$ to $b^++ {z^+}N$ on ${\mathbb {Z}}^2$ can be estimated by the local central limit theorem in (3.4)

$$\begin{aligned}&p_{\varepsilon {T_\tau }}\left( {a}^+,{b}^++{z^+}N \right) \\&\quad = \frac{1}{{T_\tau }}\mathfrak {p}_\varepsilon (a,b+Nz)\left( (1+{{\mathcal {O}} }({T_\tau }^{-1/2})\right) +A_{\varepsilon {T_\tau }}(a^+, b^++ {z^+}N), \end{aligned}$$

where ${{\mathcal {O}} }$-term arises from the rounding issue. Note that $A_t(x,y)<1/t^2$ uniformly in all $t>0$. Therefore,

$$\begin{aligned} \sum _{z\in {\mathbb {Z}}^2:|z|\le m}A_{\varepsilon {T_\tau }}(a^+, b^++{z^+}N) \end{aligned}$$

is at most ${{\mathcal {O}} }({T_\tau }^{-3/2+2\delta })$.

For the second term, by the assumption that $E\left[ |X_i|^4\right] <\infty$ we can bound by [23, Equation 2.6]

$$\begin{aligned} P_0\left( \max _{0\le j\le \varepsilon {T_\tau }}|S_j|\ge mN{T_\tau }^{1/2}\right) ={{\mathcal {O}} }\left( \frac{1}{{T_\tau }^{1+4\delta }}\right) , \end{aligned}$$

as $n\rightarrow \infty$. Now, choosing $\delta$ sufficiently small yields the claim. $\square$

Lemma 4

We have that for every $K>0$ and $\varepsilon >0$

$$\begin{aligned} \sup _{n\in {\mathbb {N}}}\sup _{|x|\le K}E_{0,x^+}\left[ \exp \left( {\frac{\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }}}\right) \right] :=C_K<\infty , \end{aligned}$$

(4.3)

with

$$\begin{aligned} \limsup _{K\rightarrow \infty } C_K> 0. \end{aligned}$$

Proof

The reasoning here follows [30]. The second statement is trivial, as $C_K\ge 1$ by construction. Denote by $E_{0,y,x}$ the measure of the random walk conditioned to be at y at time $\varepsilon {T_\tau }/2$ and at x at time $\varepsilon {T_\tau }$. We then have

$$\begin{aligned} \begin{aligned}&E_{0,x^+}\left[ \exp \left( {\frac{\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }}}\right) \right] \\&\quad = \sum _{y\in {\mathbb {Z}}^d}\frac{p_{\varepsilon {T_\tau }/2}(0,y)p_{\varepsilon {T_\tau }/2}(y,x^+) }{p_{\varepsilon {T_\tau }}(0,x^+)} E_{0,y,x^+}\left[ \exp \left( {\frac{\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }}}\right) \right] \\&\quad \le \sum _{y\in {\mathbb {Z}}^d}\frac{p_{\varepsilon {T_\tau }/2}(0,y)p_{\varepsilon {T_\tau }/2}(y,x^+) }{p_{\varepsilon {T_\tau }}(0,x^+)} E_0\left[ \mathrm{e}^{{\frac{\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}}\Big | S_{\varepsilon {T_\tau }/2}=y\right] \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \times E_{0}\left[ \mathrm{e}^{{\frac{\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}}\Big | S_{\varepsilon {T_\tau }/2}=y-x^+\right] , \end{aligned} \end{aligned}$$

where in the last step we used the subadditivity of ${{\mathcal {R}} }_n$. Then, apply Cauchy–Schwarz inequality and Jensen’s inequality to get

$$\begin{aligned} \le&\frac{1}{p_{\varepsilon {T_\tau }}(0,x^+)}\sum _{y\in {\mathbb {Z}}^d}\left( p_{\varepsilon {T_\tau }/2}(0,y)E_0\left[ \exp \left( {\frac{\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}\right) \Big | S_{\varepsilon {T_\tau }/2}=y\right] \right) ^2 \\ \le&\frac{1}{ p_{\varepsilon {T_\tau }}(0,x^+)}\sum _{y\in {\mathbb {Z}}^d} p_{\varepsilon {T_\tau }/2}(0,y)^2E_0\left[ \exp \left( {\frac{2\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}\right) \Big | S_{\varepsilon {T_\tau }/2}=y\right] . \end{aligned}$$

Employing [23, Proposition 2.4.6], we can bound $p_t(x,y)\le c/t$ for some $c>0$ neither depending on $t\in {\mathbb {N}}$ nor $x,y \in {\mathbb {Z}}^2$. Applying (3.28) and (in the second step) the Markov property, we conclude that the quantity above is bounded from above by

$$\begin{aligned} \begin{aligned} \le&\frac{2c }{\varepsilon {T_\tau }p_{\varepsilon {T_\tau }}(0,x^+)}\sum _{y\in {\mathbb {Z}}^d} p_{\varepsilon {T_\tau }/2}(0,y)E_0\left[ \exp \left( {\frac{2\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}\right) \Big | S_{\varepsilon {T_\tau }/2}=y\right] \\ \le&\, \frac{2c }{\varepsilon {T_\tau }p_{\varepsilon {T_\tau }}(0,x^+)}E_0\left[ \exp \left( {\frac{2\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}\right) \right] \\ \le&\, 2c \mathrm{e}^{K^2/(2\varepsilon )}E_0\left[ \exp \left( {\frac{2\tau }{\varepsilon {T_\tau }}{{\mathcal {R}} }_{\varepsilon {T_\tau }/2}}\right) \right] . \end{aligned} \end{aligned}$$

This finishes the proof. $\square$

Notation Glossary

In this section, we list most of the notation used throughout the paper.

