# Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $$D=diag\{d_1,...,d_n\}$$ be the degree matrix where $$d_i=\sum_{j\neq i}e^{-|i-j|}$$. Then $$L=D-A$$ is the Laplacian. Let $$L^\dagger$$ be the Moore-Penrose inverse of the Laplacian. I'm interested in the following quantity $$a_{ij}=|(e_1-e_2)^TL^\dagger(e_i-e_j)|$$ where $$e_i=(0,0...,0,1,0,...0)$$ with 1 on the ith coordinate. I conjecture that $$a_{ij}$$ will decay exponentially when both $$i$$ and $$j$$ moves away from 1 and 2. Something like $$a_{ij}\leq C _1e^{-C_2\min\{i,j\}}$$ where $$C_1,C_2$$ are some constants. From the physics point of view, $$a_{ij}$$ is the voltage potential difference between $$i$$ and $$j$$. It is intuitive that when they are far away from the source, 1 and 2, they should be very small given the structure of the graph.

In fact, my simulation shows that as long as $$i,j\neq1,2$$, $$a_{ij}$$ suddenly becomes extremely close to 0. There seems to be no decay, but an acute cut. This phenomenon holds for slight perturbation of $$A$$, keeping the decaying property.

Is this conjecture true? How can we prove it? What is the rate of decay?

Another quantity that is also interesting is $$\sum_{i\neq j}e^{-|i-j|}a_{ij}$$ which is the weighted average of potential differences. How can we bound this? For this quantity, I conjecture it is bounded by some constant instead of growing with $$n$$. The physical meaning of this quantity is the sum of all currents in each edge.

(Update)

Enlightened by the discussion with @Abdelmalek Abdesselam below. We have the Neuman series representation: $$a_{ij}=|(e_1-e_2)^TD^{-1/2}\sum_{k\geq0}\left(D^{-1/2}AD^{-1/2}-\alpha D^{1/2}JD^{1/2}\right)^kD^{-1/2}(e_i-e_j)|$$ where $$J$$ is the matrix of all 1s and $$\alpha$$ is some constant to be chosen. We want to choose $$\alpha$$ such that the power of the matrix decays fast. How can we achieve this and bound the entries of $$D^{-1/2}AD^{-1/2}-\alpha D^{1/2}JD^{1/2}$$? A possible choice is $$\alpha=1/tr(D)$$.

• I guess one should say for more precision that $C_1,C_2$ must be independent of $n$. I would have to think a bit about the use of the Moore-Penrose inverse. But for a related problem where you would add some positive quantities on the diagonal so that $L$ is invertible and you can use $L^{-1}$ instead of $L^{\dagger}$, a simple Neuman series will do the trick. It is also the Green function of a (jumping) random walk with a positive killing rate. Another idea is to use the Matrix-Tree theorem to express $L^{\dagger}$. May 22, 2020 at 17:16
• Yes. $C_1$ and $C_2$ are independent of $n$ as I posted they are constants. May 22, 2020 at 17:18
• @AbdelmalekAbdesselam Actually, they may depend on $n$, as long as the bound is sharper than they are constants. It is not clear to me that the optimal bound is independent of $n$. May 22, 2020 at 17:31
• This is a very good question BTW. I think the result should be true and the continuum limit of it is basically saying the derivative of Brownian motion is white noise. To see this generalize your conjecture to $A$ which are not equal but rather bounded by exponentials. The particular case or nearest neighbor $A$ amout to showing decorrelation of local increments of a discrete walk trying to become a Brownian motion. May 22, 2020 at 18:20
• If $L$ is replaced by $\lambda I+L$ for $\lambda>0$, the analogous result is proved as Lemma 3, page 10 of arxiv.org/abs/0901.4756 May 22, 2020 at 18:23

Edit: This turns out to be quite simple. Observe that $$a_{1i} / a_{2i} = q$$ does not depend on $$i \in \{3, 4, \ldots, n\}$$. Thus, if $$x_1 = 1$$, $$x_2 = -q$$ and $$x_i = 0$$ for $$i \in \{3, 4, \ldots, n\}$$, then we clearly have $$L x = c e_1 - c e_2$$, where $$c = \sum_{i=3}^n a_{1i}$$. It follows that $$L^\dagger (e_1 - e_2) = c^{-1} x + \operatorname{const}.$$

I leave the previous version of this answer below, as it provides a way to evaluate $$L^\dagger$$ explicitly.

