# Elastic Temporal Waveguiding

Jonatha Santini  Emanuele Riva<EMAIL_ADDRESS>Department of
Mechanical Engineering, Politecnico di Milano, Milano 20156, Italy

###### Abstract

We provide a theoretical framework to mold time-modulated lattices with
frequency conversion and wave-steering capabilities. We initially focus on 1D
lattices, whereby a sufficiently slow time-modulation of the stiffness is
employed to convert the frequency content of impinging waves. Based on the
adiabatic theorem, we demonstrate that undesired reflections, which emerge in
time-discontinuous materials, can be eliminated by a careful choice of the
modulation velocity. The concept is later explored in the context of 2D
lattices, whereby a slow time modulation of the stiffness not only induces
frequency conversion without back-scattering, but also serves as a mechanism
to steer waves. Our paper explores a new and exciting way to control wave
propagation in elastodynamics with scattering-free guiding capabilities, and
may open new avenues for the manipulation and transport of information through
elastic waves.

Time modulation, waveguiding, frequency conversion, adiabatic theorem,
metamaterials, phononic crystals, scattering

††preprint: APS/123-QED

##  I Introduction

The dynamics of space, time, and space-time tessellated materials is nowadays
explored in several realms of physics to promote rich and unprecendented
phenomena not naturally observable. In the context of elastodynamics,
materials dressed with space, time, and space-time modulations are
conveniently molded to manipulate wave motion at will. Examples have shown
different forms of waveguiding that rely on defect mechanisms [1], topological
insulators [2, 3, 4, 5, 6, 7, 8, 9, 10], metagradings [11, 12, 13, 14, 15, 16,
17], transformation optics and acoustics [18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28], which are passive strategies effectively employed to accomplish
relevant technological behaviors, such as communication, focusing, and
invisibility.

To enrich the dynamical landscape of today, exotic functionalities can be
sought by accessing the temporal degree of freedom, which consists of time-
varying materials that usually require feeding additional energy into
otherwise passive systems. The activation of time-dependent interactions has
recently revealed new physics, including studies on unidirectional waves [29,
30, 31, 32, 33], adiabatic pumping [34, 35, 36, 37], frequency conversion
[38], and parametric amplification [39].

All examples above are induced either by space or time heterogeneity, whereby
each discontinuity is capable of generating wave scattering or incremental
steering. Spatial modulations obey Snell’s law, yielding a frequency-invariant
wavenumber transformation across the discontinuities. In contrast, a temporal
version of Snell’s law requires the transformation to occur at constant
wavenumber while varying the frequency across the time discontinuity [40]. The
complex interplay between consecutive space, time, and space-time modulations,
can generate scattering patterns with attenuation, amplification,
nonreciprocity, or waveguiding capabilities observable in metamaterials and
phononic systems. Even though the implementation of space-time and time-
discontinuous materials can be challenging, the efforts done in this direction
have led to experimental papers which have propelled the research done in the
field of active phononics [41, 42].

In this paper, we focus our attention on time discontinuous 1D and 2D
lattices, providing a link between the time-modulation parameters and the
frequency conversion/wave steering capabilities. Motivated by prior works on
materials with time-varying permittivity [40, 43], we initially pursue
frequency conversion in 1D elastic lattices via adiabatic stiffness-
modulation. In contrast to Ref. [40], we herein leverage smooth time-
modulations applied to a dispersive elastic medium, whereby the gradual
variation of the stiffness is key in avoiding undesired reflections during the
frequency transformation of propagating wave-modes (i.e. the states that
populate the lattice). The adiabatic theorem is employed to grasp the implied
scattering phenomena, and allows us to provide a connection between the
modulation velocity and the energy leak between counter-propagating states. In
analogy to Ref. [43], this concept is also explored in 2D time-modulated
lattices, where the frequency conversion is accompanied by a temporal
curvature of wave motion. We show how this concept can be engineered to
produce scattering-free waveguides capable of sending an elastic wave packet
between an emitter and a receiver in a controllable manner.

Our paper explores a new way to manipulate wave motion which may promote
additional efforts in the context of time-modulated materials. The procedure
employed herein is general and applicable to more complex scenario, such as
elastic waves in the continuum, in multi-physics materials, and in more
complex metamaterial systems characterized by dispersion relations with
remarkable degree of controllability.

##  II Frequency conversion in stiffness-modulated 1D lattices

(a)

(b)

Figure 1: (a) Schematic of the time-modulated spring-mass system and (b)
graphic representation of the time-modulation function, where the scattering
and frequency conversion processes are also illustrated with white curves.

We start the discussion by considering the spring-mass system displayed in
Fig. 1(a). As mentioned, scope of this section is to perform frequency
conversion in such a system without the nucleation of back-scattered waves. We
remark that time-modulated materials are known to exhibit frequency conversion
and we hereafter present a simple and viable way to pursue this scope. As
such, our lattice is made by a number $N=300$ of unitary masses $m=1$,
separated by unitary distance $d=1$, and linked by springs which embody time-
varying characteristics $k(t)$. Without any loss of generality, we focus our
attention on linear time-modulation laws, but different functions can be
employed:

$\begin{split}&k(t)=k_{i}+v(t-t_{i})\\\ \end{split}$ (1)

here, $v=\left(k_{f}-k_{i}\right)/\left(t_{f}-t_{i}\right)$ is the modulation
speed, which is a fundamental parameter for frequency conversion and allows
for controlling the scattering phenomena within the modulated material.
$k_{i}=1$ and $k_{f}=0.2$ are the initial and final values for the stiffness,
$t_{i}$ is the activation time instant for $k\left(t\right)$ and $t_{f}$ is
the final time of the modulation. The key idea is elucidated in Fig. 2(b):
when a wave impinges across a temporal discontinuity, a pair of counter-
propagating waves are generated, whose magnitude depends on the adiabaticity
of the temporal transformation and the stiffness-modulation levels. We
hereafter provide a qualitative condition that links the reflection and
transmission characteristics of the time interface to the velocity of the
modulation $v$. To that end, we follow and adapt the derivation detailed in
our prior works, and we make it applicable to handle wave propagation problems
[37, 34].

First and foremost, the procedure requires the elastodynamic equation to be
expressed by way of a first-order differential form akin to the Schrödinger
equation. We start the derivation from the second order differential form:

$m\ddot{u}_{n}+2ku_{n}-k(u_{n+1}+u_{n-1})=0$ (2)

where $u_{n}=u_{n}(t)$ describes the longitudinal motion of the $n^{th}$ mass
and, to ease notation, the time dependence of $k$ is implicitly assumed. In
the attempt to study the coupling between counter propagating wave modes, we
focus on time-dependent dispersion properties of the lattice, which obey the
unit cell dynamics with implied Floquet-Bloch conditions. Hence, Ansatz
$u_{n}$ are expressed in the complex exponential form $u_{n}=\hat{u}e^{j\mu
n}$, where $\mu=\kappa d$ is the dimensionless wavenumber. After substitution
into Eq. 2, the relevant dynamic equation takes the following convenient form:

$\begin{split}&|{\hat{\bm{\Psi}}}\rangle_{,t}=H\left(\mu,t\right)|{\hat{\bm{\Psi}}}\rangle\hskip
28.45274pt|\hat{\bm{\Psi}}\rangle=\begin{pmatrix}\hat{p}\\\
\hat{u}\end{pmatrix}\hskip
28.45274ptH\left(\mu,t\right)=\begin{bmatrix}0&-2k(1-\cos\mu)\\\\[5.0pt]
\displaystyle\frac{1}{m}&0\end{bmatrix}\end{split}$ (3)

where $\hat{p}=m\hat{u}_{,t}$ is the linear momentum and $H\left(\mu,t\right)$
is the $2\times 2$ time-dependent dynamical matrix with incident wavenumber
$\mu$. Note that Eq. 3 describes the evolution of the amplitude coefficient
$|\hat{\bm{\Psi}}\rangle$ for a Bloch wave
$|\hat{\bm{\Psi}}_{n}\rangle=|\hat{\bm{\Psi}}\rangle{\rm e}^{j\mu}$
propagating through the time-modulated lattice. Due to a temporal version of
Snell’s law [40], the impinging wavenumber is invariant during wave
propagation and, given a set of initial conditions for $|{\bm{\Psi}}\rangle$,
the solution of Eq. 3, effectively describes the evolution of the wave in
response to time-modulation, both in terms of amplitude and frequency. This is
key for the following discussion.

To shed light on this matter, we consider a waveguide with a quasi-static
variation of the properties, i.e. whose modulation velocity is much slower
than the time-scale of the underlying dynamics, whereby the solution can be
synthesized in a complex exponential form
$|\hat{\bm{\Psi}}\rangle=|\bm{\psi}\rangle{\rm e}^{j\omega t}$. As such, for
an incident wavenumber $\mu$, the system supports a pair of states
$\omega_{1,2}=\pm\omega$ that instantaneously populate the waveguide at time
$t$ and are dictated by the right eigenvalue problem
$H\left(\mu,t\right)\left|\bm{\psi}^{R}\right>=j\omega\left|\bm{\psi}^{R}\right>$.
Such states correspond to counter propagating waves whose frequencies are
illustrated through the dispersion relations in Fig. 2(a). Now, when the
temporal modulation is sufficiently slow, a quasi-static evolution takes place
such that the impinging wave, either at $\omega_{1}$ or at $\omega_{2}$, is
frequency-converted consistently to the time-evolution of the dispersion
relation marked with colored dots. It is now assumed that a right traveling
wave packet is present before time modulation, whereby its spectral content is
schematically illustrated with the black continuous line in Fig. 2(a-b). At
this step, we remark that Eq. 3 describes the time evolution of the amplitudes
associated to an arbitrary incident wavenumber $\mu$ and hence, is also
suitable to qualify the scattering of non-monochromatic wave-packets. In other
words, any incident wave packet can be described through linear combination
truncated to $2M+1$ terms in Fourier space
$|\bm{\Psi}_{n}\rangle=\sum_{m=-M}^{M}|{\bm{\hat{\Psi}}}\rangle{\rm
e}^{jn\mu_{m}}$, where $\mu_{m}=m\frac{2\pi}{Nd}$ are the impinging
wavenumbers considered in the expansion and $|{\bm{\hat{\Psi}}}\rangle$ are
the amplitude coefficients which accommodate both momentum and displacement.
Note that the amplitude coefficients $|{\hat{\bm{\Psi}}}\rangle$ are
effectively the initial conditions at $t=0$ of the problem described in Eq. 3
and the time evolution of different amplitude coefficients
$|{\hat{\bm{\Psi}}}\rangle$ is in general variable with $\mu$. We now analyze
the time evolution of an arbitrary incident wavenumber with unitary amplitude
and, after that, we generalize to linear combination of them.

(a)

(b)

(c)

Figure 2: (a) Dimensionless dispersion relation
$\omega\left(\mu\right)/\omega_{0}$ of the spring mass lattice upon varying
time, where $\omega_{0}=\sqrt{k_{i}/m}$. Due to stiffness modulation, the
dispersion curve fattens. The black curve represents the wavenumber content of
the wavepacket employed in the numerical simulations and the colored dots
illustrate the frequency conversion that takes place for the central
wavenumber of the incident wave packet. (b) Limiting condition for
adiabaticity upon varying time $t$ and wavenumber $\mu$. The bright surface
highlights the region excited by the incident wavepacket. (c) Side view of the
limiting condition for adiabaticity. The three modulation velocities
(nonadiabatic $v_{1}$, intermediate $v_{2}$ and adiabatic $v_{3}$) employed in
the simulations are indicated alongside the diagram.

