nullity of a random matrix (infinite dimension) in $L^2$ space \begin{pmatrix} 1 & a & & \\ & 1 & a & \\ & & 1 & \dots\\ & & & \dots \end{pmatrix} This problem comes from condensed matter physics. Matrices are acting on $L^2$ space, since wavefunctions are normalizable. Right, left eigenvectors are a pair of Majorana representation of Fermions, singular values are the energy. Our interest is to look at zero energy modes.
Statement 1.a
Let's define a matrix with diagonal always equal to one.
$$ M(a,n) \equiv \begin{pmatrix}
1 & -a &  &  & \\ 
 & 1 & -a &  & \\ 
 &  & 1 & -a & \\ 
 &  &  & 1 & \dots \\ 
 &  &  &  & \dots
\end{pmatrix}_{n\times n}$$
For any finite dimension $n$, matrix $M$ is  full rank, nullity is zero.
Interesting case happens if we consider infinite dimension ($n\to \infty $):
when $|a|<1$ and  , nullity is still zero.
when $|a|>1$ and  , nullity is one!
Statement 1.b
Now let $a_i$ becomes random. ( only consider $a>0$ case ).
Statement 1.a can be modified as:
when $ \prod_i a_i <1$ , nullity is zero.
when $\prod_i a_i>1$ , nullity is one.
$$ M(a_i,n) \equiv \begin{pmatrix}
1 & -a_1 &  &  & \\ 
 & 1 & -a_2 &  & \\ 
 &  & 1 & -a_3 & \\ 
 &  &  & 1 & \dots \\ 
 &  &  &  & \dots
\end{pmatrix}_{n\times n}$$
Statement 2.a
Adding $b$ terms to statement 1.a
$$ M(a,b,n) \equiv \begin{pmatrix}
1 & -a & -b &  & \\ 
 & 1 & -a & -b & \\ 
 &  & 1 & -a & \dots \\ 
 &  &  & 1 & \dots \\ 
 &  &  &  & \dots
\end{pmatrix}_{n\times n}$$
Let's only consider $a>0$ and  $b>0$ case. The statement generalizes to:
when $b<1-a$ , nullity is still zero.
when $  1 -a < b <1 + a$  , nullity is one.
when $  1 + a < b $  , nullity is two.
My question is, what will be the Statement 2.b  when $a_i, b_i$ become random?
Statement 1.a, 1.b, 2.a are conclusions from physics. The zero singular values are in fact Majorana zero modes.
I hope to get some insights from mathematicians.
What math tools and concepts should I use?
 A: Null space, the vector has to be normalizable. i.e.  $L^2$.
Inspired by the discussion with @ancientmathematician
$$\sum_{i=1}^{+\infty} M_{ij} r_i = 0$$
The iteration relation $r_{i+2} = -\frac{a_i}{b_i} r_{i+1} + \frac{h_i}{b_i} r_{i}  $ with intial condition $r_0,r_1$  ,
diagonal $h_i \equiv 1$ now take any value $h$
The solution takes this form:
$$r_{i+2} = \big(\prod_{k=0}^i A_k \big) c_0 + \big(\prod_{k=0}^i B_k \big) c_1 $$
$$ A_i,B_i = \frac{-a_i \pm \sqrt{a^2_i + 4 h_i b_i} }{2 b_i} $$
If the solution $\{r_i\} \in L^2 $ is not exploded. There is nullity  is adding one.
Therefore the condition for
nullity=2 is :
$$|\prod_{k=0}^\infty A_k |< 1  \quad , \quad
 |\prod_{k=0}^\infty B_k |< 1 $$
nullity=1 is :
$$|\prod_{k=0}^\infty A_k |< 1  < |\prod_{k=0}^\infty B_k | $$
nullity=0 is :
$$ 1  < |\prod_{k=0}^\infty A_k |   \quad , \quad 1 < |\prod_{k=0}^\infty B_k | $$
Taking the log of the product chain, we got average:
$$ \overline{ \log | \frac{ -a \pm  \sqrt{a^2 + 4 h b} }{2 b} | } = 0 $$
This structure is Inverse hyperbolic functions.
For continues distribution $p(a), p(b), p(h)$
$$ \Lambda_{\pm} \equiv \int da \ db \ dh \log  \frac{ \sqrt{a^2 + 4 h b} \pm a }{2 b}  p(a) p(b) p(h) = 0 $$ gives the critical point.
counting the valid numbers of  $\Lambda_{\pm} < 0$ will give the nullity

Update:
My derivation for the non-constant recurrance relation is not exact.
I found (1.6) to (1.8) from this paper:
Recurrence Relations and Benford’s Law
The auxiliary functions shift one site!
$$ \mu_i + \lambda_i = A_i + B_i = \frac{a_i}{b_i} $$
$$ \mu_i  \lambda_{i-1} = A_i  B_i  = \frac{h_i}{b_i} $$
This is of course not the intial random field.
However if considering $\lambda_i$ being a constant, it is a beautiful and a strong result!
A: Here is my first guess (most likely wrong):
nullity = 0
$ (\prod_i b_i)^{\frac{1}{N}} <  1 - (\prod_i a_i)^{\frac{1}{N}}   $
nullity = 1
$1 - (\prod_i a_i)^{\frac{1}{N}}  < (\prod_i b_i)^{\frac{1}{N}}  <  1 + (\prod_i a_i)^{\frac{1}{N}}   $
nullity = 2
$(\prod_i a_i)^{\frac{1}{N}}   + 1 < (\prod_i b_i)^{\frac{1}{N}}    $
reasons:
when $b_i\equiv 0$, compatible with (1.b)
when $a_i, b_i$ are constants, compatible with (2.a)
when $a_i\equiv 0$, this matrix can be decoupled to two (1.b) matrices, nullity can only be 0 or 2