I cannot say I understand what is really going on here, but at least I have a proof that $$a_{ij} = 0$$ when $$i, j \ge 3$$. (I leave my previous comment/answer, as it contains some related stuff that is not included here.)

Notation: Every sum is a sum over $$\{1, 2, \ldots, n\}$$. We write $$q = e^{-1}$$ (and actually any $$q \in (0, 1)$$ will work). Given a vector $$x = (x_i)$$ we write $$\Delta x_i = x_{i+1} + x_{i-1} - 2 x_i$$ if $$1 < i < n$$.

Given a vector $$(x_i)$$, we have $$Lx_i = \sum_j q^{|i-j|} (x_i - x_j) = b_i x_i - \sum_j q^{|i-j|} x_j ,$$ where $$b_i = \sum_j q^{|i-j|} = \frac{1 + q - q^i - q^{n+1-i}}{1 - q}$$ Therefore, when $$1 < i < n$$, we have \begin{aligned} \Delta Lx_i & = \Delta (b x)_i - \sum_j (q^{|i-j+1|}+q^{|i-j-1|}-2q^{|i-j|}) x_j \\ & = \Delta (b x)_i - (q + q^{-1} - 2) \sum_j q^{|i-j|} x_j + (q^{-1} - q) x_i \\ & = \Delta (b x)_i + (q + q^{-1} - 2) L x_i - (q + q^{-1} - 2) b_i x_i + (q^{-1} - q) x_i \\ & = (q + q^{-1} - 2) L x_i + b_{i+1} x_{i+1} + b_{i-1} x_{i-1} - ((q + q^{-1}) b_i - (q^{-1} - q)) x_i . \end{aligned} A short calculation shows that $$b_{i+1} + b_{i-1} = ((q + q^{-1}) b_i - (q^{-1} - q))$$ (which looks somewhat miraculous, but there must be some insightful explanation for that). Thus, $$\Delta Lx_i = (q + q^{-1} - 2) L x_i + b_{i+1} (x_{i+1} - x_i) + b_{i-1} (x_{i-1} - x_i) .$$ Suppose that $$x_i = L^\dagger y_i$$ for some vector $$(y_i)$$ such that $$\sum_i y_i = 0$$. Then $$L x_i = L L^\dagger y_i = y_i$$. Write $$c = q + q^{-1} - 2$$. We then have $$\Delta y_i - c y_i = b_{i+1} (x_{i+1} - x_i) + b_{i-1} (x_{i-1} - x_i) .$$ In particular, the following claim follows.

Proposition 1: If $$1 < i < n$$, $$y_{i-1} = y_i = y_{i+1} = 0$$ and $$x_i = x_{i+1}$$, then $$x_{i-1} = x_i$$.

The above result will serve as an induction step. To initiate the induction, we need to study the $$i = n$$, which is slightly different. In that case: \begin{aligned} Lx_{n-1} - Lx_n & = b_{n-1} x_{n-1} - b_n x_n - \sum_j (q^{|n-j-1|}-q^{|n-j|}) x_j \\ & = b_{n-1} x_{n-1} - b_n x_n - (q^{-1} - 1) \sum_j q^{|n-j|} x_j + (q^{-1} - q) x_n \\ & = b_{n-1} x_{n-1} - b_n x_n + (q^{-1} - 1) L x_n - (q^{-1} - 1) b_n x_n + (q^{-1} - q) x_n \\ & = (q^{-1} - 1) L x_n + b_{n-1} x_{n-1} - (q^{-1} b_n - (q^{-1} - q)) x_n . \end{aligned} This time we have $$q^{-1} b_n - (q^{-1} - q) = b_{n-1} ,$$ and hence $$Lx_{n-1} - Lx_n = (q^{-1} - 1) L x_n + b_{n-1} (x_{n-1} - x_n) .$$ Again we consider $$x_i = L^\dagger y_i$$ for some vector $$(y_i)$$ such that $$\sum_i y_i = 0$$, and we write $$d = q^{-1} - 1$$. We then have $$(y_{n-1} - y_n) - d y_n = b_{n-1} (x_{n-1} - x_n) .$$ As a consequence, we have the following result.