We therefore assume a single value for $\mu_{m}=\mu$ and we consider a right-
traveling plane wave with unitary amplitude coefficient
$|{\bm{\hat{\Psi}}}\rangle=|\bm{\psi}_{1}^{R}\rangle{\rm e}^{j\omega_{1}t}$ as
initial condition, where $|\bm{\psi}_{1}^{R}\rangle$ is the right eigenvector
relative to $\omega_{1}$. During time modulation, the solution is synthesized
as a linear combination of the two states supported by the waveguide
$|{\bm{\hat{\Psi}}}\rangle=\sum_{n=1}^{2}a_{n}|\bm{\psi}^{R}_{n}\rangle{\rm
e}^{j\omega_{n}t}$, weighted by participation factors $a_{n}$. In other words,
if some energy is present in one state, say $\omega_{1}=+\omega$, a part of
that can leak to $\omega_{2}=-\omega$, depending on the time-modulation
parameters.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 3: Waterfall plots and corresponding spectrograms for different
modulation velocities (a-c) $v_{1}=3.5$, (d-f) $v_{2}=0.23$, (g-i)
$v_{3}=0.03$. The plots (a), (d), and (g) represent spatial snapshots of the
time history, to highlight the wavenumber-invariant transformation due to time
modulation. The plots (b), (e), and (h) are the time histories at fixed
positions, to highlight the temporal variation of frequency content. The
spectrograms (c), (f), and (i) illustrate the Fourier transformed version of
the displacement field (colored plot) superimposed to the expected temporal
evolution of the eigenvalues.

At this step, and following the procedure described in Ref. [37, 34], we
define a condition for the modulation to provide negligible energy transfer
from state $1$ to state $2$, which de facto is a condition for adiabaticity:

$\left|\displaystyle\frac{\left<\bm{\psi}_{2}^{L}\left|H(\mu,t)_{,t}\right|\bm{\psi}_{1}^{R}\right>}{\left(\omega_{2}-\omega_{1}\right)^{2}}\right|<<1$
(4)

where, in order for the expression to apply, the left and right eigenvectors
$\langle\bm{\psi}_{1,2}^{L}|$ and $|\bm{\psi}_{1,2}^{R}\rangle$ are normalized
such that $\left<\bm{\psi}_{s}^{L}|\bm{\psi}_{r}^{R}\right>=\delta_{sr}$. The
analytical expressions for the eigenstates are:

$\begin{split}&\omega_{1,2}=\pm\sqrt{\frac{2k\left(\cos{\mu}-1\right)}{m}}\hskip
28.45274pt\textit{with:}\\\\[5.0pt]
&|\psi^{R}_{1}\rangle=\begin{pmatrix}j\sqrt{mk(\cos{\mu}-1)}\\\\[6.0pt]
\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}\hskip
28.45274pt\langle\psi^{L}_{1}|=\begin{pmatrix}\displaystyle\frac{1}{j\sqrt{4mk(\cos{\mu}-1)}}&\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}\\\\[5.0pt]
&|\psi^{R}_{2}\rangle=\begin{pmatrix}-j\sqrt{mk(\cos{\mu}-1)}\\\\[6.0pt]
\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}\hskip
20.0pt\langle\psi^{L}_{2}|=\begin{pmatrix}-\displaystyle\frac{1}{j\sqrt{4mk(\cos{\mu}-1)}}&\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}\end{split}$
(5)

Merging Eqs. 4-5, we get to a limiting condition for the rate of change $v$,
in order for the modulation to induce an adiabatic transformation to the
incident state:

$\left|v_{lim}\right|<<16\left|\sin{\frac{\mu}{2}}\right|\sqrt{\frac{k^{3}}{m}}$
(6)

which is function of both incident wavenumber $\mu$ and time $t$. This
condition is mapped in Fig. 2(b), where the incident wavenumber content,
hereafter employed for simulations, is highlighted.

We now perform numerical simulations to qualify the dynamic behavior discussed
above, and in the attempt to capture limiting conditions expressed in Eq. 6.
The values employed in the simulations are $v_{1}=-3.5$, $v_{2}=-0.23$ and
$v_{3}=-0.03$, which are represented in Fig. 2(c) and superimposed to the
lateral view of Fig. 2(b). These values correspond to a nonadiabatic,
intermediate, and adiabatic modulation speeds for the entire incident
wavenumber spectrum from $t_{i}$ to $t_{f}$.

Fig. 3 illustrates results for an impinging Gaussian wavepacket with number of
periods $n=10$ and central wavenumber $\mu=0.3\pi$, which is provided as
initial condition for the simulations. The time histories are conveniently
represented with space-time diagrams in Figs. 3(a-b), 3(d-e), and 3(g-h) from
$t=0$ to $t=T$. Specifically, Figs. 3(a), 3(d), and 3(g) illustrate a
wavenumber-invariant transformation, whereby the fixed-time displacement of
the wave does not change shape during time modulation. In contrast, Figs.
3(b), 3(e), and 3(h) suitably represent the frequency change dictated by time-
modulation, whereby the temporal period of the wave packet is clearly
increased during modulation. Further relevant comments follow: (i) the wave
packet decelerates during the transformation, consistently with the dispersion
properties displayed in Fig. 2(a). (ii) The frequency and velocity changes are
accompanied by reflection and transmission of the impinging wave, whose
magnitudes are dependent upon the modulation speed. The scattering level is
reduced for lower velocity values $v$ and eventually eliminated in case of
purely adiabatic modulations.

This concept is better described by way of Figs. 3(c), 3(f), and 3(i), where
the energy content within the waveguide (colored contours) is superimposed to
the expected evolution of the states (dashed lines). Such spectrograms are
herein evaluated by windowing the time history $u\left(x_{n},t\right)$ through
a moving Gaussian function $G={\rm e}^{-\left(t-t_{0}\right)^{2}/2c^{2}}$,
where the parameter $c=T/20$ controls the width of the window and $t_{0}$ is
smoothly varied between $t_{0}\in\left[0,T\right]$, to capture the variation
of the spectral content in time. The Fourier transformed version of the
windowed displacement field $\hat{u}\left(t_{0},\omega,\mu\right)$ is further
post-processed by taking the $L^{2}$ norm in wavenumber space limited to
$\mu>0$, in order to eliminate one dimension and get to the convenient form
$\hat{u}\left(t_{0},\omega\right)$. In the figures, the energy is initially
stored in the state $\omega_{1}=+\omega$. After the modulation takes place, a
certain amount of energy is scattered toward backward propagating states, i.e.
in correspondence of $\omega_{2}=-\omega$. This energy leak is observable in
Fig. 3(c) and in Fig. 3(f) with minor intensity. In contrast, the
configuration displayed in Fig. 3(i) exhibits a clear frequency conversion
without back-scattering, whereby all the energy remains confined in
correspondence of $\omega_{1}=\omega$, and the transformation can be
considered adiabatic.

##  III Temporal steering in stiffness-modulated 2D lattices

Figure 4: (a) Schematic of the stiffness-modulated 2D spring-mass lattice, (b)
along with the visual representation of the scattering phenomena taking place
during time modulation. The colors represent time modulation, while the white
curve illustrates an incident wavepacket which undergoes frequency modulation
and steering. (c) Limiting condition for the modulation velocity, which
delineates the transition between adiabatic and non-adiabatic transformations
upon varying time. The three modulation velocities employed in the simulations
are indicated alongside the diagram.

The discussion is now focused on the elastic lattice displayed in Fig. 4(a),
which is made of $60\times 60$ masses equally spaced by the lattice constant
$d$, and connected to the nearest neighbors through linear springs $k_{x}$ and
$k_{y}$. For simplicity and without any loss of generality,
$k_{y}\left(t\right)$ accommodates the temporal degree of freedom. By
following the same line of work, we consider a linear time varying function
for $k_{y}\left(t\right)=k_{yi}+v\left(t-t_{i}\right)$, capable of driving the
stiffness value from $k_{yi}$ to $k_{yf}$ in a time interval $\Delta
t=t_{f}-t_{i}$ with modulation velocity $v$. The concept is elucidated in Fig.
4(b): an impinging wave that propagates across a temporal discontinuity
experiences a frequency transformation and, provided that the modulation is
adiabatic, the process occurs with negligible energy leak toward the back-
propagating state. We hereafter demonstrate that such a frequency
transformation is accompanied by a time-dependent curvature of wave motion,
which serves as a mechanism to functionally control and guide waves.

We start the analysis from the time-dependent elastodynamic equation, which
describes out-of-plane motion of the mass element sitting in position
$\bm{r}=\left(n,m\right)$:

$m\ddot{u}_{n,m}+2\left(k_{x}+k_{y}\right)u_{n,m}-k_{x}\left(u_{n+1,m}+u_{n-1,m}\right)-k_{y}\left(u_{n,m+1}+u_{n,m-1}\right)=0$
(7)

where to ease notation, the time dependence of $k_{y}\left(t\right)$ is
implicitly assumed. Now, wave propagation in such a system is investigated via
a dispersion analysis of the unit cell with implied Floquet-Bloch boundary
conditions, whereby the displacement $u_{n,m}\left(t\right)$ is described via
complex exponential functions $u_{n,m}=\hat{u}e^{j\bm{\mu}\cdot\bf{r}}$, where
$\bm{\mu}=\left(\mu_{x},\mu_{y}\right)$ is the dimensionless wavevector. After
a few mathematical steps we get to the following form:

$|\hat{\bm{\Psi}}\rangle,t=H\left(\mu,t\right)|\hat{\bm{\Psi}}\rangle\hskip
28.45274pt|\hat{\bm{\Psi}}\rangle=\begin{pmatrix}\hat{p}\\\
\hat{u}\end{pmatrix}\hskip
28.45274ptH\left(\bm{\mu},t\right)=\begin{bmatrix}0&-2\left[k_{x}(1-\cos\mu_{x})+k_{y}(1-\cos\mu_{y})\right]\\\\[5.0pt]
\displaystyle\frac{1}{m}&0\end{bmatrix}$ (8)

which provide suitable description of the quasi-static evolution of two
propagating eigenstates relative to an impinging wave
$|\hat{\bm{\Psi}}\rangle=|\bm{\psi}^{R}\rangle{\rm e}^{j\omega t}$ with
wavevector $\bm{\mu}$. The eigenstates, solution of the quasi-static
eigenvalue problem
$H\left(\bm{\mu},t\right)|\bm{\psi}^{R}\rangle=j\omega|\bm{\psi}^{R}\rangle$
are:

$\displaystyle\hskip
85.35826pt\omega_{1,2}=\pm\sqrt{\frac{2}{m}\left(k_{x}\left(1-\cos{\mu_{x}}\right)+k_{y}\left(1-\cos{\mu_{y}}\right)\right)}$
(9)
$\displaystyle|\bm{\psi_{1}}^{R}\rangle=\begin{pmatrix}j\sqrt{m\left(k_{x}\left(\cos{\mu_{x}}-1\right)+k_{y}\left(\cos{\mu_{y}}-1\right)\right)}\\\\[6.0pt]
\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}\hskip
14.22636pt\langle\bm{\psi_{1}}^{L}|=\begin{pmatrix}\displaystyle\frac{1}{j\sqrt{4m\left(k_{x}\left(\cos{\mu_{x}}-1\right)+k_{y}\left(\cos{\mu_{y}}-1\right)\right)}}&\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}$
$\displaystyle|\bm{\psi_{2}}^{R}\rangle=\begin{pmatrix}-j\sqrt{m\left(k_{x}\left(\cos{\mu_{x}}-1\right)+k_{y}\left(\cos{\mu_{y}}-1\right)\right)}\\\\[6.0pt]
\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}\hskip
14.22636pt\langle\bm{\psi_{2}}^{L}|=\begin{pmatrix}-\displaystyle\frac{1}{j\sqrt{4m\left(k_{x}\left(\cos{\mu_{x}}-1\right)+k_{y}\left(\cos{\mu_{y}}-1\right)\right)}}&\displaystyle\frac{\sqrt{2}}{2}\end{pmatrix}$

with suitable normalization
$\langle\bm{\psi}_{1,2}^{L}|\bm{\psi}_{1,2}^{R}\rangle=\delta_{1,2}$. Finally,
analytical forms of the group velocity
$\bm{c}_{g}=\nabla\omega\left(\bm{\mu}\right)$ for the above eigenstates can
be easily deduced:

(a)

(b)

(c)

(d)

(e)

Figure 5: (a) Dispersion relation (left) and corresponding group velocity
profiles (right) for three different time instants: before the modulation
$t=t_{i}$, during time modulation $t_{m}=(t_{i}+t_{f})/2$ and after time
modulation $t=t_{f}$. The white dot represents the central wavevector content
of the incident wavepacket employed for numerical simulations. (b) Time
histories relative to the three modulation velocities $v_{1}=93$, $v_{2}=10$,
and $v_{3}=1.6$. The black curve represents the desired trajectory predicted
through the group velocity profile. (c-e) Spectrograms to elucidate how the
frequency content within the lattice varies in time for modulation velocities
(c) $v_{1}$, (d) $v_{2}$, and (e) $v_{3}$. The colored contours are relative
to the Fourier transformed version of the displacement field, while the
expected evolution of the states is represented with black dashed lines.

