Large-scale rotational invariance in lattice space
It is often claimed among physicists that rotational invariance can emerge at large scales in lattice space. Let's focus on quantum mechanics for now. I interpret this claim as follows (I am a philosopher, so I will be relatively wordy and not technically sophisticated but I focus on conceptual clarity):
Claim. Suppose space is discrete and every point can be represented by members of $\mathbb{Z}^n$. For any quantum mechanical system $\alpha$ with initial position $x\in\mathbb{Z}^n$, for any $y,z\in\mathbb{Z}^n$ that have approximately the same Euclidean distance to $x$, the amplitudes of $\alpha$ at $y$ and $z$ are approximately the same when the Euclidean distance in question is sufficiently large.
Question: is this claim true? If it is true, what is the proof? If it is false, is there a true claim in its vicinity? I calculate the wavefunction in this lattice space, and cannot infer this claim. In addition, I did some numerical evaluations at Wolframalpha and the result suggests that the Claim is false. Here're my main steps:
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Assume that $$ \frac{\partial}{\partial x^i}\Psi (t,x) =\Psi(t, x+e_i)-\Psi(t,x) \textrm{ or} =\Psi(t,x)-\Psi(t, x-e_i)$$
2. Start from the <PERSON> equation in lattice space. The discrete Hamiltonian is $$\hat{H}\Psi(t,x)=-\frac{1}{2}\sum_i(\Psi(t, x+e_i)+\Psi(t,x-e_i)-2\Psi(t,x)).$$ Here, $\Psi$ is the position space wavefunction: $\mathbb{Z}^n\to\mathbb{C}.$
3. Its fourier series in the momentum space is: $$\tilde{\Psi}(t,p)= \sum_{x\in\mathbb{Z}^n} e^{-2\pi i p x} \Psi (t,x) $$ with $\tilde{\Psi}(t): \mathbb{R}^n\to \mathbb{C}$.
4. Since $\tilde{\Psi}(t)$ is periodical, we can just represent its domain by $B=[-1/2,1/2]\times [-1/2,1/2]$. The momentum space Hamiltonian is: $$\tilde{H}\Psi(t,p)=-\frac{1}{2}\sum_i(e^{2\pi i p_i}+e^{-2\pi i p_i}-2)\tilde{\Psi}(t,p)$$ with $p_i=p\cdot e_i$.
5. Solve the <PERSON> equation in the momentum space, we have $$\tilde{\Psi} (t,p)=(e^{-i\tilde{H}t}\tilde{\Psi})(p)=e^{-i\mathbf{Cow}(p)t}\tilde{\Psi}(0,p)$$ where $\mathbf{Cow}(p)=-\frac{1}{2}\sum_i(e^{2\pi i p_i}+e^{-2\pi i p_i}-2)$. Transform it back to the position space: $$\Psi (t,x)=\int_{p\in B} e^{2\pi i px}e^{-i\mathbf{Cow}(p)t}\tilde{\Psi}(0,p)dp=\int_{p\in B} e^{2\pi i px}e^{-i\mathbf{Cow}(p)t}dp$$
(the above integral seems to be equal to $e^{-nit}\prod_nI_{x_n}(it)$ where $I$ is modified Bessel function.)
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