Proposition 2: If $$y_{n-1} = y_n = 0$$, then $$x_{n-1} = x_n$$.

For $$y = e_1 - e_2 = (1, -1, 0, 0, \ldots)$$, we immediately obtain the desired result.

Corollary: If $$y = e_1 - e_2$$ and $$x = L^\dagger y$$, then $$x_3 = x_4 = x_5 = \ldots = x_n$$. Consequently, $$a_{ij} = x_i - x_j = 0$$ whenever $$i, j \ge 3$$.

Another consequence of the above result is that if $$L^\dagger = (u_{ij})$$, then $$u_{i+1,j+1}-u_{i,j+1}-u_{i+1,j}+u_{i,j} = 0$$ whenever $$i + 1 < j$$ or $$j + 1 < i$$. Moreover, it should be relatively easy to use Propositions 1 and 2 to evaluate $$u_{ij}$$ explicitly, and in particular to prove that $$u_{ij} = v_{\max\{i,j\}} + v_{\max\{n+1-i,n+1-j\}} + \tfrac{1}{4} |i - j|$$ when $$i \ne j$$, where $$(v_i)$$ is an explicitly given vector (in terms of products/ratios of $$b_i$$, I guess).

Final remark: there is a corresponding result in continuous variable: the Green function for the operator $$L f(x) = \int_0^1 e^{-q |x - y|} (f(x) - f(y)) dy$$ has zero mixed second-order derivative. The proof follows exactly the same line, and is in fact somewhat less technical.

• This is such a miracle! When the entries are not $e^{-|i,j|}$ exactly, for example we only have $e^{-|i-j|}\leq A_{ij}\leq 2e^{-|i-j|}$. Then I think they will not be exactly 0, can we bound $a_{kl}$ using similar arguments? May 23, 2020 at 23:21
• (1/2) A miracle — indeed, I am completely surprised by this result, and I still do not understand how this is possible. I did once look at the continuous counterpart mentioned in the end of my answer and I noticed some nice explicit expressions, but with a different boundary condition. I am kind of shocked that with this particular definition one can still get explicit expressions. May 24, 2020 at 0:14
• (2/2) Regarding perturbations: I no longer trust my intuition here, but my wild guess would be "no", I think. The above approach seems to heavily use the structure of $e^{-|i-j|}$. May 24, 2020 at 0:16
• Your updated solution is very insightful. It basically says that $e_1-e_2$ is in the span of the first two columns of $L$. When the entries of $L$ is not exactly $e^{-|i-j|}$, it seems intuitive that $e_1-e_2$ should also roughly lie within the span of the first several columns. The importance of each column to form the vector $e_1-e_2$ by linear combination seems to decay somehow. May 25, 2020 at 15:34

This is not an answer, but too long for a comment.

Consider a doubly infinite matrix $$L = (q_{ij})_{i,j \in \mathbb{Z}}$$ with entries $$q_{ij} = -e^{-|i - j|}$$ when $$i \ne j$$, and $$q_{ii} = 2 e / (1 - e)$$; here $$i, j \in \mathbb{Z}$$. The symbol of this matrix (i.e. the Fourier series with coefficients $$e^{-|j|}$$, except at $$j = 0$$) is: $$\psi(x) = \frac{e^2 - 1}{e^2 - 2 e \cos x + 1} - \frac{e + 1}{e - 1} .$$ The symbol of $$L^\dagger$$ is thus $$1 / \psi(x)$$ (in the principal value sense), which has a singularity of type $$1 / x^2$$ at $$x = 0$$. It follows that in this case $$a_{kl} = \frac{1}{2 \pi} \int_{-\pi}^{\pi} \frac{(e^{i x} - e^{2 i x}) (e^{i k x} - e^{i l x})}{\psi(x)} \, dx .$$

In general, the above expression will only have power-type decay as $$k,l \to \infty$$.