$\bm{c}_{g}=\left(\frac{k_{x}d\sin{\mu_{x}}}{\sqrt{2m\left(k_{x}\left(1-\cos{\mu_{x}}\right)+k_{y}\left(1-\cos{\mu_{y}}\right)\right)}}\hskip
7.11317pt\frac{k_{y}d\sin{\mu_{y}}}{\sqrt{2m\left(k_{x}\left(1-\cos{\mu_{x}}\right)+k_{y}\left(1-\cos{\mu_{y}}\right)\right)}}\right)$
(10)

where counter propagating states have opposite velocity sign
$\bm{c}_{g\;1,2}=\pm\bm{c}_{g}$. The direction of propagation $\phi$ follows:

$\tan{\phi(t)}=\frac{c_{gy}}{c_{gx}}=\frac{k_{y}}{k_{x}}\frac{\sin{\mu_{y}}}{\sin{\mu_{x}}}$
(11)

Interestingly, an incident state with arbitrary wavevector $\bm{\mu}$
propagates along the direction $\phi$ defined by the group velocity
$\bm{c}_{g}\left(\bm{\mu}\right)$, which is tailorable with
$k_{y}\left(t\right)$. In other words, the analytical forms for $\omega_{1,2}$
and $\phi\left(\bm{\mu}\right)$ delineate both frequency conversion and
curvature of wave motion induced by the stiffness modulation which, in turn,
can be tailored to functionally steer waves via adiabatic and nonadiabatic
transformations.

We now discuss on the dynamics of smooth temporal modulations applied to the
2D lattice, and the relative transformation that takes place when an incident
wave packet propagates in such a system. First and foremost, Eq. 8 have the
same form of Eq. 3 and, as such, for any impinging wavevector $\bm{\mu}$,
there are two counter propagating states that populate the lattice. It is
straightforward to conclude that Eq. 4 can be employed to assess adiabaticity
of the modulation, and gives the following condition for the velocity in order
for a propagating state $|\bm{\psi_{1}}^{R}\rangle{\rm e}^{j\omega_{1}t}$ not
to leak energy toward $|\bm{\psi_{2}}^{R}\rangle{\rm e}^{j\omega_{2}t}$:

$|v_{lim}|<<16\displaystyle\frac{\sqrt{\left|k_{x}\sin^{2}{\displaystyle\frac{\mu_{x}}{2}}-k_{y}\sin^{2}{\displaystyle\frac{\mu_{y}}{2}}\right|}}{\sqrt{m}\sin^{2}{\displaystyle\frac{\mu_{y}}{2}}}$
(12)

In analogy to the procedure employed in the previous section, Fig. 4(c)
illustrates the limiting condition for the speed of modulation to be
considered adiabatic. This condition is valid for the entire wavenumber
content of the incident wave packet, which consists of a prescribed
displacement with central wavnumber $\bm{\mu}=[0.5\pi,0.5\pi]$ and number of
periods $n_{x}=n_{y}=5$. To get to Fig. 4(c), the wavenumber dimension is
eliminated by taking the minimum
$\displaystyle\min_{D}\\{v_{lim}\left(\bm{\mu},t\right)\\}$ and maximum
$\displaystyle\max_{D}\\{v_{lim}\left(\bm{\mu},t\right)\\}$ values within the
domain
$D=\mu_{x}\times\mu_{y}=\left[0.3\pi,0.7\pi\right]\times\left[0.3\pi,0.7\pi\right]$,
which allowed us to include in the analysis all the relevant spectral content
of the wavepacket.

Three distinct modulation velocities $v_{1}=93$, $v_{2}=10$, and $v_{3}=1.6$
are probed to qualify consistency of Eq. 12 and Fig. 4(c) via numerical
simulation: it is expected that these values, defined across the limiting
condition, allow us to shed light on the energy leaks which manifests in time-
modulated lattices with concurrent frequency conversion and wave steering. We
firstly show dispersion relations and group velocity profiles in Fig. 5(a) for
consecutive time instants $t_{i}$, $t_{m}=\left(t_{i}+t_{f}\right)/2$ and
$t_{f}$. We focus our attention on the central wavevector content
$\bm{\mu}=\left(0.5\pi,0.5\pi\right)$ of the incident wave packet (highlighted
with a white dot in the dispersion diagram). For such a point in reciprocal
space, the corresponding group velocities are reported alongside the
dispersion, with emphasis on how the directionality evolves over time. It is
expected that the temporal evolution of a narrowband wavepacket does follow
the directionality dictated by 5(a). Results from time simulations follow.

(a)

(b)

(c)

Figure 6: (a) Snapshots of the displacement field for a 2D lattice modulated
in time via three consecutive modulation levels (colored contours). The
displacement field is superimposed to the desired trajectory (black line). (b)
RMS of the displacement field in time. (c) Time spectrogram for the time
evolution of the frequency content in time (colored contours) and expected
evolution of the states (black dashed lines).

Three relevant snapshots prior, during, and after the modulation are
illustrated in Fig. 5(b) for the three aforementioned modulation velocities.
For comparison, the expected trajectory is predicted from integration of the
group velocities in time and displayed in black underneath the displacement
fields. We observe that all configurations are capable of curving the wave
packet by the desired angle $\Delta\phi$ and, hence, are consistent with the
group velocities illustrated in Fig 5(a). In case of fast modulation $v_{1}$,
there is a relevant wave packet that separates from the incident state
$\omega_{1}$, which is blueprint of energy leak across the temporal
discontinuity. The reflected wave packet effectively embodies the energy
transferred to state $\omega_{2}$ via non-adiabatic passage, and propagates
with opposite direction as compared to $\omega_{1}$. To decrease the energy
leak, a sufficiently slow velocity $v$ is to be employed. Configurations with
$v=v_{2}$ and $v=v_{3}$ exhibit greater degree of adiabaticity and, hence, the
reflection is minimal or absent. Animations are reported in the supplementary
material.

To further verify the above considerations, the frequency spectrograms
relative to the time histories are displayed in Fig. 5 (c-e). In analogy to 1D
lattices, the wavenumber domain is eliminated by taking the $L^{2}$ norm
limited to $\mu_{x}>0$ and $\mu_{y}>0$. The results are consistent to the
theory: fast and intermediate modulations (see Fig. 5(c-d)) are not adiabatic
and, as such, there is energy transfer from the positive toward the negative
frequencies. In contrast, sufficiently slow modulations (Fig. 5(e)) induce
adiabatic transformations whereby the energy content represented in reciprocal
space does not leave the incident state. Hence, in physical space, the
trajectory of an incident wave-packet can be curved without undesired
reflections.

Finally, we design a more complex adiabatic law for $k_{y}$, which consists of
three consecutive stiffness levels $k_{y}=1$, $k_{y}=10$, and back to
$k_{y}=1$. Fig. 6(a) illustrates temporal snapshots of the wave packet that,
due to stiffness modulation, is capable of steering at will (animations are
reported in the supplementary material). The curvature of wave motion is
further highlighted by taking the $RMS$ along the temporal dimension (Fig.
6(b)), whereby the lower amplitude in the central part is attributed both to a
lower amplitude of wave motion and greater velocity of propagation. To
conclude the paper, we demonstrate adiabaticity of this transformation which,
according to the spectrogram in Fig. 6(c) occurs without energy scattering
toward counter propagating states.

##  IV conclusions

In this paper, we explored the dynamics of slow temporal modulations in the
context of 1D and 2D lattices. The behavior of such systems is driven by the
time-evolution of the underlying dispersion properties which, under quasi-
static conditions, dictate both frequency conversion and directionality of
wave motion. By way of the adiabatic theorem, we define a condition in order
for the transformation to occur without energy leakage, which is responsible
for undesired reflections. The concept can be easily extended to physical
systems with tunable dispersion relations, such as beams with piezoelectric
materials in mechanics [41, 34], and materials with tunable permittivity [43].
Although based on a very different phenomena, we herein propose a waveguiding
mechanism that allows an elastic signal to be sent from an emitter to a
receiver, similarly (but conceptually very different) to what has been
observed in many studies concerning quantum valley Hall (QVH) and quantum spin
Hall (QSH)-based waveguides.