1.1 Spaces and Projections

We will always work with the Skorokhod space D of cadlag paths from $[0,\infty )$ onto ${\mathbb {R}}^2$. The family of coordinate projections on ${\mathbb {R}}^2$ is denoted by $(S_t)_{t\ge 0}$. Given a space E, denote the space of probability measures of E by ${{\mathcal {M}} }_1(E)$.

1.2 Domains and Scalings

In general, throughout the paper, N is a cut-off constant, compactifying ${\mathbb {R}}^2$ to the torus. Furthermore, n is the inverse lattice spacing.

• 1. $x^+=\left\lfloor x{T_\tau ^{1/2}} \right\rfloor$ and $x^-=\left\lfloor x{T_\tau ^{1/2}} \right\rfloor {T_\tau }^{-1/2}$ for $x\in {\mathbb {R}}^2$.

• 2. $\tau =\log n$, the scaling of the LDP.

• 3. ${T_\tau }=n/\log n$, the mean scaling.

• 4. $Q_\tau =\sqrt{\log {T_\tau }\log \log {T_\tau }}^{-1}$, the size of the holes during the cutting procedure.

• 5. $\Lambda _N=[-N/2,N/2)^2$, the continuum torus of length N.

• 6. $\varDelta _\tau =\Lambda _N\cap {T_\tau }^{-1/2}{\mathbb {Z}}^2$, the rescaled lattice.

• 7. $B_{R}=\{x\in {\mathbb {R}}^2:|x|\le r\}$, the ball centered at zero, also $B_R(x):=x+B_R$.

1.3 Kernels and Measures

The superscripts (N) and $\langle N\rangle$ imply that the underlying process lives on the torus. The use of mathfrak and mathbb indicates continuum objects. We occasionally use the shorthand notation $p(y-x)=p(x,y)$, if a kernel p is translation invariant. Similarly, if the sub/superscript is equal to zero, occasionally we omit it, i.e., $P=P_0$.

• 1. $P_x$ the measure of the planar random walk defined on ${\mathbb {Z}}^2$ started at x, and $p_t(x,y)$ the transition kernel from point x to point y at time t associated with $P_x$.

• 2. $P^{\langle N \rangle }_x$ the measure of the planar random walk projected into $\varDelta ^\tau _N$, and $p^{\langle N\rangle }_t(x,y)$ the transition kernel associated with $P^{\langle N \rangle }_x$.

• 3. ${\mathbb {P}}_x$ the measure of Brownian motion defined on ${\mathbb {R}}^2$, and $\mathfrak {p}_t(x,y)$ the Brownian transition kernel associated with ${\mathbb {P}}_x$.

• 4. ${\mathbb {P}}_x^{(N)}$ the measure of Brownian motion projected onto $\Lambda _N$, and $\mathfrak {p}^{(N)}_t(x,y)$ the transition kernel associated with ${\mathbb {P}}_x^{(N)}$ .

• 5. ${\mathbb {P}}_{a,b}$, ${\mathbb {P}}_{a,b}^{(N)}$, $P_{a,b}$, $P_{a,b}^{\langle N\rangle }$ the bridge measures (on the whole space and on the torus) of length $\varepsilon$ for the Brownian motion, and length $\varepsilon {T_\tau }$ for the random walks. The expectation is denoted in the same style.

• 6. Define ${\mathcal {S}}_{n,\varepsilon }=\{S_{i\varepsilon {T_\tau }}\}_{1\le i\le \tau /\varepsilon }$ the skeleton walk, then denote $P^{\langle N \rangle }_{n,\varepsilon },E^{\langle N \rangle }_{n,\varepsilon }$ the conditional law/expectation given ${\mathcal {S}}_{n,\varepsilon }$ (where $S_t$ is distributed under $P^{\langle N \rangle }_0$).

• 7. $b_{n,\varepsilon }(y,z)=P^{\langle N \rangle }_{y^-}\left( \sigma \le \varepsilon {T_\tau }\mid S_{\varepsilon {T_\tau }}=z^-\right)$, and $b_{n,\varepsilon }^{\rho }(y,z)=b_{n,\varepsilon }(y,z)\mathbb {1}\{y,z\notin B_{\rho }\}$ for $y,z\in {\mathbb {R}}^2$ with $\rho >0$ the radius of the centered ball.

• 8. $\phi _\varepsilon (y,z)=\left[ {\mathfrak {p}_{\varepsilon /2}^{(N)}(z-y)}\right] ^{-1}{\int _0^\varepsilon \mathrm{d}s \mathfrak {p}_{s/2}^{(N)}(-y)\mathfrak {p}_{(\varepsilon -s)/2}^{(N)}(z)}$ with $\phi _\varepsilon ^\rho (y,z)=\phi _\varepsilon (y,z)\mathbb {1}\{y,z\notin B_{\rho }\}$.

1.4 Stopping Times

• 1. $\sigma =\min \{n\ge 0 :S_n=0\}$ and $\sigma _r=\min \{n\ge 0:S_n\in \partial B_{N/r}\}$.

• 2. $H_z=\min \{n\ge 0:S_n=z\}$, for a $z\in {\mathbb {Z}}^2$.