However, for this particular choice of $$L$$, things simplify a lot. The pseudo-inverse $$L^\dagger = (u_{ij})_{i,j \in \mathbb{Z}}$$ can be found explicitly, and its entries are $$u_{ij} = C_1 - C_2 |i - j|$$ when $$i \ne j$$ and $$u_{ii} = C_3$$ for appropriate constants $$C_1$$, $$C_2$$, $$C_3$$. Consequently, $$a_{kl} = 0$$ when $$k, l > 2$$.

I do not have a clear intuition about what happens in the one-sided case (that is, if we consider an infinite matrix $$L$$ with entries indexed by $$i, j \in \{1, 2, \ldots\}$$), let alone the bounded case (with $$i, j \in \{1, 2, \ldots, n\}$$). My wild guess would be that the symmetry breaks, and there is no hope for any closed-form formula.

However, a quick numerical experiment suggests strongly that we still have $$a_{kl} = 0$$! More precisely, the entries $$u_{ij}$$ of $$L^\dagger$$ apparently satisfy $$u_{ij} = v_{\max\{i,j\}} + v_{\max\{n+1-i,n+1-j\}}, v_{n-i} + v_j\} + \tfrac{1}{4} |i - j| \qquad (i \ne j)$$ for an appropriate vector $$v_i$$. I find this extremely surprising!

Here is the code in Octave, in case anyone is interested. First, we construct $$L$$ and its pseudo-inverse (denoted U here):

n = 10;                                  # size of the matrix
A = toeplitz(exp(-(0:n-1)));
L = diag(A * ones(n,1)) - A;             # matrix L
U = pinv(L);                             # pseudo-inverse L^\dagger


Next, we verify that the mixed second-order difference of $$L^\dagger$$ is a tri-diagonal matrix:

D = U(1:n-1, 1:n-1) - U(1:n-1, 2:n) ...
- U(2:n, 1:n-1) + U(2:n, 2:n);       # second-order difference of U


This already shows that $$L^\dagger$$ has the desired structure, but we can verify this directly. First two lines are to extract the vector $$v_i$$, the other two define the matrix Z with entries $$u_{ij} - v_{\max\{i,j\}} - v_{\max\{n+1-i,n+1-j\}}, v_{n-i} + v_j\} - \tfrac{1}{4} |i - j| \qquad (i \ne j)$$ which should be zero except on the diagonal:

X = U - 0.25 * abs(repmat(1:n, n, 1) - repmat(1:n, n, 1)');
V = X(:, 1) - 0.5 * X(n, 1);
I = repmat(1:n,n,1);
Z = X - V(max(I, I')) - V(max(n + 1 - I, n + 1 - I'));

• How did you find the expression of $L^\dagger$ explicitly? May 22, 2020 at 21:24
• We have $1/\psi(x) = c_1 (e^2 - 2 e \cos x + 1) / (1 - \cos x) = c_2 + c_3 / (1 - \cos x)$; the constant $c_2$ corresponds to $c_2 \delta_{ij}$, and $c_3 / (1 - \cos x)$ corresponds to the Green's function for the simple random walk, $-c_4 |i - j|$. May 22, 2020 at 21:37
• What if the entries of the matrix is not exactly $e^{-|i-j|}$ and the matrix is not Toeplitz, but the entries decay at the same rate. Can we still obtain explicit expression for $L^\dagger$? Is $a_{kl}$ still 0 for $k,l>2$? May 22, 2020 at 23:10
• I doubt one can get explicit expressions if $L$ is not exactly $(e^{-|i-j|})$. May 23, 2020 at 7:54
• I edited my answer/comment. I am now convinced that your $a_{ij}$ is equal to zero, but I have no clue why. May 23, 2020 at 8:42