## References

  * Casadei  _et al._ [2012] F. Casadei, T. Delpero, A. Bergamini, P. Ermanni, and M. Ruzzene, Piezoelectric resonator arrays for tunable acoustic waveguides and metamaterials, Journal of Applied Physics 112, 064902 (2012).
  * Wang  _et al._ [2015] P. Wang, L. Lu, and K. Bertoldi, Topological phononic crystals with one-way elastic edge waves, Physical review letters 115, 104302 (2015).
  * Huber [2016] S. D. Huber, Topological mechanics, Nature Physics 12, 621 (2016).
  * Miniaci  _et al._ [2018] M. Miniaci, R. Pal, B. Morvan, and M. Ruzzene, Experimental observation of topologically protected helical edge modes in patterned elastic plates, Physical Review X 8, 031074 (2018).
  * Pal and Ruzzene [2017] R. K. Pal and M. Ruzzene, Edge waves in plates with resonators: an elastic analogue of the quantum valley hall effect, New Journal of Physics 19, 025001 (2017).
  * Riva  _et al._ [2018] E. Riva, D. Quadrelli, G. Cazzulani, and F. Braghin, Tunable in-plane topologically protected edge waves in continuum kagome lattices, Journal of Applied Physics 124, 164903 (2018).
  * Qian  _et al._ [2018] K. Qian, D. J. Apigo, C. Prodan, Y. Barlas, and E. Prodan, Topology of the valley-chern effect, Physical Review B 98, 155138 (2018).
  * Jin  _et al._ [2018] Y. Jin, D. Torrent, and B. Djafari-Rouhani, Robustness of conventional and topologically protected edge states in phononic crystal plates, Physical Review B 98, 054307 (2018).
  * Liu and Semperlotti [2018] T.-W. Liu and F. Semperlotti, Tunable acoustic valley–hall edge states in reconfigurable phononic elastic waveguides, Physical Review Applied 9, 014001 (2018).
  * Riva  _et al._ [2020a] E. Riva, M. I. Rosa, and M. Ruzzene, Edge states and topological pumping in stiffness-modulated elastic plates, Physical Review B 101, 094307 (2020a).
  * De Ponti  _et al._ [2021] J. M. De Ponti, L. Iorio, E. Riva, R. Ardito, F. Braghin, and A. Corigliano, Selective mode conversion and rainbow trapping via graded elastic waveguides, Physical Review Applied 16, 034028 (2021).
  * Alshaqaq  _et al._ [2022] M. Alshaqaq, C. Sugino, and A. Erturk, Programmable rainbow trapping and band-gap enhancement via spatial group-velocity tailoring in elastic metamaterials, Physical Review Applied 17, L021003 (2022).
  * Climente  _et al._ [2010] A. Climente, D. Torrent, and J. Sánchez-Dehesa, Sound focusing by gradient index sonic lenses, Applied Physics Letters 97, 104103 (2010).
  * Tol  _et al._ [2017] S. Tol, F. L. Degertekin, and A. Erturk, Phononic crystal luneburg lens for omnidirectional elastic wave focusing and energy harvesting, Applied Physics Letters 111, 013503 (2017).
  * Allam  _et al._ [2020] A. Allam, K. Sabra, and A. Erturk, 3d-printed gradient-index phononic crystal lens for underwater acoustic wave focusing, Physical Review Applied 13, 064064 (2020).
  * Zigoneanu  _et al._ [2011] L. Zigoneanu, B.-I. Popa, and S. A. Cummer, Design and measurements of a broadband two-dimensional acoustic lens, Physical Review B 84, 024305 (2011).
  * Kim  _et al._ [2021] G. Kim, C. M. Portela, P. Celli, A. Palermo, and C. Daraio, Poroelastic microlattices for underwater wave focusing, Extreme Mechanics Letters 49, 101499 (2021).
  * Pendry  _et al._ [2006] J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, science 312, 1780 (2006).
  * Leonhardt [2006] U. Leonhardt, Optical conformal mapping, science 312, 1777 (2006).
  * Fleury  _et al._ [2015] R. Fleury, F. Monticone, and A. Alù, Invisibility and cloaking: Origins, present, and future perspectives, Physical Review Applied 4, 037001 (2015).
  * Norris [2008] A. N. Norris, Acoustic cloaking theory, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, 2411 (2008).
  * Torrent and Sánchez-Dehesa [2008] D. Torrent and J. Sánchez-Dehesa, Acoustic cloaking in two dimensions: a feasible approach, New Journal of Physics 10, 063015 (2008).
  * Xu  _et al._ [2020] X. Xu, C. Wang, W. Shou, Z. Du, Y. Chen, B. Li, W. Matusik, N. Hussein, and G. Huang, Physical realization of elastic cloaking with a polar material, Physical review letters 124, 114301 (2020).
  * Chatzopoulos  _et al._ [2022] Z. Chatzopoulos, A. Palermo, S. Guenneau, and A. Marzani, Cloaking strategy for love waves, Extreme Mechanics Letters 50, 101564 (2022).
  * Quadrelli  _et al._ [2021] D. E. Quadrelli, M. A. Casieri, G. Cazzulani, S. La Riviera, and F. Braghin, Experimental validation of a broadband pentamode elliptical-shaped cloak for underwater acoustics, Extreme Mechanics Letters 49, 101526 (2021).
  * Becker  _et al._ [2021] T. S. Becker, D.-J. Van Manen, T. Haag, C. Bärlocher, X. Li, N. Börsing, A. Curtis, M. Serra-Garcia, and J. O. Robertsson, Broadband acoustic invisibility and illusions, Science advances 7, eabi9627 (2021).
  * Popa  _et al._ [2011] B.-I. Popa, L. Zigoneanu, and S. A. Cummer, Experimental acoustic ground cloak in air, Physical review letters 106, 253901 (2011).
  * Colombi  _et al._ [2015] A. Colombi, P. Roux, S. Guenneau, and M. Rupin, Directional cloaking of flexural waves in a plate with a locally resonant metamaterial, The Journal of the Acoustical Society of America 137, 1783 (2015).
  * Nassar  _et al._ [2020] H. Nassar, B. Yousefzadeh, R. Fleury, M. Ruzzene, A. Alù, C. Daraio, A. N. Norris, G. Huang, and M. R. Haberman, Nonreciprocity in acoustic and elastic materials, Nature Reviews Materials 5, 667 (2020).
  * Trainiti and Ruzzene [2016] G. Trainiti and M. Ruzzene, Non-reciprocal elastic wave propagation in spatiotemporal periodic structures, New Journal of Physics 18, 083047 (2016).
  * Riva  _et al._ [2019] E. Riva, J. Marconi, G. Cazzulani, and F. Braghin, Generalized plane wave expansion method for non-reciprocal discretely modulated waveguides, Journal of Sound and Vibration 449, 172 (2019).
  * Palermo  _et al._ [2020] A. Palermo, P. Celli, B. Yousefzadeh, C. Daraio, and A. Marzani, Surface wave non-reciprocity via time-modulated metamaterials, Journal of the Mechanics and Physics of Solids 145, 104181 (2020).
  * Park and Min [2021] J. Park and B. Min, Spatiotemporal plane wave expansion method for arbitrary space–time periodic photonic media, Optics Letters 46, 484 (2021).
  * Xia  _et al._ [2021] Y. Xia, E. Riva, M. I. Rosa, G. Cazzulani, A. Erturk, F. Braghin, and M. Ruzzene, Experimental observation of temporal pumping in electromechanical waveguides, Physical Review Letters 126, 095501 (2021).
  * Grinberg  _et al._ [2020] I. H. Grinberg, M. Lin, C. Harris, W. A. Benalcazar, C. W. Peterson, T. L. Hughes, and G. Bahl, Robust temporal pumping in a magneto-mechanical topological insulator, Nature communications 11, 1 (2020).
  * Riva  _et al._ [2020b] E. Riva, V. Casieri, F. Resta, and F. Braghin, Adiabatic pumping via avoided crossings in stiffness-modulated quasiperiodic beams, Physical Review B 102, 014305 (2020b).
  * Riva  _et al._ [2021] E. Riva, G. Castaldini, and F. Braghin, Adiabatic edge-to-edge transformations in time-modulated elastic lattices and non-hermitian shortcuts, New Journal of Physics 23, 093008 (2021).
  * Yi  _et al._ [2017] K. Yi, M. Collet, and S. Karkar, Frequency conversion induced by time-space modulated media, Physical review B 96, 104110 (2017).
  * Trainiti  _et al._ [2019] G. Trainiti, Y. Xia, J. Marconi, G. Cazzulani, A. Erturk, and M. Ruzzene, Time-periodic stiffness modulation in elastic metamaterials for selective wave filtering: Theory and experiment, Physical review letters 122, 124301 (2019).
  * Pacheco-Peña and Engheta [2020a] V. Pacheco-Peña and N. Engheta, Antireflection temporal coatings, Optica 7, 323 (2020a).
  * Marconi  _et al._ [2020] J. Marconi, E. Riva, M. Di Ronco, G. Cazzulani, F. Braghin, and M. Ruzzene, Experimental observation of nonreciprocal band gaps in a space-time-modulated beam using a shunted piezoelectric array, Physical Review Applied 13, 031001 (2020).
  * Attarzadeh  _et al._ [2020] M. Attarzadeh, J. Callanan, and M. Nouh, Experimental observation of nonreciprocal waves in a resonant metamaterial beam, Physical Review Applied 13, 021001 (2020).
  * Pacheco-Peña and Engheta [2020b] V. Pacheco-Peña and N. Engheta, Temporal aiming, Light: Science & Applications 9, 1 (2020b).


# Scaling limit of the triangular prudent walk

P. Rigas

###### Abstract

We extend computations for the scaling limit of the prudent walk on the square
lattice, due to Beffara, Friedli, and Velenik in 2010, to obtain the scaling
limit of the prudent walk on the triangular lattice. In comparison to
computing the probability that a short excursion, which is contained with some
box of side length $L$, hits the corner of a box infinitely many times, we
incorporate symmetries of the triangular lattice for computing the scale
limit, beginning with a suitable basis. Steps for obtaining the scaling limit
include: computing the probability that the prudent walker exists a
’triangular box,’ as well as computing conditionally defined probabilities
that the prudent walker travel some fixed horizontal and/or vertical distance,
after some arbitrary, strictly positive, time, as well as demonstrating that
several probabilities asymptotically vanish as $n\longrightarrow+\infty$.
111Keywords: Prudent walk, triangular lattice, triangular prudent walker,
scaling limit

##  1 Introduction

###  1.1 Overview

Random walks are amongst the most classically studied objects in Probability
theory, with connections to scaling limits of prudent walks, which are a class
of random walks avoiding any step that the walk has previously taken [1], as
well as random walks satisfying a variety of additional criteria. Besides
other closely related types of random walks to the prudent walk, including
those discussed in [2,3,4], we answer one question raised by Beffara, Friedli,
and Velenik, raised in [1], as to whether the computations for obtaining the
scaling limit of the prudent walk over the square lattice can be extended to
obtain the scaling limit of the prudent walk over the triangular lattice. In
the case of the square lattice, the probability that some prudent walker
intersection with some point $m$, conditionally upon the path explored up to
some time $\mathscr{T}_{k}$, is equal to the probability that the prudent walk
arrives to $m$ at $\mathscr{T}_{k}$.

###  1.2 Paper organization

With the description of the prudent walker in 1.1, in the next section, 2, we
describe the prudent walk defined over the triangular lattice. After
introducing an appropriate basis for the walker over the triangular lattice,
we contrast statements to bound the excursions of the prudent walker over the
square lattice with similarly oriented statements for bounding probabilities
of excursions of the prudent walker over the triangular lattice. Depending
upon the symmetries of the triangular lattice, having arguments for upper
bounding excursions of the prudent walker over the triangular lattice
demonstrates how differences arise in the scaling limit for the prudent walker
over the square lattice, and the scaling limit of the prudent walker over the
triangular lattice.

##  2 Setting

###  2.1 Prudent walker objects

Over T, the prudent walk is defined as the random walk which avoids any of the
sites that it has visited in any previous steps of its excursion up to some
strictly positive $t$, which we denote with $\gamma_{[0,t]}$. That is, if the
prudent walk over the triangular lattice begins at the origin, given spanning
set,

$\displaystyle\mathrm{span}_{\textbf{T}}\big{\\{}\vec{\textbf{e}_{1}},\vec{\textbf{e}_{2}},\vec{\textbf{e}_{3}}\big{\\}}\text{
}\text{ , }$

of basis vectors,

$\displaystyle\textbf{e}_{1}\equiv\big{(}\frac{\sqrt{3}}{2},\frac{1}{2},0\big{)}\text{
}\text{ , }$
$\displaystyle\textbf{e}_{2}\equiv\big{(}\frac{\sqrt{3}}{2},-\frac{1}{2},0\big{)}\text{
}\text{ , }$ $\displaystyle\textbf{e}_{3}\equiv\big{(}0,0,1\big{)}\text{
}\text{ , }$

of T, one can extend the arguments from [1] to the triangular lattice by
defining the times that the prudent walker occupies at the beginning, and at
later times, of an excursion, which are respectively given by,

$\displaystyle\mathscr{T}_{0}\equiv 0\text{ }\text{ , }$
$\displaystyle\mathscr{U}_{0}\equiv\big{\\{}t>0:\mathcal{H}_{t}>1\big{\\}}-1\text{
}\text{ . }$

In the two times defined above, the latter is dependent upon the height,
$\mathcal{H}_{t}$, of the triangular prudent walker at $t$. Associated with
some rectangular box over T, $\mathscr{R}_{t}\subsetneq\textbf{T}$, the
stopping times $\eta_{L}$ introduced below will quantify the probability of
whether the triangular walker exits some region of T. Relatedly, one can also
define the following two other times associated with an excursion,

$\displaystyle\mathscr{T}_{k+1}\equiv\big{\\{}t>\mathscr{U}_{k}:\mathscr{W}_{t}>\mathscr{W}_{t-1}\big{\\}}-1\text{
}\text{ , }$
$\displaystyle\mathscr{U}_{k+1}\equiv\big{\\{}t>\mathscr{T}_{k+1}:\mathcal{H}_{t}>\mathcal{H}_{t-1}\big{\\}}-1\text{
}\text{ , }$

corresponding to the times strictly greater than $\mathscr{U}_{k}$ for which
the width of the prudent walker at time $t$, $\mathscr{W}_{t}$, is strictly
greater than the width of the excursion of the prudent walker at time $t-1$,
$\mathscr{W}_{t-1}$, and to the times strictly greater than
$\mathscr{T}_{k+1}$ for which the height of the prudent walker at time $t$,
$\mathcal{H}_{t}$, strictly exceeds the height of the prudent walker at time
$t-1$, $\mathcal{H}_{t-1}$. When $\mathscr{T}_{0}\equiv 0$, denote
$\mathscr{W}_{0}\equiv\mathcal{H}_{0}\equiv 1$.

