Definition related to homomorphic encryption In the definition of leveled FHE scheme, what is the mining of same decryption circuit?
I have read several papers about this, but I don't understand this notion.

In addition, if the scheme {$E^d$} is leveled FHE (notation based on Gentry thesis), why the scheme $E$ works correctly for circuits of some depth?
I believe you are talking about the accumulative noise associated with performing gate operations on encrypted bits.

When looking at FHE over the integers (without modular reduction), it is easy to track the noise associated with operations. 

In the 'AND' operation :
$$\begin{align*}
    c_i\cdot c_j =&amp;\left(p\cdot q_i + 2\cdot n_i + b_i\right)\cdot\left(p\cdot q_j + 2\cdot n_j + b_j\right)\\
    =&amp;p\cdot\hat{q} + 2\cdot\hat{n} + \left(b_i\cdot b_j\right)\\\\
{\#bits(\hat{n})} &lt; &amp; \#bits(\frac{p}{2})
  \end{align*} \\
$$

If this noise constraint isn't preserved, then the decrypt operation no longer holds. 
$$Dec(c_k) = b_k = \left[p\cdot q_k + 2\cdot n_k + b_k\right]_p(mod\;2)$$

The introduction of different 'Recrypt' strategies are used to increase the depth of a circuit by reducing accumulated noise.