be subdivided into two parts: for small $H$, the $\alpha^{4/3}\eps^{2/3}LH^{1/3}$ term dominates (regime VB$_1$), for large $H$, the $\alpha^4LH$ term dominates (regime VB$_2$).

5.  As in the case of phase HL, if $L$ is very small, then a long vertical interface will dominate.

The boundaries between the different regions can be easily obtained from (\[eqdefFtheo1\]), and are sketched in Figure \[fig:PD1\].

![Schematic phase diagram in case $K_2$, in the $L/\eps$, $H/\eps$ plane.[]{data-label="fig:PD3"}](figPD3-crop){width="11cm"}

In the case of a single rank-one connection the energy scaling is different, and the phase diagram simpler, see Figure \[fig:PD3\]. Here, only the three regimes A, HL and BR play a role.

\[Theorem:K2\] There is $c>0$ such that, under the assumptions (\[assumption:K2\]), (\[assumption:1\]), (\[assumption:2\]) and (\[assumption:E\]), and for $\alpha<1/2$, one has $$\frac{1}{c}g(\alpha,\eps,L,H) \leq \min_{u\in\calM} E_2^\eps[u,\Omega]  \leq c g(\alpha,\eps,L,H) \,,$$ where $$g(\alpha,\eps,L,H)=\min\Bigl\{\alpha^{6/5}\varepsilon^{4/5}L^{1/5}H+\alpha\eps L
,\alpha^2 LH\Bigr\}\,.$$ The constant $c$ does not depend on $\alpha$, $\varepsilon$, $H$ and $L$.

Follows from Lemma \[lemmaUBK2\] and Lemma \[lemmaLBK2\].

The situation in which the dependence on $\alpha$ is not analyzed is substantially simpler, in particular for the lower bounds. Indeed, a result similar to Theorem \[Theorem:K1\] without dependence on $\alpha$ and with $H=L$ was first proven in [@B9_Chan; @B9_ContiOWJun2007]. These results are discussed 