Next, with the horizontal and vertical increments of the prudent walker within
each instance of time, define the excursions of a walker,
$\mathscr{E}^{\mathrm{V}}_{k}$, and $\mathscr{E}^{\mathrm{H}}_{k}$, as the
vertical and horizontal excursions of the walker, respectively. Observe that
$\mathscr{E}^{\mathrm{V}}_{k}\neq\emptyset\Longleftrightarrow
t\in\big{(}\mathscr{T}_{k},\mathscr{U}_{k}\big{]}$, while
$\mathscr{E}^{\mathrm{H}}_{k}\neq\emptyset\Longleftrightarrow
t\in\big{(}\mathscr{U}_{k},\mathscr{T}_{k+1}\big{]}$. Furthermore, denote the
vertical and horizontal displacements of the walker over T with
$\mathscr{X}_{k}\equiv\mathscr{W}_{\mathscr{T}_{k+1}}-\mathscr{W}_{\mathscr{T}_{k}}$,
and
$\mathscr{Y}_{k}\equiv\mathcal{H}_{\mathscr{T}_{k+1}}-\mathcal{H}_{\mathscr{T}_{k}}$,
respectively.

With the vertical and horizontal displacements of the walker over
$\big{[}\mathscr{T}_{k},\mathscr{T}_{k+1}\big{]}$, introduce, for the
effective random walk, $\mathcal{S}_{n}$, over T, with,

$\displaystyle\mathcal{S}_{0}\equiv 0\text{ }\text{ , }$
$\displaystyle\mathcal{S}_{n}\equiv\underset{1\leq i\leq n}{\sum}\xi_{i}\text{
}\text{ , }$

the quantities:

  * $\bullet$

Exit times from the interval with Lebesgue measure $L-1$: $\eta_{L}$,

  * $\bullet$

The first exit time for which the triangular prudent walker exists the
interval with Lebesgue measure $L-1$:
$\big{(}\eta_{L}\big{)}_{1}\equiv\mathrm{inf}\big{\\{}n>0:\mathcal{S}_{n}\not\in\big{[}0,L-1\big{]}\big{\\}}$,

  * $\bullet$

The probability density of $\big{\\{}\xi_{i}\equiv k\big{\\}}$.
$\textbf{P}\big{[}\xi_{i}=k\big{]}=\frac{1}{3}\big{(}\frac{1}{2}\big{)}^{|k|}$.

Finally, denote the triangular normalizing constant as,

$\displaystyle
Z\bigg{(}\textbf{T},\sigma_{1},\sigma_{2},\sigma_{3},u,\alpha\bigg{)}\equiv
Z^{\sigma_{1},\sigma_{2},\sigma_{3}}_{u,\alpha}\equiv\int_{0}^{\alpha
u}\bigg{[}\sigma_{1}\textbf{1}_{\\{W(s)<0\\}}\vec{\textbf{e}_{1}}+\sigma_{2}\textbf{1}_{\\{W(s)=0\\}}\vec{\textbf{e}_{2}}+\sigma_{3}\textbf{1}_{\\{W(s)>0\\}}\vec{\textbf{e}_{3}}\bigg{]}\mathrm{d}s\text{
}\text{ , }$

for,

$\displaystyle\alpha\geq\bigg{|}\bigg{|}\int_{0}^{\alpha
u}\bigg{[}\sigma_{1}\textbf{1}_{\\{W(s)<0\\}}\vec{\textbf{e}_{1}}+\sigma_{2}\textbf{1}_{\\{W(s)=0\\}}\vec{\textbf{e}_{2}}+\sigma_{3}\textbf{1}_{\\{W(s)>0\\}}\vec{\textbf{e}_{3}}\bigg{]}\mathrm{d}s\bigg{|}\bigg{|}_{1}\text{
}\text{ . }$

###  2.2 Conditionally defined probabilities of the triangular prudent walker

Introduce the following identity for conditionally defined probabilities .

Lemma 1 (histories of the exploration paths in horizontal and vertical
directions). Conditionally upon the existence of paths
$\gamma_{[0,\mathscr{T}_{k}]}$ and $\gamma_{[0,\mathscr{U}_{k}]}$,

$\displaystyle\textbf{P}\big{[}X_{k}=m\big{|}\gamma_{[0,\mathscr{T}_{k}]}\big{]}=\textbf{P}\big{[}\eta_{X_{[\mathscr{T}_{k}]}}=m\big{]}\text{
}\text{ , }$
$\displaystyle\textbf{P}\big{[}Y_{k}=m\big{|}\gamma_{[0,\mathscr{U}_{k}]}\big{]}=\textbf{P}\big{[}\eta_{Y_{[\mathscr{U}_{k}]}}=m\big{]}\text{
}\text{ , }$

$\forall k\geq 0,m\geq 1$, where $\eta$ denotes a point process that is
dependent upon either $\mathcal{H}_{\mathscr{T}_{k}}$, or upon
$\mathcal{W}_{\mathscr{U}_{k}}$.

Over $\mathscr{T}_{k}$ or $\mathscr{U}_{k}$, the fact that the value of
$X_{k}$ and $Y_{k}$ at the k $th$ step are only dependent upon the vertical,
and horizontal, segments of the prudent walker, the first probability
conditioned on $\gamma_{[0,\mathscr{T}_{k}]}$ can be expressed as,

$\displaystyle\textbf{P}\big{[}X_{k}=m\big{|}\gamma_{[0,\mathscr{T}_{k}]}\big{]}=\underset{\\{\alpha_{1},\cdots,\alpha_{n},\beta\\}}{\sum}\bigg{(}\underset{1\leq
i\leq
m-1}{\prod}\textbf{P}\big{[}X_{k}=\alpha_{i}\big{]}\textbf{P}\big{[}X_{k}=\beta\big{]}\bigg{)}\text{
}\text{ , }$

for times
$\big{\\{}\gamma_{[0,\mathscr{T}_{\alpha_{i}}]}\big{\\}}_{i\in\\{i_{1},\cdots,i_{n},i_{n+1}\\}}$.
The main task will revolve around evaluating the probability, for each
$\alpha_{i}$, that the event that the prudent walker achieves some height
$\alpha_{i}$ conditionally upon the history of the path at time
$\mathscr{T}_{\alpha_{i}}$ occurs. Moreover, the product of probabilities
admits the identity,

$\displaystyle\bigg{(}\underset{1\leq i\leq
m-1}{\prod}\textbf{P}\big{[}X_{k}=\alpha_{i}\big{]}\bigg{)}\textbf{P}\big{[}X_{k}=\beta\big{]}=\bigg{(}\underset{1\leq
i\leq m-1}{\prod}p\big{(}\alpha_{i}\big{)}\bigg{)}p\big{(}\beta\big{)}\text{
}\text{ , }$

for a suitable transformation $p$ of each $\alpha_{i}$ and $\beta$.

Above, from the final increment of the prudent walker corresponding to
$\beta$, one has,

$\displaystyle p\big{(}\beta\big{)}=\textbf{P}\big{[}\text{ }\text{selecting
}\xi_{l}\text{ }\text{from the iid sequence
}\\{\xi_{1},\cdots,\xi_{k}\\}\text{
}\big{]}=\frac{1}{3}\big{(}\frac{1}{2}\big{)}^{|l|}\text{ }\text{ , }$

The intersection of each $\alpha_{i}$ occurring under $p$ can be expressed as,

$\displaystyle
p\big{(}\alpha_{1},\cdots,\alpha_{n}\big{)}=p\bigg{(}\underset{1\leq i\leq
m-1}{\bigcap}\alpha_{i}\bigg{)}=p\bigg{(}\underset{1\leq i\leq
m-1}{\bigcap}\big{\\{}\xi_{i}=l_{i}\big{\\}}\bigg{)}=p\big{(}\xi_{1}=l_{1},\cdots,\xi_{m-1}=l_{m-1}\big{)}\text{
}\text{ , }$

for increments $\xi_{i}$.

###  2.3 Excursions of the effective triangular walk

Denote the effective triangular walk with $S_{n}$.

To demonstrate that the first Lemma holds, below we argue that each
$\big{\\{}\xi_{i}=l_{i}\big{\\}}$ occurring implies the two desired
expressions for the probabilities on the RHS that are conditioned in terms of
$\gamma_{[0,\mathscr{T}_{k}]}$, and $\gamma_{[0,\mathscr{U}_{k}]}$.

Proof of Lemma 1. Write,

$\displaystyle\underset{\\{\alpha_{1},\cdots,\alpha_{m-1},\beta\\}}{\sum}\bigg{(}\text{
}\underset{i\in\\{\alpha_{1},\cdots,\alpha_{m-1},\beta\\}}{\prod}p\big{(}i\big{)}\text{
}\bigg{)}\text{ }\text{ , }$

corresponding to
$\textbf{P}\big{[}X_{k}=m\big{|}\gamma_{[0,\mathscr{T}_{k}]}\big{]}$. Over
each $\xi_{i}$, the summation of probabilities above over
$\big{\\{}\alpha_{1},\cdots,\alpha_{m}\big{\\}}$ satisfies the identity,

$\displaystyle\underset{i\in\\{\alpha_{1},\cdots,\alpha_{m-1}\\}}{\prod}p\big{(}i\big{)}=p\bigg{(}\underset{1\leq
i\leq m-1}{\bigcap}\big{\\{}\xi_{i}-l_{i}\big{\\}}\bigg{)}\equiv
p\bigg{(}\xi_{1}=l_{1},\cdots,\xi_{m-1}=l_{m-1}\bigg{)}\text{ }\text{ . }$

From a property of the probability measure on the previous page, namely the
expression for the probability mass of $\big{\\{}\xi_{i}\equiv k\big{\\}}$,
putting together the previous two identities implies that the conditional
probability,

$\displaystyle\textbf{P}\big{[}X_{k}=m\big{|}\gamma_{[0,\mathscr{T}_{k}]}\big{]}\text{
}\text{ , }$

can be decomposed as the intersection of all previous steps that the prudent
walker takes from $\big{[}0,\mathscr{T}_{k}\big{]}$ over T, in which,

$\displaystyle\textbf{P}\bigg{[}\bigg{(}\text{ }\underset{i_{1}\leq k\leq
i_{k}}{\bigcap}\big{\\{}X_{k}=m\big{\\}}\text{
}\bigg{)}\bigg{|}\gamma_{[0,\mathscr{T}_{k}]}\bigg{]}\approx\textbf{P}\bigg{[}\bigg{(}\big{\\{}X_{1}=l_{1}\big{\\}}\cap\cdots\cap\big{\\{}X_{\mathscr{T}_{k}}=l_{k}\big{\\}}\bigg{)}\bigg{|}\gamma_{[0,\mathscr{T}_{k}]}\bigg{]}\text{
}\text{ , }$

for,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{1\leq i\leq
m-1}{\bigcup}\big{\\{}S_{i}\in\big{[}0,h-1\big{]}\big{\\}},S_{m}\not\in\big{[}0,h-1\big{]}\text{
}\bigg{)}\text{ }\text{ , }$

which itself can further be expressed as,

$\displaystyle\textbf{P}\big{[}\eta_{X_{\mathscr{T}_{k}}}=m\big{]}\text{
}\text{ . }$

The second identity follows from the same arguments applied to
$\eta_{Y_{\mathscr{U}_{k}}}$ instead of to $\eta_{X_{\mathscr{T}_{k}}}$, from
which we conclude the argument.

