I’ve been studying a proof of the discrete Liouville theorem (see Jack D’Aurizio’s answer) which proceeds as follows, as I understand it:
Let
$$ \Gamma_N =\max_{|m|+|n|\le N} f(m,n), $$
where $f\colon\Bbb Z^2\to\Bbb R$ is discrete harmonic and bounded. $f(0,0)=0$. $\Gamma_N$ is non‑decreasing in $N$.
The maximum on $B_N$ is attained on the boundary $\partial B_N = \{(m, n) : |m| + |n| = N\}$, $$ \Gamma_N =\max_{|m|+|n|=N} f(m,n), $$ 2. Consider a point $(m, n) \in \partial B_N$. The neighbors fall into $B_{N-1}$ or $B_{N+1}$.
- For "interior" boundary points (not vertices), two neighbors are in $B_{N-1}$ and two in $B_{N+1}$, so $$ f(m, n) \le \frac{2\Gamma_{N+1} + 2\Gamma_{N-1}}{4} = \frac{\Gamma_{N+1} + \Gamma_{N-1}}{2}. $$
- For vertex points (e.g., $(N, 0)$), three neighbors are in $B_{N+1}$ and one in $B_{N-1}$, so
$$
f(m, n) \le \frac{3\Gamma_{N+1} + \Gamma_{N-1}}{4}.
$$
Since $\Gamma_{N+1} \ge \Gamma_{N-1}$ and the maximum may be at a vertex, the controlling bound is the weaker one: $$ \Gamma_N \le \frac{3\Gamma_{N+1} + \Gamma_{N-1}}{4}. $$ Rearranging gives $$ \Gamma_{N+1} - \Gamma_N \ge \frac{1}{3} (\Gamma_N - \Gamma_{N-1}). $$ Let $d_N = \Gamma_N - \Gamma_{N-1} \ge 0$ (with $\Gamma_0 = 0$, so $d_1 = \Gamma_1$). Then $d_{N+1} \ge \frac{1}{3} d_N$, and by induction, $$ d_{N+1} \ge \frac{\Gamma_1}{3^N}. $$
Summing the increments, $$ \Gamma_N = \sum_{k=1}^N d_k \ge \sum_{k=1}^N \frac{\Gamma_1}{3^{k-1}} = \Gamma_1 \frac{1 - (1/3)^N}{1 - 1/3} = \frac{3}{2} \Gamma_1 \left(1 - \frac{1}{3^N}\right). $$ If $\Gamma_1 > 0$, then $\Gamma_N \ge \frac{3}{2} \Gamma_1 (1 - 1/3^N)$, approaching $\frac{3}{2} \Gamma_1$ as $N \to \infty$, so for any $\epsilon>0$ we can find a point $(a_1,b_1)$ such that $f(a_1,b_1)\geq\frac{3}{2}\Gamma_1-\varepsilon>\frac{10}{7}\Gamma_1.$
This contradicts boundedness of $f$, and hence $f$ constant.
My doubt: the total possible increase $\tfrac32\Gamma_1$ is finite, so there is no obvious contradiction with $f$ being bounded (indeed, a sequence satisfying the same recurrence would converge to $1.5\Gamma_1$, exactly as @Labo pointed out). Why does D’Aurizio conclude unbounded growth rather than merely bounded growth up to $\tfrac32\Gamma_1$?
Could someone clarify why these finite geometric‑series increments still force a contradiction in his argument? Is there a subtlety or additional step that ensures $\Gamma_N$ cannot stabilize within a finite bound?