For the upcoming result, introduce the event that the triangular random walker
at $\mathscr{T}_{k}$ is at the bottom (resp. top) corner of the box, and is
then at the top (resp. bottom) corner of the box at $\mathscr{U}_{k}$, or that
the triangular walker is at the bottom (resp. top) corner of the box at
$\mathscr{U}_{k}$, and then is at the top (resp. bottom) corner of the box at
$\mathscr{U}_{k}$. We denote this event with $\mathscr{A}_{k}$.

Proposition (the probability of an A type crtossing event occurring is bound
above with a strictly positive function of the box length). With the
quantities defined above, there exists suitable $c_{1}$ for which,

$\displaystyle\textbf{P}\big{(}\mathscr{A}_{k}\big{)}\leq\frac{c_{1}}{k^{\frac{3}{2}}}\text{
}\text{ , }$

given $k$ large enough.

Proof of Proposition. To upper bound
$\textbf{P}\big{(}\mathscr{A}_{k}\big{)}$, write the following superposition,

$\displaystyle\textbf{P}\big{[}\mathcal{H}_{\mathscr{T}_{k}}<c_{1}k^{\frac{3}{2}}\big{]}+\textbf{P}\big{[}\mathscr{A}^{\mathrm{v}}_{k}\big{|}\mathcal{H}_{\mathscr{T}_{k}}\geq
c_{1}k^{\frac{3}{2}}\big{]}\text{ }\text{ , }$

of probabilities, in which the first term is the chance of
$\big{\\{}\mathcal{H}_{\mathscr{T}_{k}}<c_{1}k^{\frac{3}{2}}\big{\\}}$
occurring, while the second term is conditioned on the occurrence of
$\big{\\{}\mathcal{H}_{\mathscr{T}_{k}}\geq c_{1}k^{\frac{3}{2}}\big{\\}}$
occurring. From this superposition, to bound the conditionally defined
probability from above, observe,

$\displaystyle\textbf{P}\big{[}\mathscr{A}^{\mathrm{v}}_{k}\big{|}\mathcal{H}_{\mathscr{T}_{k}}\geq
c_{1}k^{\frac{3}{2}}\big{]}\leq\frac{\big{(}c_{1}\big{)}^{\prime}}{k^{\frac{3}{2}}}\text{
}\text{ , }$

for a suitable $\big{(}c_{1}\big{)}^{\prime}$. The $\frac{1}{k^{\frac{3}{2}}}$
bound holds from the fact that the upper bound,

$\displaystyle\textbf{P}\big{[}\mathscr{A}^{\mathrm{v}}_{k}\big{|}\mathcal{H}_{\mathscr{T}_{k}}\geq
c_{1}k^{\frac{3}{2}}\big{]}\approx\textbf{P}\big{[}\mathscr{A}^{\mathrm{v}}_{k}\big{|}\underset{i\in\textbf{R}:1\leq
i\leq\frac{3}{2}}{\cap}\big{\\{}\mathcal{H}_{\mathscr{T}_{k}}\geq
c_{1}k^{i}\big{\\}}\big{]}\leq\textbf{P}\bigg{(}\mathscr{A}^{\mathrm{v}}_{k}\text{
}\bigg{|}\text{ }\bigg{(}\underset{i}{\mathrm{sup}}\text{
}\big{(}\mathcal{H}_{\mathscr{T}_{k}}:{\mathrm{diam}}\big{(}\mathcal{H}_{\mathscr{T}_{k}}\big{)}\geq
c_{1}k^{i}\big{)}\text{ }\bigg{)}\text{ }\bigg{)}\text{ }\text{ , }$

can be decomposed into the collection of events,

$\displaystyle\textbf{P}\bigg{(}\underset{i\in\textbf{R}:1\leq
i\leq\frac{3}{2}}{\cap}\mathscr{A}^{\mathrm{v}}_{i}\text{
}\bigg{|}\bigg{(}\underset{i}{\mathrm{sup}}\text{
}\big{(}\mathcal{H}_{\mathscr{T}_{k}}:{\mathrm{diam}}\big{(}\mathcal{H}_{\mathscr{T}_{k}}\big{)}\geq
c_{1}k^{i}\big{)}\text{ }\bigg{)}\text{ }\bigg{)}\text{ }\text{ . }$

The collection of events above has the upper bound, given a suitable constant,

$\displaystyle\underset{i\in\textbf{R}:1\leq
i\leq\frac{3}{2}}{\prod}\mathrm{exp}\big{(}-c_{i}\sqrt{k}\big{)}\leq\underset{i\in\textbf{R}:1\leq
i\leq\frac{3}{2}}{\prod}\bigg{(}\text{ }\underset{1\leq
j\leq\frac{3}{2}}{\underset{j\in\textbf{R}:i\neq j}{\mathrm{sup}}}\text{
}\mathrm{exp}\big{(}-c_{i}\sqrt{k}\big{)}\bigg{)}=\bigg{(}\text{
}\underset{1\leq j\leq\frac{3}{2}}{\underset{j\in\textbf{R}:i\neq
j}{\mathrm{sup}}}\text{
}\mathrm{exp}\big{(}-c_{i}\sqrt{k}\big{)}\bigg{)}^{|i\in\textbf{R}:1\leq
i\leq\frac{3}{2}|}\leq\frac{c_{5}}{k^{\frac{3}{2}}}\text{ }\text{ , }$

implying,

$\displaystyle\textbf{P}\big{(}\mathscr{A}_{k}\big{)}-\textbf{P}\big{(}\mathcal{H}_{\mathscr{T}_{k}}<c_{1}k^{\frac{3}{2}}\big{)}\leq\frac{c_{5}}{k^{\frac{3}{2}}}\text{
}\text{ . }$

To demonstrate that the remaining probability that is subtracted from
$\textbf{P}\big{(}\mathscr{A}_{k}\big{)}$ is also of approximately the same
order as $\frac{c_{5}}{k^{\frac{3}{2}}}$, observe that there exists a suitable
constant for which,

$\displaystyle\textbf{P}\big{(}\mathcal{H}_{\mathscr{T}_{k}}<c_{1}k^{\frac{3}{2}}\big{)}\leq\frac{c^{\prime}_{5}}{k^{\frac{3}{2}}}\text{
}\text{ . }$

Hence,

$\displaystyle\textbf{P}\big{(}\mathscr{A}_{k}\big{)}\leq\textbf{P}\big{[}\mathcal{H}_{\mathscr{T}_{k}}<c_{1}k^{\frac{3}{2}}\big{]}+\textbf{P}\big{[}\mathscr{A}^{\mathrm{v}}_{k}\big{|}\mathcal{H}_{\mathscr{T}_{k}}\geq
c_{1}k^{\frac{3}{2}}\big{]}\leq\big{(}c_{5}+c^{\prime}_{5}\big{)}\frac{1}{k^{\frac{3}{2}}}\leq\frac{c^{\prime\prime}_{5}}{k^{\frac{3}{2}}}\equiv\frac{c_{1}}{k^{\frac{3}{2}}}\text{
}\text{ , }$

from which we conclude the argument.

The final result of the section below provides a lower bound for the
probability that the prudent walker takes an excursion of length $n$, where
$n_{L}\geq n$.

Lemma 1.5 (gambler’s ruin estimate). Fix some $L\geq 1$. One has that,

$\displaystyle\textbf{P}\big{(}\eta_{L}\geq n\big{)}\geq
c\big{(}n\big{)}\text{ }\text{ , }$

for all integers $n$ satisfying,

$\displaystyle n\leq cL^{\frac{3}{2}}\text{ }\text{ , }$

for suitable $c\equiv c\big{(}n\big{)}$.

Proof of Lemma 1.5. From the definition of $\eta_{L}$ provided in 2.1, as
demonstrated in the arguments for Lemma 3.1 in [1], introduce a similar
quantity,

$\displaystyle\vec{\eta_{L}}\equiv\mathrm{inf}\big{\\{}n>0:S_{n}\geq
L-1\big{\\}}\text{ }\text{ , }$

from which one can obtain the desired lower bound for
$\textbf{P}\big{(}\eta_{L}\geq n\big{)}$ by observing that,

$\displaystyle\textbf{P}\big{(}\eta_{L}\geq
n\big{)}\equiv\textbf{P}\bigg{(}\text{ }\big{\\{}\eta_{\infty}\geq
n\big{\\}},\big{\\{}\vec{\eta_{L}}\geq n\big{\\}}\text{
}\bigg{)}\equiv\textbf{P}\big{(}\eta_{\infty}\geq
n\big{)}-\textbf{P}\bigg{(}\big{\\{}\vec{\eta_{L}}\geq
n\big{\\}},\big{\\{}\eta_{\infty}\geq n\big{\\}}\bigg{)}$
$\displaystyle\geq\textbf{P}\big{(}\eta_{\infty}\geq
n\big{)}-\textbf{P}\big{(}\vec{\eta_{L}}<n\big{)}\text{ }\text{ , }$

where in the first equality for $\textbf{P}\big{(}\eta_{L}\geq n\big{)}$,
$\eta_{\infty}$ denotes the infinite volume limit of $\eta_{L}$ as
$L\longrightarrow+\infty$, which is given by,

$\displaystyle\eta_{\infty}\equiv\mathrm{inf}\big{\\{}n>0:S_{n}\geq+\infty\big{\\}}\text{
}\text{ . }$

From previous results on the gambler’s run estimate mentioned in [1], one
obtains the desired estimate by lower bounding each of the probabilities from
the observation above, with the upper bound for the second term obtained by
the Doob-Kolmogorov Inequality, from which we conclude the argument.

###  2.4 Upper bounds on the probabilities of $\mathcal{W}_{\mathscr{T}_{k}}$
and of $\mathcal{H}_{\mathscr{T}_{k}}$

Below, it is shown that the probability of $\mathcal{W}_{\mathscr{T}_{k}}$,
and of $\mathcal{H}_{\mathscr{T}_{k}}$ exceeding some constant times a
polynomial factor of the triangular box length is exponentially small.

Lemma 2 (exponentially small upper bounds). There exists two constants for
which,

$\displaystyle\textbf{P}\big{(}\mathcal{W}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{)}\leq\mathrm{exp}\big{(}-c_{3}k^{\frac{3}{2}}\big{)}\text{
}\text{ , }$
$\displaystyle\textbf{P}\big{(}\mathcal{H}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{)}\leq\mathrm{exp}\big{(}-c_{3}k^{\frac{3}{2}}\big{)}\text{
}\text{ , }$

in which
$\big{\\{}\mathcal{W}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{\\}}$, and
$\big{\\{}\mathcal{H}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{\\}}$,
occurring are each bounded above by an exponentially decaying constant.

Proof of Lemma 2. Set,

$\displaystyle m=\lfloor\frac{k}{2}\rfloor\text{ }\text{ . }$

The probability of
$\big{\\{}\mathcal{W}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{\\}}$
occurring, from the first equality, can be bounded above as desired from the
fact that,

$\displaystyle\textbf{P}\big{(}\text{
}\big{|}\mathcal{I}_{j}\big{|}=1\big{|}\gamma_{[0,\mathscr{T}_{j}]}\text{
}\big{)}=\textbf{P}\big{(}\text{
}\big{|}\mathcal{I}_{j}\big{|}=1\big{|}\mathcal{H}_{\mathscr{T}_{j}}\text{
}\big{)}=\textbf{P}\big{(}\eta_{\mathcal{H}_{\mathscr{T}_{j}}}\leq
c_{2}k^{\frac{3}{2}}\big{)}\geq\textbf{P}\big{(}\eta_{m}\geq
c_{2}k^{\frac{3}{2}}\big{)}$ $\displaystyle=\textbf{P}\bigg{(}\text{
}\underset{i\in\textbf{R}:1\leq
i\leq\frac{3}{2}}{\bigcap}\big{\\{}\eta_{m}\geq c_{2}k^{i}\big{\\}}\text{
}\bigg{)}$ $\displaystyle=\underset{i\in\textbf{R}:1\leq
i\leq\frac{3}{2}}{\prod}\textbf{P}\big{(}\eta_{m}\geq c_{2}k^{i}\big{)}$
$\displaystyle\geq\bigg{(}\text{ }\underset{i}{\mathrm{sup}}\text{
}\textbf{P}\big{(}\eta_{m}\geq
c_{2}k^{i}\big{)}\bigg{)}^{|i\in\textbf{R}:1\leq i\leq\frac{3}{2}|}$
$\displaystyle\geq\mathrm{exp}\big{(}-\sqrt{c_{1}}k^{3}\big{)}\text{ }\text{ ,
}$ (*)

for,

$\displaystyle m\geq\mathcal{H}_{\mathscr{T}_{j}}\text{ }\text{ , }$

at $\mathscr{T}_{j}$, and the region $\mathcal{I}_{j}$ explored up to time
$\mathscr{T}_{j}$ in the path $\gamma_{[0,\mathscr{T}_{j}]}$, which in turn
implies,

$\displaystyle\textbf{P}\big{(}\mathcal{W}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{)}\leq
1-\textbf{P}\big{(}\text{
}\big{|}\mathcal{I}_{j}\big{|}=1\big{|}\mathcal{H}_{\mathscr{T}_{j}}\text{
}\big{)}\overset{(*)}{\leq}\bigg{(}1-\mathrm{exp}\big{(}-\sqrt{c_{1}}k^{3}\big{)}\bigg{)}^{m}\approx\bigg{(}-\text{
}\underset{n\geq
0}{\sum}\frac{\big{(}-1\big{)}^{n}}{n!}\big{(}\sqrt{c_{1}}k^{3}\big{)}^{n}\text{
}\bigg{)}^{m}$ $\displaystyle\leq\bigg{(}-\text{ }\underset{n\geq
0}{\sum}\frac{\big{(}-1\big{)}^{n}}{n!}\big{(}c_{3}k^{\frac{3}{2}}\big{)}^{n}\text{
}\bigg{)}^{m}$
$\displaystyle\approx\mathrm{exp}\big{(}-c_{3}k^{\frac{3}{2}}\big{)}\text{
}\text{ . }$

Pursuing the same line of argument implies the second desired inequality,

$\displaystyle\textbf{P}\big{(}\mathcal{H}_{\mathscr{T}_{k}}<c_{2}k^{\frac{3}{2}}\big{)}\leq\mathrm{exp}\big{(}-c_{3}k^{\frac{3}{2}}\big{)}\text{
}\text{ , }$

from which we conclude the argument.

###  2.5 Corners process coupling

First, denote,

$\displaystyle
m^{\mathrm{v}}_{k}=\hat{\gamma_{\mathscr{T}_{k-1}}}\big{(}2\big{)}-\underset{\mathscr{T}_{k-1}\leq
t\leq\mathscr{U}_{k-1}}{\mathrm{min}}\hat{\gamma_{t}}\big{(}2\big{)}\text{
}\text{ , }$ $\displaystyle
m^{\mathrm{h}}_{k}=\hat{\gamma_{\mathscr{U}_{k-1}}}\big{(}1\big{)}-\underset{\mathscr{U}_{k-1}\leq
t\leq\mathscr{T}_{k}}{\mathrm{min}}\hat{\gamma_{t}}\big{(}1\big{)}\text{
}\text{ , }$

corresponding to the vertical, and horizontal, trajectories of the corners
process $\hat{\gamma}$. Also, introduce,

$\mathcal{E}^{\mathrm{v}}_{1}\equiv\text{
}\left\\{\\!\begin{array}[]{ll@{}>{{}}l}\hat{\mathcal{E}^{\mathrm{v}}_{1}}\text{
}\text{if}\text{ }m^{\mathrm{v}}_{1}\leq H_{0}\text{ , }\\\
\mathrm{Trunc}_{H_{0}}\big{(}\hat{\mathcal{E}^{\mathrm{v}}_{1}}\big{)}\text{
}\text{if}\text{ }m^{\mathrm{v}}_{1}>H_{0}\text{ . }\\\ \end{array}\right.$

In the piecewise defined function above, the truncation of the
$\mathcal{E}^{\mathrm{v}}_{1}$, as explorations over T, satisfy,

$\displaystyle\mathrm{Trunc}_{H_{0}}\big{(}\hat{\mathcal{E}^{\mathrm{v}}_{1}}\big{)}=\hat{\mathcal{E}^{\mathrm{v}}_{1}}\cap
H_{0}\text{ }\text{ , }$

for,

$\displaystyle H_{0}\equiv 0\text{ }\text{ . }$

Instead of the special cases for $\mathcal{E}$ above, for general $k+1$,
$\mathcal{E}$ takes the form,

$\mathcal{E}^{\mathrm{v}}_{k+1}\equiv\text{
}\left\\{\\!\begin{array}[]{ll@{}>{{}}l}\hat{\mathcal{E}^{\mathrm{v}}_{k+1}}\text{
}\text{if}\text{ }m^{\mathrm{v}}_{1}\leq H_{k}\text{ , }\\\
\mathrm{Trunc}_{H_{k}}\big{(}\hat{\mathcal{E}^{\mathrm{v}}_{k+1}}\big{)}\text{
}\text{if}\text{ }m^{\mathrm{v}}_{1}>H_{k}\text{ . }\\\ \end{array}\right.$

Instead of truncations taken with respect to $H_{0}$, truncation with respect
to $H_{k}$, have the following action,

$\displaystyle\mathrm{Trunc}_{H_{k}}\big{(}\hat{\mathcal{E}^{\mathrm{v}}_{k+1}}\big{)}=\hat{\mathcal{E}^{\mathrm{v}}_{k+1}}\cap
H_{k}\text{ }\text{ , }$

for,

$\displaystyle H_{k}\equiv\big{\\{}k\in\textbf{Z}:H_{0}\geq k,\cdots,H_{k}\geq
k\big{\\}}\text{ }\text{ . }$

Similarly, with respect to horizontal excursions of the triangular prudent
walker, from analogous definitions of $\mathcal{E}^{\mathrm{h}}_{1}$ and
$\mathcal{E}^{\mathrm{rh}}_{k+1}$, which takes the form,

$\mathcal{E}^{\mathrm{h}}_{k+1}\equiv\text{
}\left\\{\\!\begin{array}[]{ll@{}>{{}}l}\hat{\mathcal{E}^{\mathrm{h}}_{k+1}}\text{
}\text{if}\text{ }m^{\mathrm{h}}_{1}\leq W_{k}\text{ , }\\\
\mathrm{Trunc}_{W_{k}}\big{(}\hat{\mathcal{E}^{\mathrm{h}}_{k+1}}\big{)}\text{
}\text{if}\text{ }m^{\mathrm{h}}_{1}>W_{k}\text{ . }\\\ \end{array}\right.$

for,

$\displaystyle W_{k}\equiv\text{ }\big{\\{}k\in\textbf{Z}:W_{0}\geq
k,\cdots,W_{k}\geq k\big{\\}}\text{ , }$

given some initial $W_{0}>0$.

The coupling with the corners process ensures,

$\displaystyle\textbf{P}\bigg{(}\text{
}\big{\\{}\mathcal{E}^{\mathrm{v}}_{k+1}\overset{\mathrm{a.s.}}{=}\hat{\mathcal{E}^{\mathrm{v}}_{k+1}}\big{\\}},\big{\\{}\mathcal{E}^{\mathrm{h}}_{k+1}\overset{\mathrm{a.s.}}{=}\hat{\mathcal{E}^{\mathrm{h}}_{k+1}}\big{\\}}\text{
}\bigg{)}=1\text{ }\text{ . }$

###  2.6 The conditionally defined probabilities from stationary behavior of
the walker’s trajectory

Conditionally upon the triangular walker settling in the first quadrant of T,
the result below states that the probability of $\big{\\{}\underset{0\leq
s\leq
t}{\mathrm{sup}}\big{|}\big{|}\frac{1}{t}\big{(}\hat{\gamma_{s}}-\gamma_{s}\big{)}\big{|}\big{|}_{2}\geq\epsilon\big{\\}}$
is zero, for arbitrarily small $\epsilon$.

Theorem (the probability of the supremum exceeding some sufficiently small
parameter, conditional upon the prudent walker settling in the first quadrant
of the triangular lattice, is zero). Denote $\mathcal{Q}_{1}$ as the event
that the prudent walker’s last step of an excursion is in the first quadrant
of the triangular lattice. For $\epsilon$ sufficiently small, one has that,

$\displaystyle\underset{t\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq s\leq
t}{\mathrm{sup}}\bigg{|}\bigg{|}\frac{1}{t}\big{(}\hat{\gamma_{s}}-\gamma_{s}\big{)}\bigg{|}\bigg{|}_{2}\geq\epsilon\text{
}\big{|}\text{ }\mathcal{Q}_{1}\text{ }\bigg{)}\text{ }\text{ , }$

vanishes.

Proof of Theorem. Given some $\gamma$, if the supremum of the $\mathrm{l}$-2
norm between $\hat{\gamma_{s}}$ and $\gamma_{s}$ exceeds $\epsilon$, then
there exists a coupling between $\hat{\gamma_{s}}$ and $\gamma_{s}$ for which:

  * $\bullet$

Property 1: There exists a mapping between $\hat{\gamma_{s}}$ and $\gamma_{s}$
for which,

$\displaystyle\phi:\hat{\gamma_{s}}\longrightarrow\gamma_{s}:\text{
}\text{there exists countably many }k\text{ }\text{for which
}\big{\\{}\hat{\gamma_{s}}\neq\gamma_{s}\big{\\}}\text{ }\text{ . }$

  * $\bullet$

Property 2: Dependent upon the occurence of the event $\mathcal{Q}_{1}$,

$\displaystyle\frac{\textbf{P}\bigg{(}\big{\\{}\underset{0\leq s\leq
t}{\mathrm{sup}}\big{|}\big{|}\frac{1}{t}\big{(}\hat{\gamma_{s}}-\gamma_{s}\big{)}\big{|}\big{|}_{2}\geq\epsilon\big{\\}}\cap\big{\\{}\mathcal{Q}_{1}\big{\\}}\bigg{)}}{\textbf{P}\big{(}\mathcal{Q}_{1}\big{)}}\leq\epsilon\text{
}\text{ . }$

  * $\bullet$

Property 3. Almost surely, there exists a coupling between paths for which,

$\displaystyle\underset{t\geq
0}{\mathrm{sup}}\big{|}\big{|}\hat{\gamma_{t}}-\gamma_{t}\big{|}\big{|}_{1}<+\infty\text{
}\text{ . }$

With such properties of the two paths, the supremum of the $\mathrm{l}$-2 norm
is almost surely finite, implying that the desired result holds, from which we
conclude the argument.

###  2.7 Triangular scaling limit

In infinite time, the following probability below also vanishes.

Theorem 2 (the supremum of the l2 norm between the time re-weighted path and
the normalizing constant exceeding an arbitrarily small parameter vanishes).
With previously defined quantities, one has that,

$\displaystyle\underset{t\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq s\leq
t}{\mathrm{sup}}\bigg{|}\bigg{|}\frac{1}{t}\hat{\gamma_{s}}-Z^{1,1,1}_{u,\alpha}\bigg{|}\bigg{|}_{2}\geq\epsilon\text{
}\bigg{)}\text{ }\text{ , }$

vanishes.

To demonstrate that the above result holds, we will introduce, and manipulate,
the initial time,

$\displaystyle\tau_{0}\equiv 0\text{ }\text{ , }$

with the following two stopping times for $S$, with,

$\displaystyle\tau_{2k+1}\equiv\big{\\{}n>\tau_{2k}:S_{n}<S_{\tau_{2k}}\big{\\}}\text{
}\text{ , }$
$\displaystyle\tau_{2k+2}\equiv\big{\\{}n>\tau_{2k+1}:S_{n}>S_{\tau_{2k+1}}\big{\\}}\text{
}\text{ . }$

At any instance of time, $S_{n}$ and $S_{\tau_{2k_{1}}}$, or $S_{n}$ and
$S_{\tau_{2k+2}}$ can have overshoots with each other depending upon whether
the horizontal/vertical displacement of one random walk exceeds that of the
other, which can be computed by taking,

$\displaystyle\Delta_{2k+1}=-1-\big{(}S_{\tau_{2k+1}}-S_{\tau_{2k}}\big{)}\text{
}\text{ , }$
$\displaystyle\Delta_{2k+2}=1-\big{(}S_{\tau_{2k+2}}-S_{\tau_{2k+1}}\big{)}\text{
}\text{ . }$

###  2.8 Probability of $\big{|}S_{k}-\hat{S_{k}}\big{|}$ exceeding a
threshold

The result below determines the order of deviation between $S_{k}$ and
$\hat{S_{k}}$.

Lemma 3 (vanishing probability as n approaches infinity). For any $\delta>0$,
the probability,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq
n^{\frac{1}{3}+\delta}\text{ }\bigg{)}\text{ }\text{ , }$

vanishes.

Proof of Lemma 3. To prove the claim, it suffices to demonstrate the existence
of some threshold, $N$, for which,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq N\bigg{)}=0\text{ }\text{
, }$

beyond which it will also hold that,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq
n^{\frac{1}{2}+\delta}\bigg{)}=0\text{ }\text{ , }$

given some $m$, with,

$\displaystyle n\approx\frac{1}{N^{\frac{1}{3}+\delta}}\text{ }\text{ . }$

The first statement above can be phrased in terms of the observation that,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq N^{\prime}\text{
}\bigg{)}=0\text{ }\text{ , }$

for some strictly positive $N^{\prime}$, then necessarily,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq N^{\prime\prime}\text{
}\bigg{)}=0\text{ }\text{ , }$

for some $N^{\prime\prime}>N^{\prime}$.

To demonstrate that the claim above holds, observe,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq
n^{\frac{1}{3}+\delta}\text{
}\bigg{)}\leq\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq
L\text{ }\bigg{)}$
$\displaystyle\leq\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq L\text{
}\bigg{)}$
$\displaystyle\equiv\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\big{\\{}\underset{0\leq
k\leq n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq
L\big{\\}}\cap\big{\\{}\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq\frac{L}{2}\big{\\}}\bigg{)}$
$\displaystyle\leq\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq L\text{
}\bigg{)}+\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq\frac{L}{2}\text{ }\bigg{)}$
$\displaystyle\leq\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq L\text{
}\bigg{)}+\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq\frac{L}{6}\text{ }\bigg{)}$
$\displaystyle\equiv\underset{n\longrightarrow+\infty}{\mathrm{lim}}\bigg{(}\text{
}\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq L\text{
}\bigg{)}+\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq\frac{L}{6}\text{ }\bigg{)}\text{
}\bigg{)}$
$\displaystyle\leq\underset{n\longrightarrow+\infty}{\mathrm{lim}}\bigg{(}\text{
}\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq L\text{
}\bigg{)}+\textbf{P}\bigg{(}\text{
}N_{0}\big{(}n\big{)}\geq\bigg{(}\frac{L}{6}\bigg{)}^{2}\text{ }\bigg{)}\text{
}\bigg{)}\text{ }\text{ . }$

for $L$ sufficiently large given some choice of $\delta$, where $\hat{S_{k}}$
is a martingale, and $N_{0}\big{(}n\big{)}$ denotes the number of edges
surrounding the origin of T contained within the box of side length $k$. From
reasons previously described in the result of Lemma 4.1 of [1], each of the
probabilities in the limit as $n\longrightarrow+\infty$ above converge to $0$,
in which,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}\hat{S_{k}}\big{|}\geq L\text{ }\bigg{)}\longrightarrow
0\text{ }\text{ , }$

and also that,

$\displaystyle\textbf{P}\bigg{(}\text{
}N_{0}\big{(}n\big{)}\geq\bigg{(}\frac{L}{6}\bigg{)}^{2}\text{
}\bigg{)}\longrightarrow 0\text{ }\text{ , }$

hence implying,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq
n^{\frac{1}{3}+\delta}\text{ }\bigg{)}\leq
0\Longleftrightarrow\textbf{P}\bigg{(}\text{ }\underset{0\leq k\leq
n}{\mathrm{sup}}\big{|}S_{k}-\hat{S_{k}}\big{|}\geq
n^{\frac{1}{3}+\delta}\text{ }\bigg{)}\equiv 0\text{ }\text{ , }$

as $n\longrightarrow+\infty$, from which we conclude the argument.

###  2.9 Probability that the maximum of the difference exceeding an
arbitrarily small constant vanishes

Lemma 4 (vanishing probability statement). Fix some $\delta>0$ sufficiently
small, and another constant $\epsilon$ which can also be taken sufficiently
small. One has that,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{1\leq k\leq n}{\mathrm{sup}}\bigg{|}\text{
}\frac{1}{n}\bigg{(}\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{\widetilde{S_{i}}\geq 0\\}}-\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{S_{i}\geq
n^{\frac{1}{3}}+\delta\\}}\bigg{)}\bigg{|}>\epsilon\text{ }\bigg{)}\text{
}\text{ , }$

and that,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{1\leq k\leq n}{\mathrm{sup}}\bigg{|}\text{
}\frac{1}{n}\bigg{(}\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{\widetilde{S_{i}}<0\\}}-\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{S_{i}<-n^{\frac{1}{3}}+\delta\\}}\bigg{)}\bigg{|}>\epsilon\text{
}\bigg{)}\text{ }\text{ , }$

each vanish.

Proof of Lemma 4. As $n\longrightarrow+\infty$ each statement holds from the
fact that the probability that number of $i$ for which the magnitude of each
$S_{i}$ is strictly less than some fractional power of $n$ vanishes. This is
given by the statement,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{1\leq i\leq n}{\sum}\textbf{1}_{\\{|S_{i}|\leq
n^{\frac{1}{3}+\delta}\\}}\geq n^{\frac{2}{3}+2\delta}\text{ }\bigg{)}=0\text{
}\text{ . }$

From the previous result the supremum of the overlap between $S_{k}$ and
$\hat{S_{k}}$, $\big{|}S_{k}-\hat{S_{k}}\big{|}$, implies,

$\displaystyle\textbf{P}\bigg{(}\text{ }\underset{1\leq k\leq
n}{\mathrm{sup}}\bigg{|}\text{ }\frac{1}{n}\bigg{(}\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{\hat{S_{i}}\geq 0\\}}-\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{S_{i}\geq n^{\frac{1}{3}+\delta}\\}}\bigg{)}\text{
}\bigg{|}>\epsilon\text{ }\bigg{)}$ $\displaystyle\leq\textbf{P}\bigg{(}\text{
}\underset{\epsilon n\leq k\leq n}{\mathrm{sup}}\bigg{|}\text{
}\frac{1}{n}\bigg{(}\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{\hat{S_{i}}\geq 0\\}}-\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{S_{i}\geq n^{\frac{1}{3}+\delta}\\}}\bigg{)}\text{
}\bigg{|}>\frac{\epsilon}{2}\text{ }\bigg{)}\text{ }\text{ , }$

for,

$\displaystyle n<\epsilon\text{ }\text{ . }$

Therefore, the infinite limit of each of the two probabilities above vanish,
from which we conclude the argument.

In the remaining results below before returning to the proof of Theorem 2,
there exists a probability space for which the supremum of the difference
between $S_{k}$ and $\sigma B_{k}$ exceeds $n^{\frac{1}{3}}$ is exponentially
small, for which,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{k\leq n}{\mathrm{sup}}\big{|}\hat{S_{k}}-\sigma
B_{k}\big{|}>n^{\frac{1}{3}+\delta}\text{ }\bigg{)}\text{ }\text{ . }$

Following the statement of this result, the remaining two results state that
the probability of two other conditions occurring also vanishes; the first of
the conditions relates the l-2 distance between $\hat{\gamma_{t(m)}}$ and
$\Gamma_{m}$, while the second of the conditions relates the l-1 distance
between $t\big{(}m\big{)}$ and $\alpha m$.

Lemma 5 (probability of the supremum exceeding epsilon vanishes). Fix some
integer $k$. One has that,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{k\leq n}{\sup}\text{ }\bigg{|}\text{ }\frac{1}{n}\bigg{(}\text{
}\int_{0}^{k}\textbf{1}_{\\{B_{s}\geq 0\\}}\mathrm{d}s-\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{S_{i}\geq n^{\frac{1}{3}+\delta}\\}}\text{
}\bigg{)}\text{ }\bigg{|}>\epsilon\text{ }\bigg{)}\text{ }\text{ , }$

and,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{k\leq n}{\sup}\text{ }\bigg{|}\text{ }\frac{1}{n}\bigg{(}\text{
}\int_{0}^{k}\textbf{1}_{\\{B_{s}<0\\}}\mathrm{d}s-\underset{1\leq i\leq
k}{\sum}\textbf{1}_{\\{S_{i}<n^{-\frac{1}{3}+\delta}\\}}\text{ }\bigg{)}\text{
}\bigg{|}>\epsilon\text{ }\bigg{)}\text{ }\text{ , }$

vanish.

Proof of Lemma 5. Apply the argument of Lemma 4.3 in [1].

Lemma 6 (supremum exceeding $\epsilon$ occurs with zero probability). For any
$\epsilon>0$,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq m\leq
n}{\mathrm{sup}}\bigg{|}\bigg{|}\frac{1}{n}\bigg{(}\hat{\gamma_{t(m)}}-\Gamma_{m}\bigg{)}\bigg{|}\bigg{|}_{2}>\epsilon\text{
}\bigg{)}\text{ }\text{ , }$

vanishes.

Proof of Lemma 6. Directly apply the argument from Lemma 4.4 in [1].

Lemma 7 (supremum exceeding $\epsilon n$ occurs with zero probability). For
any $\epsilon>0$,

$\displaystyle\underset{n\longrightarrow+\infty}{\mathrm{lim}}\textbf{P}\bigg{(}\text{
}\underset{0\leq m\leq n}{\mathrm{sup}}\big{|}t\big{(}m\big{)}-\alpha
m\big{|}>\epsilon n\text{ }\bigg{)}\text{ }\text{ , }$

vanishes.

Proof of Lemma 7. Directly apply the argument of Lemma 4.5 in [1] with
$\alpha$ in place of $\frac{7}{3}$.

We return to the proof of the final remaining item with the results from this
section.

Proof of Theorem 2. From the definition of the triangular renormalizing
constant, write,

$\displaystyle\hat{S_{n}}\equiv S_{n}+\underset{j\geq
0}{\sum}\Delta_{j}\textbf{1}_{\\{\tau_{j}\leq n\\}}\text{ }\text{ , }$

for the object $\hat{S_{n}}$ obtained from $S_{n}$ by adding on a summation
term of $\Delta_{j}\textbf{1}_{\\{\tau_{j}\leq n\\}}$ for all $j\geq 0$.
Equipped with the three previous results, to demonstrate that Theorem 2 holds,
apply the results from Lemma 4, Lemma 5, Lemma 6, and Lemma 7, from which we
conclude the argument.

##  3 References

[1] Beffara, V., Friedli, S., Velenik, Y. Scaling Limit of the Prudent Walk.
Electron. Comm. Probab. 15: 44-58 (2010).

[2] Bousquet-Melou, M. Families of prudent self-avoiding walks. Journal of
Combinatorial Theory, Series A 117: 313-344 (2010).

[3] Bousquet-Melou, M. Lattice paths and random walks. Course lecture notes.

[4] Petrelis, N., Sun, R., Torri, N. Scaling limit of the uniform prudent
walk. Elec. J. Probab. 22: 1-19 (2017).